Non-Abelian Localization and U(1) Chern-Simons Theory · Brendan Donald Kenneth McLellan Doctor of...
Transcript of Non-Abelian Localization and U(1) Chern-Simons Theory · Brendan Donald Kenneth McLellan Doctor of...
Non-Abelian Localization and U(1) Chern-Simons Theory
by
Brendan Donald Kenneth McLellan
A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics
University of Toronto
Copyright c© 2010 by Brendan Donald Kenneth McLellan
Abstract
Non-Abelian Localization and U(1) Chern-Simons Theory
Brendan Donald Kenneth McLellan
Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
2010
This thesis studies U(1) Chern-Simons theory and its relation to the results of Chris
Beasley and Edward Witten, [BW05]. Using the partition function formalism, we are
led to compare U(1) Chern-Simons theory as constructed in [Man98] to the results of
[BW05]. This leads to an explicit calculation of the U(1) Chern Simons partition function
on a closed Sasakian three-manifold and opens the door to studying rigorous extensions
of this theory to more general gauge groups and three-manifold geometries.
The first main part of this thesis studies an analogue of the work of Beasley and Witten
[BW05] for the Chern-Simons partition function on a Sasakian three-manifold for U(1)
gauge group. A key point is that our gauge group is not simply connected, whereas this
is an essential assumption in Beasley and Witten’s work. We are still able to use Beasley
and Witten’s results, however, to derive a definition of a U(1) Chern-Simons partition
function. We then compare this result to a definition of the U(1) Chern-Simons partition
function given by Mihaela Manoliu [Man98], and find that the two definitions agree up
to some undetermined multiplicative constant. These results lead to a natural interpre-
tation of the Reidemeister-Ray-Singer torsion as a symplectic volume form on the moduli
space of flat U(1) connections over a Sasakian three-manifold.
The second main part of this thesis studies U(1) Chern-Simons theory and its relation
ii
to a construction of Chris Beasley and Edward Witten, [BW05]. The natural geometric
setup here is that of a three-manifold with a Sasakian structure. We are led to study the
stationary phase approximation of the path integral for U(1) Chern-Simons theory after
one of the three components of the gauge field is decoupled. This gives an alternative
formulation of the partition function for U(1) Chern-Simons theory that is conjecturally
equivalent to the usual U(1) Chern-Simons theory, [Man98]. We establish this conjectural
equivalence rigorously using appropriate regularization techniques.
iii
Acknowledgements
There are several people whom I would like to thank here. First, I would like to thank my
thesis advisor Lisa Jeffrey for her patience, understanding, and her insights throughout
the years. This work would not have been possible without her. I would also like to thank
Yael Karshon for several useful discussions. Her guidance and support are greatly ap-
preciated. I would also like to thank Dror Bar-Natan for his unique perspective, insights
and for several helpful discussions. I would especially like to thank Frederic Rochon for
taking the time to explain elliptic PDE theory, the Atiyah-Patodi-Singer theorem, and
the method of heat kernels and eta-invariants in the elliptic case. His help regarding the
study of hypoelliptic operators was also invaluable. I would particularly like to thank
Raphael Ponge and Michel Rumin for their correspondence relating to hypoelliptic opera-
tors and the contact Laplacian. I am indebted to Paul Selick for explaining several useful
concepts related to (co)homology and homotopy theory. Thanks also to Eckhard Mein-
renken for several useful discussions regarding Chern-Weil theory and generally helping
with any questions that I had. I would like to thank Ben Burrington for taking the
time to review the computation of the gravitational Chern-Simons term in [GIJP03]. His
expert understanding of these types of computations proved invaluable for my own com-
putation. My sincerest thanks to John Bland for meeting with me on more than several
occasions to discuss contact and CR-geometry. His keen interest in my work also forced
me to look at some deeper questions more carefully and has certainly deepened my own
understanding. Lastly, I would like to thank Edward Witten for originally suggesting the
topic of my thesis. I appreciate him taking the time to correspond on questions related
to my thesis and also for an invaluable meeting that helped to answer some of my more
general questions.
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Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Geometry 9
2.1 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Contact and CR Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Seifert Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Vielbein Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 The Space of Connections . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.2 The U(1)-Bundle Case . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Heisenberg Calculus on Contact Manifolds 43
3.1 Heisenberg Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Heisenberg Calculus and the Rockland Condition . . . . . . . . . . . . . 48
3.3 Some Results in the Heisenberg Calculus . . . . . . . . . . . . . . . . . . 51
4 U(1) Chern-Simons Theory 54
4.1 Induced Principal Bundles and Induced Connections . . . . . . . . . . . 54
v
4.2 The U(1) Chern-Simons Action . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 The U(1) Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Shift Symmetry and the U(1) Partition Function . . . . . . . . . . . . . . 68
5 Non-Abelian Localization for U(1) Chern-Simons Theory 75
5.1 Symplectic Formulation of the U(1) Partition Function . . . . . . . . . . 78
5.2 Orbifolds and Symplectic Volume . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Reidemeister Torsion and Symplectic Volume . . . . . . . . . . . . . . . 86
5.4 The Adiabatic Eta-Invariant . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Eta-Invariants and Anomalies 98
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Structure Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.4 The Contact Operator D . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.5 Gauge Group and the Isotropy Subgroup . . . . . . . . . . . . . . . . . . 111
6.6 The Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.7 Zeta Function Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.8 The Eta-Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.9 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.10 Regularizing the Eta-Invariants . . . . . . . . . . . . . . . . . . . . . . . 122
7 Gravitational Chern-Simons and the Adiabatic Limit 130
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.2 Local Formulation of Gravitational Chern-Simons Term . . . . . . . . . . 134
7.3 Computation of Gravitational Chern-Simons Term . . . . . . . . . . . . . 136
A Localization 143
A.1 Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
vi
B Constructions and Computations 148
B.1 Finite Dimensional Analogue of the Shift Symmetry . . . . . . . . . . . . 148
B.2 Gravitational Chern-Simons Calculations . . . . . . . . . . . . . . . . . . 156
B.2.1 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . 156
B.2.2 Spin Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B.2.3 Reduced Spin Connection . . . . . . . . . . . . . . . . . . . . . . 162
B.2.4 Reduced Gravitational Chern-Simons . . . . . . . . . . . . . . . . 166
C Miscellaneous Results 169
C.1 Horizontal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C.2 T dRS = T dC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.3 A Standard Result in Cohomology . . . . . . . . . . . . . . . . . . . . . 180
Bibliography 182
vii
Chapter 1
Introduction
1.1 Background
This thesis is generally concerned with U(1) Chern-Simons theory on a closed Sasakian
three-manifold X. We will in fact be interested in a particular choice of quasi-regular K-
contact structure on a given Sasakian three-manifold, and most of our results are stated
assuming that such a choice has been made.
With the exception of §4.3 and §4.4, chapters 2, 3 and 4 are completely review and
serve only to establish notation and some standard definitions and results. There is a
slight novelty in §4.3, which is the extension of the definition of the U(1) Chern-Simons
partition function contained in [Man98] to take into account a dependence on a choice
of framing. §4.4 provides a new heuristic definition of what we call the shifted U(1)
Chern-Simons partition function.
For an excellent overview of the history and background of Chern-Simons theory in
general, we refer the interested reader to [Fre95] and [Fre09]. The background that is
most relevant for this thesis is contained in the papers [Man98] and [BW05]. In par-
1
Chapter 1. Introduction 2
ticular, [Man98] provides a rigorous foundation to study U(1) Chern-Simons theory and
[BW05] provides a heuristic motivation for our work. One of the main aspects of [Man98]
that is crucial for us is the derivation of a rigorous definition of the U(1) Chern-Simons
partition function starting from a heuristically defined path integral. Since the work of
[BW05] uses the path integral approach, this allows us to connect the work of [BW05]
with that of [Man98] very naturally.
1.2 Thesis Results
The first part of this thesis is concerned with the direct comparison of the partition
function of Manoliu [Man98] to the non-abelian localization results of Beasley and Witten
[BW05]. We first derive a definition for the U(1) Chern-Simons partition function that
is closely related to the definition of [Man98]. We obtain the following (see Def. 4.3.22
and §4.3 for all the relevant background)
1.2.1 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented three-
manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k), is the quantity
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k) (1.2.2)
where,
ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2 (1.2.3)
where mX = 12(dimH1(X; R)− dimH0(X; R)).
In §5.1 we derive a new definition of the partition function for U(1) Chern-Simons
theory using the methods of [BW05]. We make the following (see Def. 5.1.13 and §5.1
for all the relevant background)
Chapter 1. Introduction 3
1.2.4 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented Seifert
three-manifold,
U(1) // X
Σ
,
where Σ = |Σ|,U is an orbifold with underlying space |Σ| a Riemann surface of genus
g. The symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), is the quantity
ZSU(1)(X, k) =
∑p∈TorsH2(X;Z)
ZSU(1)(X, p, k) (1.2.5)
where,
ZSU(1)(X, p, k) = kmXeπikSX,P (AP )eiπ(
14− 1
2η0)∫MP
ωP . (1.2.6)
One of our main objectives in this thesis is to establish the equivalence of Def. 1.2.1
and Def. 1.2.4. As a first step in this direction, we compare the square root of the Ray-
Singer torsion (T dRS)1/2 (see Appendix C.2) to the symplectic volume form ωP in §5.3.
We establish the following (see Prop. 5.3.19)
1.2.7 Proposition. Let X be a closed, oriented Siefert three-manifold,√τ(X) the square
root of the R-torsion of X, and ωP the symplectic volume form on the moduli space of
flat connections MP ' U(1)2g for some flat U(1)-bundle P over X. Then, there exists
C ∈ R∗, such that √τ(X) = C · ωP .
We then make the following (see Conj. 5.3.20)
1.2.8 Conjecture. C = 1 in Prop. 1.2.7 above.
The next step that we take to establish the equivalence of Def. 1.2.1 and Def. 1.2.4
is to compare the eta-invariants that arise therein. In §5.4 we study the adiabatic eta-
invariant, η0. This invariant is studied in [BW05], [Nic00] and [Bea07], for example. We
find that η0 agrees with the results that we obtain in §6.10, after some observations. On
Chapter 1. Introduction 4
the one hand, we have (see §5.4)
− η0
2=
[−c1(X)
12+
N∑j=1
s(αj, βj)
], (1.2.9)
and on the other we have (see §6.10 and Prop. 6.10.27)
η(?d)
4+
1
12
CS(Ag)
2π=η0(X, κ)
4=
1
4− c1(X)
12+
N∑j=1
s(αj, βj), (1.2.10)
where,
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)(1.2.11)
is the classical Rademacher-Dedekind sum and [n; (α1, β1), . . . , (αN , βN)] (for (αi, βi) = 1
relatively prime) are the Seifert invariants of X. We observe in §5.4 that the eta-invariant
dependent parts of Def.’s 1.2.1 and 1.2.4 are exactly the same. That is, we have shown
(See Prop. 5.4.11 and §5.4)
1.2.12 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then,1 we have
η(?d)
4+
1
12
CS(Ag)
2π=
1
4− 1
2η0,
=1
4− d
12+
N∑j=1
s(αj, βj).
The second part of this thesis is contained in our paper [JM10]. Our main objective
here is the rigorous confirmation of a heuristic result utilized in [BW05]. Our results
involve some fairly deep facts about the “contact operator” studied by Michel Rumin,
[Rum94]. Recall that this is the second order operator “D” that fits into the complex
(see §6.4),
C∞(X)dH−→ Ω1(H)
D−→ Ω2(V )dH−→ Ω3(X), (1.2.13)
and in our geometric situation2 can be written as follows:
Dα = κ ∧ [Lξ + dH ?H dH ]α, α ∈ Ω1(H). (1.2.14)
1This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
2When X is a closed, oriented, quasi-regular K-contact three-manifold.
Chapter 1. Introduction 5
A somewhat surprising observation is that this operator shows up quite naturally in
U(1) Chern-Simons theory, and this leads us to make several conjectures motivated by
the heuristic constructions of [BW05]. Our main result is the following (see Chapter 6
and Prop. 6.10.24):
1.2.15 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three
manifold. If,
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (η(?D)+ 1
512
∫X R2 κ∧dκ)
∫MP
(T dC)1/2 (1.2.16)
and,
ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2 (1.2.17)
then,3 we have
ZU(1)(X, k) = ZU(1)(X, k)
as topological invariants.
The first main component of this result is the fact that the Ray-Singer analytic
torsion of X, T dRS, is identically equal to the contact analytic torsion T dC . This result is
in fact already known and follows directly from [RS08, Theorem 4.2]. The second main
component of this result is the existence, and the explicit identification, of regularizations
of the eta-invariants of the de-Rham and contact operators, d and D, respectively. This
result is contained in the following (see Chapter 6 and Prop. 6.10.23):
1.2.18 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-
manifold. Then,4 there exists a counterterm, CT , such that eπi4
[η(?D)+CT ] is a topological
invariant that is identically equal to the topological invariant eπi[η(?d)
4+ 1
12CS(Ag)
2π
]. In fact,
3This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
4This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 1. Introduction 6
we have
CT =1
512
∫X
R2 κ ∧ dκ,
where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X.
We prove this theorem by appealing to a result that we establish using a “Kaluza-
Klein” dimensional reduction technique, modeled after the paper [GIJP03], for the grav-
itational Chern-Simons term (see Chapter 7 and Prop. 7.3.40).
1.2.19 Theorem. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-
manifold,
U(1) // X
Σ
.
Let gε := ε−1 κ⊗ κ+ π∗h. After a particular choice of Vielbein,5 then,
CS(Agε) =
(ε−1
2
)∫Σ
r ω +
(ε−2
2
)∫Σ
f 2 ω (1.2.20)
where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)
Hodge form of (Σ, h), and f ∈ Ω0orb(Σ) is the invariant field strength on (Σ, h).6 In
particular, the adiabatic limit of CS(Agε) vanishes:
limε→∞
CS(Agε) = 0. (1.2.21)
Finally, as a consequence of these investigations, we are able to compute the U(1)
Chern-Simons partition function fairly explicitly (see Chapter 6 and Prop. 6.10.27).
5See Equations (7.3.20) and (7.3.21).6In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx
0 + ϕ1dx1 + dx2, and dκ =
(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,
and fαβ =√h εαβ f, where f ∈ Ω0
orb(Σ) is called the invariant field strength on (X,h). See Chapter 7and §2.4 for more background.
Chapter 1. Introduction 7
1.2.22 Theorem. [JM10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-
manifold. Then,7 we have
η(?d) +1
3
CS(Ag)
2π= η(?D) +
1
512
∫X
R2 κ ∧ dκ
= 1− d
3+ 4
N∑j=1
s(αj, βj),
where 0 < d = c1(X) = n+∑N
j=1βjαj∈ Q and
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)∈ Q
is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for gcd(αj, βj) =
1) are the Seifert invariants of X. In particular, we have computed the U(1) Chern-
Simons partition function as:
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dC)1/2,
= kmXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dRS)1/2.
These results provide a rigorous confirmation of Beasley and Witten’s heuristic shift
symmetry construction in the case that G = U(1). This work also puts some of the work
of Biquard, Herzlich, Rumin, Seshadri, and Ponge into a physics context. For example,
our work shows why it is natural to expect that the contact Laplacian should be “nice.”8
Our work implies that it is natural to expect that the contact analytic torsion equals the
Ray-Singer analytic torsion in our geometric situation, and that the topological regular-
izations of the contact and de-Rham eta-invariants are in a precise sense equivalent. This
work also establishes a rigorous and explicit computation of the U(1) Chern-Simons par-
tition function for the general class of closed quasi-regular K-contact three-manifolds.9
7This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
8In fact, it is maximally hypoelliptic and invertible in the Heisenberg symbolic calculus. See Chapter3.
9This work applies to all closed K-contact three-manifolds, and hence by [Bla76, Corollary 6.5] to allclosed Sasakian three manifolds.
Chapter 1. Introduction 8
1.3 Future Work
One of my main interests for future work is the extension of these results beyond the
case of quasi-regular K-contact manifolds. Ultimately, I am interested in obtaining an
analogue of these results for any three-manifold with a chosen contact form. It has re-
cently been suggested by Witten that a possible approach to this problem is to consider
the one-loop computation for the shift invariant partition function. He suspects that this
invariant would depend only on the contact structure and not necessarily on the Seifert
structure. It would be interesting to study general contact metric manifolds that do not
necessarily possess a Seifert structure, for example. Additionally, it would be interesting
to look at extending these results to three-manifolds with boundary.
Another interesting direction for future research is the generalization of these results
from a U(1) gauge group to SU(2). As observed in [Wit89], the stationary phase approx-
imation for SU(2) Chern-Simons theory is a straightforward generalization of the U(1)
case. The work of Braverman and Kappeler [BK08] on refined analytic torsion seems to
contain exactly the right tools for such an extension. We note that there is still much work
to be done for other gauge groups (e.g. SU(2)) and future work will be concerned with
finding appropriate generalizations of the results of this thesis to these other cases. As is
noted in [BW05], Chern-Simons theory is intimately related to the theory of knot invari-
ants and the topology of three-manifolds [Wit89], [Ati90a], to two-dimensional rational
conformal field theory [MS90], to three-dimensional quantum gravity [DJT82], [Wit88],
to open string field theory on the topological A-model [Wit95] and to the Gromov-Witten
theory of non-compact Calabi-Yau threefolds [GV98], [GV99].
Chapter 2
Geometry
2.1 Orbifolds
In this section we review orbifolds. We follow primarily [BG08]. Other good references
are [Nic00], and [Sat57].
We begin with the following
2.1.1 Definition. [BG08, Def. 4.1.1] Let |Σ| be a paracompact Hausdorff space. An
orbifold chart or local uniformizing system on |Σ| is a triple (U ,Γ, φ), where U ⊂ Rn is a
connected open subset of Rn, Γ is a finite group acting effectively on U , and φ : U → U
is a continuous map onto an open set U ⊂ |Σ| such that φ γ = φ for all γ ∈ Γ and the
induced natural map of U/Γ onto U is a homeomorphism. An injection or embedding
between two such charts (U ,Γ, φ) and (U ′,Γ′, φ′) is a smooth (or holomorphic) embedding
λ : U → U ′ such that φ′ λ = φ. An orbifold atlas on |Σ| is a family U = Ui,Γi, φi of
orbifold charts such that
i. |Σ| = ∪iφi(Ui),
ii. given two charts (Ui,Γi, φi) and (Uj,Γj, φj) in U with Ui = φi(Ui) and Uj = φj(Uj)
and a point p ∈ Ui ∩ Uj, there exists an open neighbourhood Uk of p, and a
9
Chapter 2. Geometry 10
chart (Uk,Γk, φk) such that there are injections λik : (Uk,Γk, φk)→ (Ui,Γi, φi) and
λjk : (Uk,Γk, φk)→ (Uj,Γj, φj).
An atlas U is a refinement of an atlas V if there exists an injection of every chart of U into
some chart of V . Two orbifold atlases are equivalent if they have a common refinement.
A smooth (complex) orbifold (or V-manifold) is a paracompact Hausdorff space |Σ| with
an equivalence class of smooth (holomorphic) orbifold atlases and we write Σ = (|Σ|,U).
If every finite group Γ consists of orientation preserving diffeomorphisms and there is an
atlas such that all the injections are orientation preserving then the orbifold is orientable.
The finite groups Γi are called local uniformizing groups.
2.1.2 Example. [BH99] Let us define an orbifold structure on S2. We take |Σ| = S2 as
our underlying paracompact Hausdorff space as in Def. 2.1.1 and we define an orbifold
Σ = (S2,U). We view S2 = C ∪ ∞ = U0 ∪ ∞ as the Riemann sphere with two
marked points 0,∞, the south and north pole respectively. Define two orbifold charts,
(U0,Γ0, φ0) = (C,Zm, φ0 : C→ S2\∞),
where φ0(z) = zm, and,
(U∞,Γ∞, φ∞) = (C,Zn, φ∞ : C→ S2\0),
where φ∞(w) = w−n. e2πirm ∈ Zm acts on α ∈ C by rotations e
2πirm : α 7→ e
2πirm ·α in the first
coordinate chart, and similarly for the second. Given any point p ∈ U0∩U∞ = S2\0,∞,
let Up ⊂ U0 ∩ U∞ be any open subset Up ⊂ U0 = C, viewed as a subset of U0 with
coordinate z ∈ C, that is homeomorphic to a disk D ⊂ C. For such sets, let the
corresponding charts be (Up ' Up ⊂ C,Γp := e, φp(z) := z). We claim that the
collection V consisting of the collection of such charts (Up,Γp, φp) along with the charts
(U0,Γ0, φ0), and (U∞,Γ∞, φ∞) forms an orbifold atlas on |Σ| = S2. Condition (ii) in Def.
2.1.1 is clearly satisfied for any two charts of the form (Up,Γp, φp). Let us check this
condition for the pair (U0,Γ0, φ0), and (U∞,Γ∞, φ∞), and leave the other cases to the
Chapter 2. Geometry 11
reader. Consider, p ∈ U0 ∩ U∞ = S2\0,∞. Let U∩ ⊂ U0 ∩ U∞ be any subset of the
form Up ⊂ U0 = C as described above, with chart:
(U∩ ' U∩ ⊂ C,Γ∩ := e, φ∩(z) := z).
Then,
λ0∩ : (U∩,Γ∩, φ∩)→ (U0,Γ0, φ0),
defined by λ0∩(z) = z1m is our first injection, and
λ∞∩ : (U∩,Γ∩, φ∩)→ (U∞,Γ∞, φ∞),
defined by λ∞∩(z) := (1/z)1n = w
1n is our second injection. Clearly φ∩(z) = φ∞
λ∞∩(z) = z holds, for example. Finally, let U to be the unique maximal atlas containing
V and obtain the orbifold Σ = (S2,U).
2.1.3 Remark. We note that the points 0,∞ in Ex. 2.1.2 are the only points fixed
under the actions of the groups Γ, when |Γ| > 1, in any of the orbifold coordinate charts
contained in the maximal atlas U . The set of points x ∈ |Σ| whose isotropy subgroup
Γx 6= e are called singular points. Those points with Γx = e are called regular points.
See [BH99] for examples of orbifold structures on Riemann surfaces of genus g ≥ 1.
Next we define a natural notion of a “bundle” over an orbifold, which will be of
particular importance in our study of Seifert manifolds. We make the following
2.1.4 Definition. [BG08, Def. 4.2.7] A V-bundle or orbibundle over an orbifold Σ =
(|Σ|,U) consists of a fibre bundle BU over U for each orbifold chart (Ui,Γi, φi) ∈ U with
Lie group G and fiber F a smooth G-manifold which is independent of Ui together with
a homomorphism hUi : Γi → G satisfying:
i. if b lies in the fiber over xi ∈ Ui then for each γ ∈ Γi, bhUi(γ) lies in the fiber over
γ−1xi,
Chapter 2. Geometry 12
ii. if the map λji : Ui → Uj is an injection, then there is a fixed bundle map λ∗ji :
BUj|λji(Ui) → BUi
satisfying the condition that if γ ∈ Γi, and γ′ ∈ Γj is the unique
element such that λji γ = γ′ λji, then hUi(γ) λ∗ji = λ∗ji hUj(γ′), and if
λkj : Uj → Uk is another such injection then (λkj λji)∗ = λ∗ji λ∗kj.
If the fiber F is a vector space of dimension r and G acts on F as linear transformations
of F , then the V -bundle is called a vector V-bundle of rank r. Similarly, if F is the Lie
group G with its right action, then the V -bundle is called a principal V -bundle.
2.1.5 Remark. Note that the total space E of a V -bundle over an orbifold Σ inherits an
orbifold structure from Σ. We do not describe this here and instead refer the reader to
[BG08, Remark 4.2.1]. We will denote this orbifold structure as (E,U∗).
The notion of sections of orbibundles also makes sense and is important for us since
these are necessary to define vector fields, differential forms, and other such notions. We
have the following
2.1.6 Definition. [BG08, Def. 4.2.9] Let E = (E,U∗) be a V -bundle over an orbifold
Σ. Then a section σ of E over an open set W ⊂ |Σ| is a section σU of the bundle BU for
each orbifold chart (U ,Γ, φ) such that U ⊂ W and for each x ∈ U we have
i. For each γ ∈ Γ, σU(γ−1x) = σU(x)hU(γ).
ii. If λ : (U ,Γ, φ)→ (U ′,Γ′, φ′) is an injection, then λ∗σU ′(λ(x)) = σU(x).
One of the most important examples of an orbibundle for us will the tangent bundle.
We describe this in the following
2.1.7 Example. [BG08, Example 4.2.10] Given an orbifold Σ = (|Σ|,U), we briefly de-
scribe the tangent V -bundle of Σ. For an orbifold chart (Ui,Γi, φi) ∈ U take BUi= TUi
the usual tangent bundle of Ui ⊂ Rn. The fiber is F = Rn with G = GL(n,R) acting via
the Jacobian Jλji for each injection λji : (Ui,Γi, φi) → (Uj,Γj, φj). Since each element
γ ∈ Γi defines an injection (via x 7→ γ · x), the homomorphisms hUi are injective into
Chapter 2. Geometry 13
G = GL(n,R) and therefore satisfy condition (i) of Def. 2.1.4. The total space of TΣ is
T |Σ| with orbifold atlas U∗ consisting of orbifold charts of the form (Ui×Rn,Γ∗i , φ∗i ), where
Γ∗i = Γi acts on Ui × Rn via (x, v) 7→ (γ−1x, hUi(γ)v), and φ∗i : Ui × Rn → (Ui × Rn)/Γ∗i
is the natural quotient projection. A smooth1 invariant2 section of TΣ is called a vector
field on Σ.
2.1.8 Remark. The notions of cotangent V -bundle, and tensor bundles are constructed
similarly to the construction of Ex. 2.1.7. One can then talk about Riemannian metrics,
symplectic forms, connections, etc., without concern. For example, an orbifold real n-
form ω ∈ Ωnorb(Σ,R) is a collection of Γi invariant n-forms ωi ∈ Ωn(Ui,R) such that
λ∗jiωj = ωi, (2.1.9)
for all injections λji : (Ui,Γi, φi)→ (Uj,Γj, φj).
An important notion for us is contained in the following
2.1.10 Definition. [BG08, Def. 4.2.11] A Riemannian metric g on an orbifold Σ =
(|Σ|,U) is a Riemannian metric gi on each orbifold chart (Ui,Γi, φi) ∈ U that is invariant
under the local uniformizing group Γi and such that the injections λji : (Ui,Γi, φi) →
(Uj,Γj, φj) are isometries, i.e. λ∗ji(gj|λji(Ui)) = gi. Similarly, if Σ is a complex orbifold,
then a Hermitian metric h is a Γi-invariant Hermitian metric on each neighbourhood Ui
such that the injection maps are Hermitian isometries. An orbifold with a Riemannian
(Hermitian) metric is called a Riemannian (Hermitian) orbifold.
The fact that every smooth manifold admits a Riemannian metric also holds in the
orbifold case. Using a modification of the partition of unity argument, we have the
following
1A section σ of a V -bundle is called smooth, continuous, holomorphic, etc. if each of the local sectionsσUi
are smooth, continuous, holomorphic, etc., respectively.2An invariant section σ of a V -bundle is a section such that the local sections can be taken as well-
defined maps from the underlying space |Σ|, σU : U = φ(U)→ BU . Such invariant sections can always beconstructed from local sections σU by “averaging over the group,” Γ, and extended to a global invariantsection.
Chapter 2. Geometry 14
2.1.11 Proposition. [MM03] Every orbifold admits a Riemannian metric, and every
complex orbifold admits a Hermitian metric.
One of the most important notions that goes through in the orbifold case is that of
integration. Choose a partition of unity (αi) subordinate to the open cover Ui of |Σ|
and define integration of an orbifold n-form ω ∈ Ωnorb(Σ,R) (See Remark 2.1.8) on an
n-dimensional orbifold Σ by ∫Σ
ω :=∑i
1
|Γi|
∫Ui
φ∗i (αi)ωi. (2.1.12)
We have the following nice classification result for orbifolds.
2.1.13 Proposition. [Sat57] Every orbifold Σ = (|Σ|,U) can be presented as a quotient
space of a locally free action of a compact Lie group G on a manifold X.
In light of Prop. 2.1.13, we will in fact be interested in studying manifolds X that
possess a natural U(1) action and may therefore be viewed as principal U(1)-bundles
over an orbifold surface Σ. We note that our main interest in this thesis are complex
and Kahler orbifolds since these structures are essential for understanding quasi-regular
K-contact manifolds (See §2.3). On a complex orbifold there is a Γ-invariant tensor field
J of type (1, 1) which describes the complex structure on the tangent V -bundle TΣ.
Recall the following
2.1.14 Definition. Let Σ = (|Σ|,U) be a complex orbifold with a Hermitian metric
g (See Def. 2.1.10) and corresponding two-form ωg defined by ωg(Y1, Y2) = g(Y1, JY2),
for Y1, Y2 ∈ Γ(TΣ). Then Σ is called a Kahler orbifold if ωg is closed. We call a
Kahler orbifold a Hodge orbifold if [ωg] lies in the image of the coefficient homomorphism
C : H2(|Σ|,Q)→ H2(|Σ|,R).
Finally, we recall that Chern-Weil theory works equally well on orbifolds. Note that
we primarily follow [BG08, §4.3] here. For orbifolds it is also true that every principal
V -bundle P has a connection, κ say, which can be described by an invariant connection
Chapter 2. Geometry 15
one-form on P with values in the Lie algebra g of G and is a smooth section of Λ1P ⊗ g.
Let I(G) =∑
k Ik(G) denote the algebra of polynomials on g that are invariant under
the adjoint action of G on g. To each polynomial f ∈ Ik(G) one defines a 2k-form on
the orbifold P by
f(Ω)(Y1, . . . , Y2k) =1
2k!
∑σ∈S2k
(−1)|σ|f(Ω(Yσ(1), Yσ(2)), . . . ,Ω(Yσ(2k−1), Yσ(2k))),
for Y1, . . . , Y2k vector fields on P , Ω the curvature of κ, and |σ| denotes the sign of σ as a
permutation. As in the manifold case, this 2k-form is basic and projects to a closed 2-form
on the underlying space |Σ| of the orbifold. Its de Rham cohomology class is independent
of the choice of connection κ as well. This describes the Weil homomorphism
w : I(G)→ H∗DR(|Σ|,R) ' H∗(|Σ|,R).
It turns out that for orbifolds w(I(G)) lies in the image of the coefficient homomorphism
C : H2(|Σ|,Q)→ H2(|Σ|,R).
2.2 Contact and CR Manifolds
In this section we review contact and CR-geometry. We will also review the Tanaka-
Webster torsion and curvature in this section. Our main references are [DT06], [Bla76],
[Gei08], [BG08], and [Bog91].
We begin this section with the following
2.2.1 Definition. [BG08, Def. 6.1.7 ] A (2n + 1)-dimensional manifold X is a contact
manifold if it is equipped with a one-form κ ∈ Ω1(X,R), called a contact one-form, on
X, such that
κ ∧ (dκ)n 6= 0,
Chapter 2. Geometry 16
everywhere on X. A contact structure on X is an equivalence class of such one-forms,
where κ′ ∼ κ ⇐⇒ ∃ 0 6= f ∈ C∞(X) such that κ′ = fκ. The subbundle ker(κ) =: H ⊂
TX will be called the contact subbundle of X.
2.2.2 Remark. Note that the condition κ ∧ (dκ)n 6= 0 is equivalent to dκ being non-
degenerate on H. There are several different perspectives and more general approaches
to defining a contact structure on an odd dimensional manifold X that we will not pursue
here, as in [Bla76], and [Gei08]. Let the line bundle LH be defined as the annihilator
bundle of the contact subbundle ker(κ) =: H ⊂ TX, i.e. LH := H0. We only note
that a more general definition involves allowing the line bundle LH ⊂ T ∗X over X to be
non-trivial. In Def. 2.2.1 we have assumed that LH is trivial and κ ∈ Γ(LH) represents
a choice of trivializing section. In order to distinguish the two cases, we will refer to
the case where LH is trivial as a strict (or co-orientable) contact structure, and the case
where LH is non-trivial as a non-strict contact structure. Note that the two-fold cover of
a non-strict contact manifold is strict, and in particular, every simply connected contact
manifold is strict.
2.2.3 Example. [BG08, Ex. 6.1.15] One of the most important examples of a contact
manifold is R2n+1 with contact form given by κ = dt−∑
i yidxi. The contact subbundle
H is spanned by ∂∂xi
+ yi∂∂t, ∂∂yi. This clearly defines a contact structure on R2n+1
according to Def. 2.2.1. For example dκ = dxi ∧ dyi is easily seen to be non-degenerate
on H (See Remark 2.2.2). This is called the standard contact structure on R2n+1. By the
contact version of Darboux’s theorem every contact manifold is locally contactomorphic3
to R2n+1 with the standard contact structure, [Bla76].
2.2.4 Example. [BG08, Ex. 6.1.16] Let X = S2n+1, the unit (2n + 1)-sphere. Let
α :=∑n
i=0(xidyi − yidxi) ∈ Ω1(R2n+2,R), where R2n+2 is given the standard Cartesian
3A map Ψ : (X,κ) → (X ′, κ′) between contact manifolds is called a contactomorphism if it is adiffeomorphism that preserves the contact structure, Ψ∗κ′ = f · κ, for some 0 6= f ∈ C∞(X). If thereexists an contactomorphism Ψ : (X,κ)→ (X ′, κ′), then (X,κ) ' (X ′, κ′) are said to be contactomorphic.
Chapter 2. Geometry 17
coordinates (x0, . . . , xn, y0, . . . , yn). Define
κ := α|S2n+1 .
It is straightforward to see that κ ∧ (dκ)n 6= 0 everywhere on S2n+1. This defines the
standard contact structure on S2n+1.
There is a very interesting generalization of Ex. 2.2.4 above due to Gray, [Gra59].
2.2.5 Proposition. [BG08, Prop. 6.1.17] Let X be an immersed hypersurface in R2n+2
such that no tangent space of X contains the origin of R2n+2. Then X has a contact
structure.
It is interesting to note that the following provides an example of a contact manifold
for which the results of this thesis do not apply. One goal of future work is to find an
analogue of Prop. 6.10.24 in this case.
2.2.6 Example. [BG08, Ex. 6.1.23] Let X = T3 := R3/Z3. Let (x, y, z) denote the
standard Cartesian coordinates in R3 and let
κ := sin(y)dx+ cos(y)dz.
Then κ ∧ dκ = −dx ∧ dy ∧ dz, and the contact subbundle is spanned by ∂∂y, cos(y) ∂
∂x−
sin(y) ∂∂z.
Recall that every oriented surface possesses a symplectic structure. The analogue of
this fairly trivial fact in the three-manifold case is one of the most important facts for
this thesis and is the reason why studying contact structures on three-manifolds is so
natural. We have the following
2.2.7 Theorem. [Mar71] Every orientable three-manifold admits a contact structure.
We recall the following standard fact:
Chapter 2. Geometry 18
2.2.8 Lemma. [BG08, Lemma 6.1.24] On a contact manifold (X, κ) there is a unique
vector field ξ ∈ Γ(TX) called the Reeb vector field, satisfying the two conditions
ιξκ = 1, ιξdκ = 0.
We note that the Reeb vector field ξ depends strongly on the choice of contact form
κ and one can obtain very different Reeb vector fields for equivalent choices of contact
forms within a contact structure.
2.2.9 Example. [BG08, Ex. 7.1.12] Consider (S2n+1, κ) with the standard contact struc-
ture as in Ex. 2.2.4. Let Hi := xi∂∂yi− yi ∂
∂xi. It is straightforward to see that the Reeb
vector field of κ is given by
ξ =∑i
Hi.
Let w = (w0, . . . , wn) ∈ Rn+1 be some positive vector so that wi > 0 ∀ 0 ≤ i ≤ n. Let
fw(x) :=1∑n
i=0 wi(x2i + y2
i ), for x ∈ S2n+1.
Define a deformed contact form
κw := fw κ.
It is easy to see that
ξw =∑i
wiHi,
is the corresponding Reeb field for κw. Clearly, the Reeb field changes drastically de-
pending on the choice of vector w. If the components of w are rational numbers, for
example, the orbits of the Reeb field turn out to be all circles. If we choose one of the
components to be irrational, however, we may obtain Reeb orbits that do not close. Yet,
since κw := fw κ differ by a non-zero function fw, these different choices amount to the
same underlying contact structure.
The Reeb vector field is sometimes called the characteristic vector field and the one-
dimensional foliation Fξ uniquely determined by ξ is called the characteristic foliation of
Chapter 2. Geometry 19
(X, κ).
Let us move on and make the following
2.2.10 Definition. [BG08, Def. 6.2.5] An almost contact structure on a differentiable
manifold X is a triple (ξ, κ, φ), where φ : TX → TX is a tensor field of type (1, 1), ξ is
a vector field, and κ ∈ Ω1(X,R) is a one-form which satisfy
κ(ξ) = 1, φ2 = −I + ξ ⊗ κ,
where I is the identity endomorphism of TX. A smooth manifold with such a structure
is called an almost contact manifold. An almost contact structure is said to be normal if
[φ, φ] + 2dκ⊗ ξ = 0,
where,
[φ, φ](Y1, Y2) := φ2[Y1, Y2]− [φY1, φY2]− φ[φY1, Y2]− φ[Y1, φY2],
is the Nijenhuis torsion of φ.
We make the following
2.2.11 Definition. [BG08, Def. 6.3.1] Let X be an almost contact manifold. A Rie-
mannian metric g on X is said to be compatible with the almost contact structure if for
any vector fields Y1, Y2 ∈ Γ(TX), we have
g(φY1, φY2) = g(Y1, Y2)− κ(Y1)κ(Y2).
An almost contact structure with a compatible metric is called an almost contact metric
structure.
We have the following
2.2.12 Proposition. [BG08] Every almost contact manifold admits a compatible metric.
Chapter 2. Geometry 20
The following theorem is important for this thesis as it allows us to conclude a par-
ticularly nice form for the metric structures on our contact manifolds.
2.2.13 Theorem. [BG08, Theorem 6.3.6] Let (Σ, J) be an almost complex orbifold (See
§2.1) and let π : X → Σ be a principal U(1) V -bundle over Σ. Suppose that ξ is a
generator of the U(1)-action that corresponds to 1 in the Lie algebra g ' R, and that κ
is a connection one-form on X. Then (ξ, κ, φ), with φ defined as in Eq. 6.3.3 of [BG08],
defines an almost contact structure on X. Furthermore, if h is a Hermitian metric on Σ
compatible with J then
g = κ⊗ κ+ π∗h
is a (bundle-like) Riemannian metric on X compatible with the almost contact structure,
and ξ is a Killing field for g.
We will need the following
2.2.14 Definition. [BG08, Def. 6.4.1] Let (X, κ) be a contact manifold with contact
distribution H. Then an almost contact structure (ξ, κ′, φ) is said to be compatible with
the contact structure if κ = κ′, ξ is the Reeb vector field, and the endomorphism φ
satisfies
dκ(φY1, φY2) = dκ(Y1, Y2), for all Y1, Y2 ∈ Γ(TX),
and,
dκ(φY0, Y0) > 0, for all Y0 ∈ Γ(H).
Denote by AC(κ) the set of compatible almost contact structures on (X, κ).
2.2.15 Proposition. [BG08, Prop. 6.4.3] Let (X, κ) be a contact manifold. The set of
associated Riemannian metrics are in one-to-one correspondence with the set of compat-
ible almost contact structures, AC(κ), on (X, κ).
Finally, the following is the basic definition that we need for this thesis.
Chapter 2. Geometry 21
2.2.16 Definition. [BG08, Def. 6.4.4] A contact manifold (X, κ) with a compatible
almost contact metric structure (ξ, κ, φ, g) such that
g(Y1, φY2) = dκ(Y1, Y2), for all Y1, Y2 ∈ Γ(TX),
is called a contact metric structure, and (X, ξ, κ, φ, g) is called a contact metric manifold.
We will also need the following
2.2.17 Definition. A normal4 contact metric manifold (X, ξ, κ, φ, g) is called a Sasakian
manifold.
We now move on to study CR structures. We start with the following
2.2.18 Definition. [DT06, Def. 1.1 and 1.2] An almost CR structure on a manifold
X (dim(X) = m) is a subbundle T(1,0) = T(1,0)(X) ⊂ TCX of complex rank n of the
complexified tangent bundle such that
T(1,0)(X) ∩ T(0,1)(X) = 0,
where T(0,1)(X) := T(1,0)(X) the complex conjugate. An almost CR structure is called a
CR structure if
[T(1,0)(X), T(1,0)(X)] ⊂ T(1,0)(X),
so that T(1,0)(X) is an integrable subbundle of TCX. The integers n and k = m − 2n
are called the CR dimension and CR codimension of the almost CR structure and (n, k)
denotes its type. The pair (X,T(1,0)) is called an (almost) CR manifold of type (n, k).
We are mainly interested in almost CR structures of type (1, 1) in this thesis. Consider
the following
2.2.19 Example. Let (X, κ, ξ, φ) be an almost contact manifold with distribution H =
kerκ. The restriction of φ to H determines the decomposition
HC = H(1,0) ⊕H(0,1) ⊂ TCX,
4See Def. 2.2.10.
Chapter 2. Geometry 22
where H(1,0) and H(0,1) are the +i and −i eigenbundles of φ|H , respectively. Taking
T(1,0)X = H(1,0) determines an almost CR structure on X. This construction clearly
also applies to the case where (X, κ) is a contact manifold and (X, κ, ξ, φ) is choice of
compatible almost complex structure. Whether or not this construction yields a CR
structure (i.e. integrable distribution) has been determined by S. Tanno in [Tan89]. In
this thesis we will always be concerned with the case where this construction yields a CR
structure. This follows because our contact structures will be assumed normal (See Def.
2.2.10).
Notice in the above example that we may recover H from HC by taking the real part
of HC:
H = R(H(1,0) ⊕H(0,1)).
The following is a generalization of this idea.
2.2.20 Definition. Let (X,T(1,0)) be an (almost) CR manifold of type (n, k). Its maximal
complex, or Levi distribution is the real rank 2n subbundle defined as
L(X) = R(T(1,0) ⊕ T(1,0)).
L(X) carries the complex structure JL : L(X)→ L(X) defined by
JL(Y + Y ) = i(Y + Y ),
for any Y ∈ T(1,0).
As noted in Remark 2.2.2 above, given a contact manifold (X, κ) with contact dis-
tribution H, a contact form is naturally viewed a section of the annihilator bundle H0.
Generalizing this to CR manifolds of type (n, 1),5 we let H0 denote the annihilator bun-
dle of the Levi distribution H = L(X). It is easy to see that H0 is a subbundle of T ∗X
that is isomorphic to TX/H. Assume X is orientable. Then since H is oriented by the
5Also called CR manifolds of hypersurface type.
Chapter 2. Geometry 23
complex structure JL, it follows that H is orientable. Any orientable real line bundle
over a connected manifold is trivial, so there exist globally defined nowhere vanishing
sections θ ∈ Γ(H).
2.2.21 Definition. [DT06, Def. 1.6] Let (X,T(0,1)) be an oriented CR manifold of
type (n, 1) with H = L(X). Then any a choice of θ ∈ Γ(H) is referred to as a pseudo-
Hermitian structure on X. Given a pseudo-Hermitian structure θ on X the Levi form
Lθ is defined by
Lθ(Z,W ) = −idθ(Z,W ),
for any Z,W ∈ T(1,0).
We now make the following
2.2.22 Definition. [DT06, Def. 1.7] Let (X,T(0,1)) be an oriented CR manifold of type
(n, 1) with H = L(X). We say that (X,T(0,1)) is nondegenerate if the Levi form Lθ is
non-degenerate for some (and hence any) choice of pseudo-Hermitian structure θ on X.
If Lθ is positive definite (i.e. Lθ(Z,Z) > 0, ∀ 0 6= Z ∈ T(0,1)) for some θ,6 then (X,T(0,1))
is said to be strictly pseudoconvex.
The main point that we would like to make is that on a contact manifold (X, κ), the
choice of contact form κ defines a pseudo-Hermitian structure on X relative to the almost
CR structure defined in Ex. 2.2.19 above. Going in the other direction, one may obtain
a natural contact metric structure (X, ξ, κ, φ, g) from a CR structure (X,T(0,1)) of type
(n, 1) with pseudo-Hermitian structure κ. First we need the following
2.2.23 Proposition. [DT06, Prop. 1.2] Given (X,T(0,1)) a type (n, 1) CR manifold with
pseudo-Hermitian structure κ, there exists a unique globally defined nowhere zero tangent
vector field ξ on X such that
ιξκ = 1, ιξdκ = 0,
6This does not apply to all choices of θ since Lθ positive definite implies that L−θ is negative definite.
Chapter 2. Geometry 24
and ξ is transverse to the Levi distribution H = L(X).
We also have
2.2.24 Proposition. [DT06, Prop. 1.4] Given (X,T(0,1)) a type (n, 1) CR manifold
with pseudo-Hermitian structure κ, Levi distribution H = L(X) and ξ as in Prop. 2.2.23
above, then
TX ' H ⊕ Rξ.
By setting φ(Y ) = JLY for all Y ∈ H, and φξ = 0, one can show that (X, κ, ξ, φ)
defines an almost contact manifold. If the Levi form Lκ is non-degenerate then (X, κ) is
a contact manifold. Let
g(Y1, Y2) = dκ(Y1, JLY2).
Then g(JLY1, JLY2) = g(Y1, Y2) since the Nijenhuis tensor of JL vanishes when X is a
CR manifold. We may now extend g to all of TX by using the splitting TX ' H ⊕ Rξ
and defining g(Y, ξ) = 0 and g(ξ, ξ) = 1. The resulting form g is called the Webster
metric of (X, κ). If X is strictly pseudoconvex, then g defines a Riemannian metric and
(X, ξ, κ, φ, g) defines a contact metric structure on X. We may now state the following
2.2.25 Theorem. [DT06, Theorem 1.3] Given (X,T(0,1)) a type (n, 1) CR manifold with
pseudo-Hermitian structure κ, Levi distribution H = L(X), ξ as in Prop. 2.2.23 above,
and JL the complex structure on H = L(X) (extended to φ ∈ End(TX) by requiring that
φ(ξ) = 0). Let g be the Webster metric on (X, κ). There is a unique linear connection
∇ on X satisfying the following axioms:
i. H is parallel with respect to ∇, that is,
∇Y Γ(H) ⊂ Γ(H),
for any Y ∈ Γ(TX).
ii. ∇J = 0, and ∇g = 0.
Chapter 2. Geometry 25
iii. The torsion T∇7 of ∇ is pure, i.e.
T∇(Y1, Y2) = dκ(Y1, Y2)ξ, and, T∇(ξ, JLY ) + JLT∇(ξ, Y ) = 0,
for all Y1, Y2 ∈ Γ(H).
We then make the following
2.2.26 Definition. [DT06, Def. 1.25] Given (X,T(0,1)) as in Theorem 2.2.25 above, the
connection ∇ in Theorem 2.2.25 is called the Tanaka-Webster connection. The Tanaka-
Webster scalar curvature R ∈ C∞(X) is the scalar curvature8 associated to the Tanaka-
Webster connection.
2.3 Seifert Manifolds
We review the relevant definitions and results in Seifert geometry. We will primarily
study the class of Seifert manifolds that come from the orbifold version of the Boothby-
Wang theorem; that is, we study the class of quasi-regular K-contact manifolds. We will
also study Seifert manifolds in general, and recall some basic facts about their funda-
mental groups and related invariants. Our main references are [BG08], [BW05], [FS92]
and [Orl72].
In this section we briefly review our geometric situation. In particular, we recall the
7Recall that the torsion of a linear connection is defined as
T∇(Y1, Y2) = ∇Y1Y2 −∇Y2Y1 − [Y1, Y2].
8Recall that the scalar curvature is computed via contraction by the Webster metric g of the Riccicurvature
Rij = Rkikj ,
where R is the curvature tensor of the connection ∇,
R(Y1, Y2)Y3 = ∇Y1∇Y2Y3 −∇Y2∇Y1Y3 −∇[Y1,Y1]Y3.
That is, R = Rii = gijRij .
Chapter 2. Geometry 26
definition of a quasi-regular K-contact manifold and review some standard facts about
these structures in the case of dimension three.
2.3.1 Remark. Our three manifolds X are assumed to be closed throughout this thesis.
2.3.2 Definition. A K-contact manifold is a manifold X with a contact metric structure
(φ, ξ, κ, g) such that the Reeb field ξ is Killing for the associated metric g, Lξg = 0.
where,
• κ ∈ Ω1(X) contact form, ξ = Reeb vector field.
• H := kerκ ⊂ TX denotes the horizontal or contact distribution on (X, κ).
• φ ∈ End(TX), φ(Y ) = JY for Y ∈ Γ(H), φ(ξ) = 0 where J ∈ End(H) complex
structure on the contact distribution H ⊂ TX.
• g = κ⊗ κ+ dκ(·, φ·)
2.3.3 Remark. Note that we will assume that our contact structure is “co-oriented,”
meaning that the contact form κ ∈ Ω1(X) is a global form. Generally, one can take the
contact structure to be defined only locally by the condition H := kerκ, where κ ∈ Ω1(U)
for open subsets U ∈ X contained in an open cover of X.
2.3.4 Definition. The characteristic foliation Fξ of a contact manifold (X, κ) is said
to be quasi-regular if there is a positive integer j such that each point has a foliated
coordinate chart (U, x) such that each leaf of Fξ passes through U at most j times. If
j = 1 then the foliation is said to be regular.
Definitions 6.9.10 and 6.9.11 together define a quasi-regular K-contact manifold, (X,φ, ξ, κ, g).
The following result provides several different perspectives on K-contact structures:
2.3.5 Proposition. [BG08, Prop. 6.4.8] On a contact metric manifold (X,φ, ξ, κ, g),
the following conditions are equivalent:
Chapter 2. Geometry 27
i. The characteristic foliation Fξ is a Riemannian foliation.
ii. g is bundle-like.
iii. The Reeb flow is an isometry.
iv. The Reeb flow is a CR-transformation.
v. The contact metric structure (φ, ξ, κ, g) is K-contact.
Quasi-regular K-contact three-manifolds are necessarily “Seifert” manifolds that fiber
over a two dimensional orbifold Σ (See §2.1) with some additional structure. Recall:
2.3.6 Definition. A Seifert manifold is a three manifold X equipped with a locally free
U(1)-action.
Thus, a Seifert manifold is simply a U(1)-bundle over an orbifold Σ,
U(1) // X
Σ
.
with smooth total space X. The orbifold base Σ of X is taken to be a Riemann surface
of genus g with N marked points pjNj=1, the exceptional points, and the local model is
C/Zαj centered about the point pj. Here Zαj acts on the local coordinate z at pj as
z 7→ ξ · z, ξ = e2πi/αj
We follow [FS92], and describe a general Seifert manifold X as the total space associated
to the S1 fibration of a line V-bundle over Σ. A line V-bundle over Σ (See Def. 2.1.4) can
be described as a complex line bundle, where the local trivialization over each orbifold
point pj of Σ is now modeled on C × C/Zαj , where Zαj acts on the local coordinates
(z, s) as
z 7→ ξ · z, s 7→ ξβj · s, ξ = e2πi/αj
Chapter 2. Geometry 28
for some integers 0 ≤ βj < αj. We require that X be smooth, which implies that each
pair of integers (αj, βj) are relatively prime, so that the local action of Zαj on C× S1 is
free (in particular, βj 6= 0). The U(1) action on X is again rotations of the fibres over Σ,
but the points in the S1 fiber over each point pj are fixed by the cyclic subgroup Zαj of
U(1).
It is well known that9 the topological isomorphism class of a Seifert manifold X is given
by the Seifert invariants
[g, n; (α1, β1), . . . , (αN , βN)], gcd(αj, βj) = 1
where g is the genus of Σ, and n is the degree of the line V-bundle.
For later use, we record now the following assumption, that
c1(L) = n+N∑j=1
βjαj
> 0
where L denotes the line V-bundle over Σ which describes X, and c1(L) is the orbifold
first Chern number of L.
2.3.7 Remark. We take c1(L) to be positive by convention, and non-zero so that L is
non-trivial (See Example 2.3.9 for the reason behind this).
Recall also the following description of the fundamental group of X, [Orl72]. π1(X)
is generated by the following elements
ap, bp, p = 1, . . . , g
cj, j = 1, . . . , N
h
9See [Orl72] for example.
Chapter 2. Geometry 29
which satisfy the relations,
[ap, h] = [bp, h] = [cj, h] = 1 (2.3.8)
cαjj h
βj = 1g∏p=1
[ap, bp]N∏j=1
cj = hn
The generator h is associated to the generic S1 fiber over Σ, the generators ap, bp come
from the 2g non-contractible cycles on Σ, and the generators cj come from the small one
cycles in Σ around each of the orbifold points pj.
We have the following classification result: X is a quasi-regular K-contact three manifold
⇐⇒
• [BG08, Theorem 7.5.1, (i)] X is a U(1)-Seifert manifold over a Hodge orbifold
surface, Σ.
• [BG08, Theorem 7.5.1, (iii)] X is a U(1)-Seifert manifold over a normal projective
algebraic variety of real dimension two.
2.3.9 Example. All 3-dimensional lens spaces, L(p, q) and the Hopf fibration S1 → S3 →
CP1 admit quasi-regular K-contact structures. Note that any trivial U(1)-bundle over a
Riemann surface Σg, X = U(1)×Σg, admits no K-contact structure,10 however, and our
results do not apply in this case.
2.3.10 Remark. Note that in fact our results apply to the class of all closed Sasakian
three-manifolds. This follows from the observation that every Sasakian three manifold is
K-contact, [Bla76, Corollary 6.5], and every K-contact manifold possesses a quasi-regular
K-contact structure, [BG08, Theorem 7.1.10].
A useful observation for us is that for a quasi-regular K-contact three-manifold, the metric
tensor gε must take the following form, [BG08, Theorem 6.3.6]:
gε = ε κ⊗ κ+ π∗h (2.3.11)
10See [Ito97] for example.
Chapter 2. Geometry 30
where π : X → Σ is our quotient map, and h represents any (orbifold) Kahler metric
(See Def. 2.1.10) on Σ which is normalized so that the corresponding (orbifold) Kahler
form (See Def. 2.1.14), ω ∈ Ω2orb(Σ,R), pulls back to dκ.
2.4 Vielbein Formalism
In this section we outline the basic construction of a Vielbein on an orientable Riemannian
three-manifold (X,G). Our primary references for this are [FF03] and [Car04]. We note
that the contents of this chapter are essential for our computation in Chapter 7 and are
well established in the physics literature.11
2.4.1 Definition. LetM be an orientable manifold of dimension n, Θ an SO(n)-principal
bundle over M , and L(M) the GL(n)-frame bundle of M . A Vielbein is a principal
morphism E : Θ→ L(M) (if one exists).
2.4.2 Remark. Note that if we fix Θ arbitrarily, then it could happen that no principal
morphism E : Θ → L(M) exists. For example, if M is non-parallelizable12 and Θ =
Θtriv := M × SO(n), then there is no global principal morphism E : Θ → L(M). If
such a morphism did exist then E σ : M → L(M) would be a global section for some
trivialization σ of Θtriv, which is impossible since M is non-parallelizable. Of course, if
M is parallelizable, so that σ : M → L(M) is a trivialization, then a Vielbein always
exists by choosing Eσ : Θtriv → L(M) the standard inclusion.
Let E : Θ → L(M) be a Vielbein. Since Θ is an SO(n)-principal bundle over M ,
then by construction E induces a reduction of the structure group of the tangent bundle
TM from GL(n) to SO(n), and hence induces an associated Riemannian metric on M .
Conversely, if a Riemannian metric G is given on M , then the corresponding reduction
of the structure group induces a natural Vielbein.
11See [GIJP03] for example.12A manifold M is called parallelizable if the frame bundle L(M) is trivializable.
Chapter 2. Geometry 31
Let us henceforth assume that M = (X,G) is an oriented Riemannian three-manifold.
Given a local chart U ⊂ X, a Vielbein may therefore be expressed as a triple
E0, E1, E2 (2.4.3)
where EA ∈ Γ|U(TX) such that
G(EA, EB) = ηAB (2.4.4)
whereA,B ∈ 0, 1, 2. Note that in a Lorentzian spacetime, ηAB represents the Minkowski
metric, of signature (−,+,+) say, and in a Euclidean spacetime it represents the positive-
definite Euclidean metric (so ηAB = δAB, is the Kronecker pairing in this case). We
implicitly assumed a Euclidean signature in our definition of a Vielbein by choosing to
work with SO(n)-bundles. More generally we could have allowed SO(p, q)-bundles of
arbitrary signature. We will work in a Euclidean signature in this thesis. Given a choice
of local coordinate system x0, x1, x2 on X, we define
Gµν := G(∂µ, ∂ν) (2.4.5)
and we define the notation EµA to represent the coordinates of EA in the coordinate
system basis ∂0, ∂1, ∂2:
EA =2∑
µ=0
EµA∂µ. (2.4.6)
Note that we also adopt the Einstein summation convention and write
EA = EµA∂µ, (2.4.7)
for example, where it is understood that a sum is taken over the repeated raised and
lowered indices. We may then express Eq. (2.4.4) in local coordinates:
Gµν EµA E
νB = ηAB (2.4.8)
Chapter 2. Geometry 32
where µ, ν ∈ 0, 1, 2 are thought of as indexing the spacetime coordinates related to
the manifold coordinates x0, x1, x2, and A,B ∈ 0, 1, 2 are thought of as indexing the
tangent space coordinates that label the Vielbein. By dualizing to the cotangent bundle,
we also consider
EA := EAµ dx
µ ∈ Γ|U(T ∗X). (2.4.9)
which are defined by requiring
EA(EB) = δAB. (2.4.10)
Let Gµν denote the inverse of Gµν , so that GµλGλν = GνλGλµ = δµν . Our relevant
relations for the Vielbein are then:
EµAE
Aν = δµν , EA
µ EµB = δAB (2.4.11)
Gµν EµA E
νB = ηAB , Gµν = EA
µEBν ηAB (2.4.12)
EAµ = Gµνη
ABEνB. (2.4.13)
Note that every closed, oriented three-manifold is parallelizable.13 Thus, if σ : X → L(X)
is a trivialization, one can take Eσ : Θtriv → L(M) the standard inclusion. An important
point for us to note is that there are many possible choices of trivialization σ : X → L(X),
and it is a standard fact that the gravitational Chern-Simons term
CS(A) :=1
4π
∫X
Tr(A ∧ dA+2
3A ∧ A ∧ A), (2.4.14)
where A the Levi-Civita connection on the spin bundle of X, is sensitive to the choice of
trivialization σ up to homotopy equivalence. This dependence is explicitly computable
as the winding number W (E) of the Vielbein, [GIJP03, Eq. 2.21]:
W (E) =1
24π2
∫X
d3x εµνλ tr(VµVνVλ) ∈ Z, (2.4.15)
where (Vµ)σρ := EσA∂µE
Aρ .
13See [LM89].
Chapter 2. Geometry 33
2.5 The Space of Connections
2.5.1 The General Case
Let X be a compact smooth manifold and G a compact Lie group. Since it will be useful
to deal in both the general case and the special G = U(1) case, we will assume no further
restrictions on X and G, and specialize to the G = U(1) case in the next section. Most
of the material in this section can be found in [AB83].
Let P be a principal G-bundle over X. P is a manifold equipped with a proper and free
action of G for which X = P/G. Define the group of gauge transformations, GP , of P to
be the set of G-equivariant diffeomorphisms of P that preserve fibres:
GP := ψ ∈ (Diff(P, P ))G | π ψ = π (2.5.1)
where,
π : P → X = P/G (2.5.2)
is the standard projection map. We will sometimes identify GP with the infinite dimen-
sional Lie group of G-equivariant smooth functions from P to G,
(C∞(P,G))G := u ∈ C∞(P,G) | u(g · p) = gu(p)g−1 (2.5.3)
The identification is given by,
(C∞(P,G))G → GP (2.5.4)
u 7→ ψ (2.5.5)
where ψ(p) = p · u is a Lie group isomorphism. For a fixed g ∈ G, let ψg ∈ Diff(P, P ) be
the action map,
ψg(p) := p · g (2.5.6)
This G action on P induces an action on the space Ω(P ) of differential forms
g · ω := ψ∗g−1(ω) (2.5.7)
Chapter 2. Geometry 34
Define the set of horizontal m-forms on P to be
Ωmhor.(P ) := ω ∈ Ωm(P ) | ιξ](ω) = 0, ∀ ξ ∈ g (2.5.8)
We combine the action of G on Ω(P ) with the adjoint action of G of g to obtain an action
of G on the space of g-valued m-forms on P , Ωm(P )⊗ g,
g ·
(∑j
ωj ⊗ ξj
)=∑j
ψ∗g−1(ωj)⊗ Adg(ξj) (2.5.9)
Denote by (Ωm(P ) ⊗ g)G the space of G equivariant g-valued m-forms on P , and by
(Ωmhor.(P ) ⊗ g)G the space of horizontal G-equivariant g-valued m-forms on P . Given a
principal G-bundle P over X, we define the space of connections to be:
AP := A ∈ (Ω1(P )⊗ g)G | A(ξ]) = ξ, ∀ ξ ∈ g (2.5.10)
We have the following,
2.5.11 Proposition. If P is an arbitrary principal G-bundle, then the space of connec-
tions AP is non-empty, and if AP ∈ AP , then AP = AP + (Ω1hor.(P )⊗ g)G
That is, AP is an affine space modeled on the linear space
aP := (Ω1hor.(P )⊗ g)G (2.5.12)
Let g(P ) denote the associated bundle for the adjoint action of G on g,
g(P ) = P ×G g = (g× P )/ ∼ (2.5.13)
where (ξ, p) ∼ (Adg(ξ), g · p). Observe that g(P ) is a vector bundle over X with fibre g.
It is sometimes useful to make the identification
(Ωmhor.(P )⊗ g)G = Ωm(X, g(P )) (2.5.14)
where Ωm(X, g(P )) are vector bundle valued forms onX. Recall that for A =∑
j αj⊗ξj ∈
Ωp(P )⊗ g and B =∑
j βj ⊗ ζj ∈ Ωq(P )⊗ g, that
dA = (d⊗ Idg)(A) =∑j
dαj ⊗ ξj (2.5.15)
Chapter 2. Geometry 35
and
[A,B] = (∧ ⊗ [·, ·])(A,B) =∑j
∑k
αj ∧ βk ⊗ [ξj, ζk] (2.5.16)
2.5.17 Definition. A connection A ∈ AP will be called flat if its curvature FA =
dA+ 12[A,A] ∈ Ω2(P, g) vanishes, i.e. FA = 0.
Recall that the group of gauge transformations GP acts on the space of connections AP
in the following way;
2.5.18 Proposition. Let u = uψ for the associated gauge transformation ψ ∈ GP . Then
GP acts on AP as follows: ψ ·A = uψ ·A = (ψ)∗A = Adu−1ψA+(uψ)∗ϑ, where ϑ ∈ Ω1(G)⊗g
is the Maurer-Cartan form.14
2.5.20 Remark. Since,
(u∗ϑ)γ(0)
(d
dt|t=0γ(t)
)=
d
dt|t=0((u(γ(0)))−1(u(γ(t)))) (2.5.21)
one often writes u∗ϑ = u−1 · du. Using this notation, the action of the gauge group on a
connection is given by
u · A = Adu−1A+ u−1 · du. (2.5.22)
2.5.23 Proposition. GP acts on (Ω1hor.(P )⊗ g)G as follows: if B ∈ (Ω1
hor.(P )⊗ g)G and
ψ(p) = p · uψ, then ψ ·B = ψ∗B = Adu−1ψB.
The Lie algebra of the gauge group GP is
Lie(GP ) = (Xvert.(P ))G = Z ∈ X(P ) | dπp(Zp) = 0, ψ∗gZ = Z (2.5.24)
and the Lie algebra of (C∞(P,G))G is:
Lie((C∞(P,G))G) = (C∞(P, g))G ' (Ω0hor.(P )⊗ g)G (2.5.25)
We have the following:
14Recall that the Maurer-Cartan form ϑ ∈ Ω1(G)⊗ g is defined by:
ϑg(ξ) := (Lg−1)∗ξ (2.5.19)
for ξ ∈ TgG.
Chapter 2. Geometry 36
2.5.26 Proposition. The map given by
(C∞(P, g))G → (Xvert.(P ))G (2.5.27)
f 7→ Z (2.5.28)
where Zp = ddt|t=0 p · (exp(−tf(p))) = −(f(p))]p, is a Lie algebra isomorphism.
2.5.29 Proposition. If Z ∈ (Xvert.(P ))G and Zp = −(f(p))]p, define φt ∈ GP by setting
φt(p) = p · exp(−t(f(p))). Then φtt∈R is the flow of Z.
2.5.30 Proposition. The adjoint action of ψ ∈ GP on Z ∈ Lie(GP ) is
ψ · Z = ψ∗Z (2.5.31)
Also, if ψ(p) = p · uψ and Zp = −(f(p))]p, then
(ψ · Z)p = −(Adu(p)f(p))]p. (2.5.32)
2.5.2 The U(1)-Bundle Case
In this section we specialize to G = U(1). We primarily follow sections §2 and §3 of
[Man98].
First, observe that
C∞(X,U(1)) = (C∞(P,G))G (2.5.33)
h ↔ u (2.5.34)
where an element h of C∞(X,U(1)) defines a map uh : P → U(1) by uh(p) = h(π(p)),
and conversely a map u : P → U(1) defines a map hu : X → U(1) by hu(x) = u(p) for
any p ∈ π−1(x). Thus, GP = C∞(X,U(1)). Since GP is independent of the bundle P in
the U(1) case, we define:
GX := C∞(X,U(1)) (2.5.35)
Chapter 2. Geometry 37
We also have that aP = (Ω1hor.(P ) ⊗ R)U(1) = (Ω1
hor.(P ))U(1) = Ω1(X), where the last
equality comes from the fact that π∗ : Ω1(X)→ Ω1(P ) is an isomorphism onto its image
(Ω1hor.(P ))U(1). In summary, when G = U(1) we identify:
aP = Ω1(X) (2.5.36)
Let AX denote the space of all U(1)-connections on X. An element A ∈ AX is a
connection on a principal U(1)-bundle P . We may write
AX =⊔P
AP (2.5.37)
as the union over all principal U(1)-bundles over X. Recall the notion of equivalence
among principal U(1)-bundles. Two principal U(1)-bundles over X, say P1 and P2, are
equivalent if there exists a U(1)-equivariant map ψ : P1 → P2 that covers the identity
map on X. Two elements A1, A2 ∈ AX are called gauge equivalent if there exists and
isomorphism ψ : P1 → P2 such that A1 = ψ∗A2. This defines an equivalence relation on
AX , where
A1 ∼ A2 ⇐⇒ A1 = ψ∗A2 (2.5.38)
The action of GX = C∞(X,U(1)) on AP can be written out for any U(1)-bundle P
over X. If h : X → U(1) is an element of GX with associated bundle automorphism
ψh : P → P defined by ψh(p) = p · uh(p), the action of h on A ∈ AP is described by:
h · A = (ψh)∗A = A+ u∗hϑ, (2.5.39)
since the adjoint action is trivial is in the U(1) case. Note that ϑ is the Maurer-Cartan
form of U(1), so that ϑ generates H1(U(1),Z) ⊂ H1(U(1),R).
We will now recall some basic facts about curvature and connections for principal U(1)-
bundles. Fix a principal U(1)-bundle π : P → X and let A ∈ AP be a connection on P .
We will be slightly ambiguous in our terminology regarding the curvature of a connection
and we aim to describe this here. First we will need the following
Chapter 2. Geometry 38
2.5.40 Lemma. Given a connection A ∈ AP on a principal U(1)-bundle π : P → X,
there exists a two form F on X such that
π∗F = dA (2.5.41)
Proof. We shall prove this by showing that the form dA is U(1)-invariant and horizontal,
i.e. that dA is basic. Clearly dA is invariant because A is invariant. dA is horizontal since
for any Y ∈ u(1), ι(Y ]P )dA = dι(Y ]
P )A = 0 since by definition of a connection ι(Y ]P )A = Y
is constant (where Y ]P denotes the vector field on P generated by the infinitesimal action
of Y on P ).
2.5.42 Remark. The ambiguity in our terminology will be the following. The curvature
of a connection A ∈ AP in the U(1)-bundle case will refer to both
FA := dA (2.5.43)
and the form F that satisfies the above lemma 2.5.40
π∗F = dA. (2.5.44)
We will usually use the same notation for both. When there is a chance of confusion, we
will write FA for the form dA and reserve the notation F for the two form on the base
manifold X in lemma 2.5.40.
The notation for the curvature F ∈ Ω2(X,R), being written as independent of the original
connection A used to define it, is partially justified by the following
2.5.45 Lemma. The cohomology class of the curvature F is independent of the connec-
tion form A used to define it.
Proof. Recall from Prop. 2.5.11 that the space of connections is an affine space modeled
on the linear space aP = Ω1(X). Thus, for any two connection forms A,A′ ∈ AP , there
exists a form β ∈ Ω1(X) such that
A− A′ = π∗β.
Chapter 2. Geometry 39
Taking F, F ′ to be the associated curvature forms of A,A′ respectively, we compute
π∗(F − F ′) = dA− dA′
= π∗(β)
and so F = F ′ + dβ since π∗ : Ω2(X,R)→ Ω2basic(P, u(1)) is an isomorphism, and hence
[F ] = [F ′] ∈ H2DR(X,R).
The following is a standard result in Chern-Weil theory.
2.5.46 Proposition. [MS74] The curvature class [F ] ∈ H2(X,R) is the image of the
first chern class c1(P ) ∈ H2(X,Z) under the natural homomorphism
ψ : H2(X,Z)→ H2(X,R)
so that ψ(c1(P )) = [F ].
Recall that an element in a Z-module is said to be torsion if some integral multiple of it
is zero. We have
2.5.47 Proposition. Let X be a compact 3-manifold. A principal U(1)-bundle π : P →
X has flat connections if and only if the first Chern class, c1(P ) ∈ H2(X,Z), is torsion
c1(P ) ∈ TorsH2(X,Z). (2.5.48)
Proof. Let us first prove that the existence of a flat connection A ∈ AP implies that
c1(P ) ∈ TorsH2(X,Z). By Prop. 2.5.46, we know that ψ(c1(P )) = [F ] and since A is flat
we have F = 0. Thus, c1(P ) ∈ ker(ψ), and by the Universal Coefficient Theorem’s and
the definition of ψ is it elementary to see that ker(ψ) = TorsH2(X,Z). This establishes
the first direction.
To see the other direction, assume that c1(P ) ∈ TorsH2(X,Z) and observe that since
ker(ψ) = TorsH2(X,Z), we must have that F ∈ Ω1(X,R) is exact, i.e. F = dβ for some
Chapter 2. Geometry 40
β ∈ Ω1(X,R). Since π∗F = dA for some connection A ∈ AP we see that we can find a
flat connection A′ ∈ AP by setting
A′ = A− π∗β
since dA′ = d(A− π∗β) = dA− π∗dβ = π∗(F − F ) = 0. This completes the proof.
Recall that for A =∑
j αj ⊗ ξj ∈ Ωp(P )⊗ g and B =∑
j βj ⊗ ζj ∈ Ωq(P )⊗ g, that
dA = (d⊗ Idg)(A) =∑j
dαj ⊗ ξj (2.5.49)
and
[A ∧B] = (∧ ⊗ [·, ·])(A,B) =∑j
∑k
αj ∧ βk ⊗ [ξj, ζk] (2.5.50)
2.5.51 Definition. We decompose the subspace AfX ⊂ AX of flat connections as:
AfX =⊔P
c1(P )∈TorsH2(X,Z)
AfP (2.5.52)
where AfP ⊂ AP is the space of flat connections A on P with curvature
FA = dA+1
2[A ∧ A] = dA = 0 (2.5.53)
In the case that G = U(1), we have g = R, so that
[ξ, ζ] = 0, ∀ ξ, ζ ∈ g = R (2.5.54)
and hence,
FA = dA+1
2[A ∧ A] = dA = 0 (2.5.55)
as above.
2.5.56 Definition. Let P be a principal U(1)-bundle with c1(P ) ∈ TorsH2(X,Z). The
moduli space of flat connections on P is the quotient space,
MP := AfP/GP (2.5.57)
Chapter 2. Geometry 41
Note that the action of the gauge group GP on AfP is well defined since if h ∈ GP = GX =
C∞(X,U(1)), and AP ∈ AfP , then
Fh·AP = d(h · AP ) +1
2[h · AP , h · AP ] (2.5.58)
= d(h · AP ), since [h · AP , h · AP ] = 0 for G = U(1) (2.5.59)
= d(AP + u∗hϑ), by definition of h · AP (2.5.60)
= dAP + u∗hdϑ (2.5.61)
= 0, since AP is flat and dϑ = 0 for G = U(1) (2.5.62)
If P1 and P2 are bundles over X with equal first Chern classes, c1(P1) = c1(P2) ∈
TorsHm(X,Z), then MP1 ' MP2 are canonically isomorphic. This can be seen by
choosing a bundle isomorphism, φ : P1 → P2. This induces an isomorphism φ∗ : AP2 →
AP1 , which can be pushed down to an isomorphism of the of the quotients, φ∗ : AP2/GP →
AP2/GP . It can be shown that the isomorphism AP2/GP ' AP2/GP is independent of the
choice of bundle isomorphism φ : P1 → P2.
For each torsion class p ∈ TorsH2(X,Z), define
AfX,p =⊔P
c1(P )=p
AfP (2.5.63)
Once we mod out by the equivalence relation on AX , “∼” as defined in Eq. (2.5.38), the
space
MX,p := AfX,p/ ∼ (2.5.64)
is naturally isomorphic toMP for any principal U(1)-bundle P with c1(P ) = p. We then
have that the moduli space of gauge equivalence classes of flat U(1)-connections on X,
MX = AfX/ ∼, is equal to
MX =⊔
p∈TorsH2(X,Z)
MX,p (2.5.65)
We have the following characterization of MX ,
2.5.66 Proposition. [Man98] Let X be a smooth manifold.
Chapter 2. Geometry 42
1. There is a natural identification
MX = H1(X,U(1)) (2.5.67)
2. π0(MX) ' TorsH2(X,Z) and each connected component of MX is diffeomorphic
to the torus
H1(X,R)/H1(X,Z). (2.5.68)
Chapter 3
Heisenberg Calculus on Contact
Manifolds
In this chapter we provide the background, notation and terminology that is requisite
for working in the Heisenberg calculus. Since the operators that we will be interested
in studying are not elliptic, we cannot apply the usual pseudodifferential calculus of
elliptic theory to our case, and we use instead the Heisenberg calculus. First we review
general Heisenberg manifolds and some standard operators in this case, and then we
briefly develop the Heisenberg calculus and state some results from this theory that are
needed for this thesis.
3.1 Heisenberg Manifolds
In this section we review Heisenberg manifolds and some standard operators in this set-
ting. We mainly follow [BG88] and [Pon08]. We begin with the following
3.1.1 Definition. 1. A Heisenberg manifold (X,H) consists of a manifold X together
with a distinguished hyperplane bundle H ⊂ TX.
43
Chapter 3. Heisenberg Calculus on Contact Manifolds 44
2. A Heisenberg diffeomorphism φ from a Heisenberg manifold (X,H) to another
Heisenberg manifold (X ′, H ′) is a diffeomorphism φ : X → X ′ such that φ∗H = H ′.
3.1.2 Definition. Let (Xd+1, H) be a Heisenberg manifold. Then:
1. A (local) H-frame for TX is a (local) frame Y0, Y1, . . . , Yd so that Y1, . . . , Yd span
H.
2. A local Heisenberg chart is a local chart with a local H-frame of TX over its
domain.
The main examples of Heisenberg manifolds are the following.
3.1.3 Example. • Heisenberg Group: The (2n+ 1)-dimensional Heisenberg group
H(2n+1) is R(2n+1) = R× R2n equipped with the group law,
x.y := (x0 + y0 +n∑j=1
(xn+jyj − xjyn+j), x1 + y1, . . . , x2n + y2n) (3.1.4)
A left-invariant basis for its Lie algebra h2n+1 is then given by the vector fields
Y0 =∂
∂x0
, (3.1.5)
Yj =∂
∂xj+ xn+j
∂
∂x0
, (3.1.6)
Yn+j =∂
∂xn+j
+ xj∂
∂x0
(3.1.7)
for 1 ≤ j ≤ n. This left-invariant basis for h2n+1 satisfies the relations,
[Yj, Yn+k] = −2δjkY0, [Y0, Yj] = [Yj, Yk] = [Yn+j, Yn+k] = 0 (3.1.8)
for 1 ≤ j, k ≤ n and j 6= k. In particular, the subbundle spanned by the vector
fields Y1, Y2, . . . , Y2n, yields a left-invariant Heisenberg structure on H(2n+1).
• Foliations: Recall that a (smooth) foliation is a manifold X together with a sub-
bundle F ⊂ TX which is integrable in the sense of Frobenius, so that [F ,F ] ⊂ F .
Thus, any codimension one foliation is a Heisenberg manifold.
Chapter 3. Heisenberg Calculus on Contact Manifolds 45
• Contact Manifolds: Opposite to foliations are contact manifolds, (X(2n+1), H)
where H = kerκ for some one-form κ ∈ Ω1(X). In fact, by Darboux’s theorem
any contact manifold is locally contactomorphic to the Heisenberg group H(2n+1)
equipped with its standard contact form κ0 = dx0 +∑n
j=1(xjdxn+j − xn+jdxj).
• CR Manifolds: Recall that a CR-structure on an orientable manifold X2n+1 is
given by a rank n complex subbundle T1,0 ⊂ TCX which is integrable in Frobenius’
sense and such that T1,0 ∩ T0,1 = 0, where T0,1 = T 1,0. Equivalently, the subbundle
R(T1,0⊗T0,1) (where R denotes the real part) has the structure of a complex bundle
of (real) dimension 2n. In particular, (X,H) is a Heisenberg manifold.
An important object of study for us will be the tangent Lie group bundle of a Heisenberg
manifold (X,H). Let
L : H ×H → TX/H,
be defined by
Lx(Yx, Yx) = [Y, Y ](x) modHx,
for sections Y, Y ∈ Γ(H) near a point x ∈ X. L is well defined by [Pon08, Lemma 2.1.3]
and is called the Levi form of (X,H). The Levi form allows us to define the bundle gX
of graded Lie algebras by endowing (TX/H) ⊕ H with a Lie bracket and grading. Let
Y0, Y0 ∈ Γ(TX/H), Y1, Y1 ∈ Γ(H) and t ∈ R. Then
[Y0 + Y1, Y0 + Y1]x = Lx(Y1, Y1), t · (Y0 + Y1) = t2Y0 + tY1.
One can check that gX is a bundle of two-step nilpotent Lie algebras1 which contains
the normal bundle TX/H in its center. The associated Lie group bundle GX is also
(TX/H)⊕H with the identity as exponential map. Since gX is two-step nilpotent, the
Campbell-Baker-Hausdorff formula implies that the group law is given by
exp(Y )exp(Y ) = exp(Y + Y +1
2[Y, Y ]).
1Recall that a Lie algebra L is said to be two-step nilpotent if its commutator ideal [L,L] is nontrivialand contained in the center of L.
Chapter 3. Heisenberg Calculus on Contact Manifolds 46
For sections Y0, Y0 ∈ Γ(TX/H), Y1, Y1 ∈ Γ(H), we then have explicitly the group law
(Y0 + Y1) · (Y0 + Y1) = Y0 + Y1 + Y0 + Y1 +1
2L(Y1, Y1).
3.1.9 Definition. The bundles gX and GX are called the tangent Lie group bundle and
the tangent Lie group of X, respectively. We let gx := gX|x and Gx := GX|x denote the
fibres of these bundles with their respective structures.
3.1.10 Remark. It is a basic fact that on a contact manifold the point-wise groups Gx
are isomorphic to the Heisenberg group.2
Our main interest is the study of the contact Laplacian of [RS08]. It is interesting,
however, to note some examples of other differential operators on Heisenberg manifolds.
3.1.11 Example. • Hormander’s sum of squares on a Heisenberg manifold (M,H) of
the form
∆ := ∇∗Y1∇Y1 + · · ·+∇∗Ym∇Ym , (3.1.12)
where the (real) vectors fields Y1, . . . , Ym span H and ∇ is a connection on a vector
bundle E over X and the adjoint is taken with respect to a smooth positive measure
on X and a Hermitian metric on E .
• The Kohn Laplacian 2b;p,q acting on (p, q)-forms on a CR manifold X2n+1 endowed
with a CR compatible Hermitian metric (not necessarily a Levi metric).
• The horizontal sublaplacian ∆b;k acting on horizontal differential forms of degree
k on a Heisenberg manifold (X,H). When X2n+1 is a CR manifold the horizontal
sublaplacian preserves the bidegree and we can consider its restriction ∆b;p,q to
forms of bidegree (p, q).
• The contact Laplacian on a contact manifold X2n+1 associated to the contact com-
plex of [RS08]. There are actually two different working definitions of this Lapla-
cian. One definition can be found in [Rum94], where the contact Laplacian is
2See [Erp10].
Chapter 3. Heisenberg Calculus on Contact Manifolds 47
defined as a differential operator of order two in degree k 6= n, n + 1 and of order
four in degree n, n+ 1. Recall, [Rum94, pg. 290 Theorem],
∆q =
2d∗HdH if q = 0,
−dHd∗H if q = 3,
D∗D + (dHd∗H)2 if q = 1.
DD∗ + (d∗HdH)2 if q = 2.
(3.1.13)
The definition that we will use is as defined in [RS08] where the contact Laplacian
is a uniformly fourth order operator in all degrees. Recall, [RS08, Eq. 10],
∆q =
(d∗HdH + dHd
∗H)2 if q = 0, 3,
D∗D + (dHd∗H)2 if q = 1.
DD∗ + (d∗HdH)2 if q = 2.
(3.1.14)
We will need the following
3.1.15 Definition. [HN86] Let X be a compact manifold. A differential operator P =∑|α|≤d aα(x)Y α of order d on X is called maximal hypoelliptic at x ∈ X if there exists a
neighbourhood U of x and a constant C ∈ R>0 such that for every distribution u on U ,
∑|α|≤d
||Y αu||L2 ≤ C(||u||L2 + ||Pu||L2).
Recall that maximal hypoellipticity of P implies that P is hypoelliptic, i.e.
Pu smooth⇒ u smooth.
It turns out that the operators above are known to be hypoelliptic in certain cases. For
example, Hormanders sum of squares is hypoelliptic provided that the following bracket
condition is satisfied3:
[Yj1 , [Yj2 , . . . , Yj1 ] . . .],
3See [Hor67].
Chapter 3. Heisenberg Calculus on Contact Manifolds 48
spans the tangent bundle TX at every point. Kohn, [Koh65], proved that under the
“Y (q)” condition that the Kohn Laplacian 2b;p,q is hypoelliptic. The most important
example for us is the contact Laplacian of Eq. (3.1.14), which Rumin showed in [Rum94]
is maximal hypoelliptic.
3.2 Heisenberg Calculus and the Rockland Condi-
tion
In this section we provide a general overview of the Heisenberg calculus. This theory
was independently introduced by Beals-Greiner [BG88] and Taylor [Tay84]. Our aim is
to develop the basic definitions, terminology, results and ideas that we will need for this
thesis. We primarily follow [BG88], [Pon07], [Pon08], [Erp10] and we refer the reader to
[Pon08] for a much more in depth discussion of the details in the more general case of
Heisenberg manifolds. We will work with differential operators P : C∞(X) → C∞(X)
acting on functions, and defer to [Pon08] the general case of psedodifferential operators,
ΨHDO’s. Also note that we are ultimately interested in contact manifolds (X, κ) in this
thesis and throughout this section the notation gx and Gx should be thought of as the
Heisenberg Lie algebra and Heisenberg group, respectively (See Remark 3.1.10).
The main idea of subelliptic theory on contact manifolds is that vector fields transversal
to the contact hyperplane bundle H are treated as second order operators. The Heisen-
berg calculus is derived quite naturally from this basic idea. This ordering of vector
fields generates a filtration on the space of differential operators P on X. One then as-
sociates a graded algebra D to the filtered algebra P and defines the principal symbol of
an operator P ∈ P as the image of its highest order part in D. The first main deviation
from elliptic theory is the identification of the principal part of the operator P at a point
x ∈ X as an element in a noncommutative algebra Ux. U := Ux |x ∈ X is a bundle
Chapter 3. Heisenberg Calculus on Contact Manifolds 49
of graded algebras over X, and Ux is realized as the universal enveloping algebra of the
graded nilpotent Lie algebra gx (See Def. 3.1.9). This leads to the idea of replacing the
notion of constant coefficient operators in elliptic theory with Rockland operators and is
why the invertibility of an operator in the Heisenberg calculus involves the representation
theoretic Rockland condition (See Def. 3.2.1). This, for us, summarizes the basic ideas
of the Heisenberg calculus.
First, let us consider the space of differential operators
P = span∏j
Yj |Yj ∈ Γ(TX).
It is natural to define a filtration on P by defining ord(Y ) = 1 for any Y ∈ Γ(H) ⊂ Γ(TX)
(where H ⊂ TX denotes the contact distribution), and ord(Y ) = 2 for any Y /∈ Γ(H).
Then we define,
Pd := span∏j
Yj |∑j
ord(Yj) ≤ d ⊂ P ,
where P0 := C∞(X). We define the (Volterra-Heisenberg) order of an operator P ∈ P
in the Heisenberg calculus to be the integer d such that P ∈ Pd\Pd−1. Recall that the
algebra of symbols for the Heisenberg calculus is the graded algebra D associated to the
filtered algebra P defined in the usual way
D :=⊕Dd, where Dd := Pd/Pd−1.
The principal part of a degree d operator P : C∞(X)→ C∞(X) is the image of P under
the degree d quotient mapping
σdH : Pd → Dd.
As noted above, the principal symbol of an operator P at x ∈ X, σdH(P )(x), is viewed
as an element of an algebra Ux. The algebra Ux is generated by operators in P and
consists of the linear span of their principal symbols at x ∈ X, σdH(P )(x). Thus, Ux =
spanσdH(P )(x) ∈ Ddx | d ∈ N, P ∈ P with the following graded algebra structure. Let
Chapter 3. Heisenberg Calculus on Contact Manifolds 50
Ax, Bx ∈ Ux be monomials, so that Ax = σdH(P )(x) and Bx = σqH(Q)(x) for some
P,Q ∈ P . Then we define Ax ·Bx ∈ D(d+q)x ⊂ Ux by:
Ax ·Bx := σ(d+q)H (PQ)(x).
It is easy to verify that this is a well defined graded algebra structure on Ux.4 Now ob-
serve that we may naturally view gx ⊂ Ux (See Def. 3.1.9). The identification comes from
observing that the vector fields Γ(TX) ⊂ P are a subset of P and gx = (TX/H)x ⊕Hx
with the Lie algebra structure defined in §3.1. Since the vector fields Γ(TX) generate P ,
gx generates Ux as an algebra. By the Poincare-Birkoff-Witt theorem one can also see
that Ux is naturally isomorphic to the universal enveloping algebra of gx.
The last thing we review in this section is the Rockland condition for differential op-
erators P ∈ P . Our main reason for reviewing this here is that the Rockland condition is
the crucial condition that the contact Laplacian ∆q of Eq. (3.1.14) must satisfy in order
to carry out our analysis in Chapter 6. First we recall that a unitary representation π of
a Lie group G on a Hilbert space Hπ induces a representation dπ of the Lie algebra g on
the space of smooth vectors C∞π ,5 defined by:
dπ(Y )ν =d
dt
∣∣∣t=0π(exp(tY ))ν,
for Y ∈ g and ν ∈ C∞π . We may now make the following, originally due to Rockland
[Roc78]:
3.2.1 Definition. [Pon08, Def. 3.3.8] We say that an operator P ∈ P satisfies the
Rockland condition at x ∈ X if for a nontrivial unitary irreducible representation π of
GxX the operator dπ(P ) is injective on C∞π .
4See [Erp10].5Recall that the space of smooth vectors C∞π ⊂ Hπ is the set of vectors ν ∈ Hπ such that g 7→ π(g)ν
is smooth from G to Hπ.
Chapter 3. Heisenberg Calculus on Contact Manifolds 51
3.3 Some Results in the Heisenberg Calculus
In this section we list several key results that are needed throughout this thesis. These
results will be applied to the hypoelliptic contact Laplacian, ∆q, defined in Eq. (3.1.14).
In order to be as self contained as possible we have developed most of the requisite
notation and terminology for this section in §3.1 and §3.2. Our first result is the following
3.3.1 Proposition. [Pon08, Chapter 5] Let V be a vector bundle over a compact contact
manifold (X, κ) of dimension 2n + 1. Let P : C∞(X,V) → C∞(X,V) be a differential
operator of even Volterra-Heisenberg order v that is self-adjoint and bounded from below.
If P satisfies the Rockland condition at every point then the principal symbol of P + ∂t is
an invertible Heisenberg symbol and as t 0 the heat kernel kt(x, x) of P on the diagonal
has the following asymptotics in C∞(X, (EndV)⊗ |Λ|(X)):
kt(x, x) ∼∞∑j=0
t2(j−n−1)
v aj(P )(x).
The next proposition is crucial to our construction in Chapter 6 and allows us to
define regularized determinants rigorously using zeta functions.
3.3.2 Proposition. [Pon07, §4] Let P be as in Prop. 3.3.1. Then the zeta function
ζ(P )(s) = dim kerP + Tr∗(P−s), s ∈ C,
is a well defined holomorphic function for Re(s) 1 and admits a meromorphic extension
to C with at worst simple poles occurring at s ∈ 2(n+1−j)v
| j ∈ N\(−N). Moreover
ζ(P )(0) =
∫X
tr(an+1(P ))κ ∧ (dκ)n
is the constant term in the development of Tr(e−tP ) as t 0.
Our main observation is that Prop.’s 3.3.1 and 3.3.2 apply to the contact Laplacian
∆q. That is, ∆q satifies the Rockland condition. As is noted in [RS08], this is observed
Chapter 3. Heisenberg Calculus on Contact Manifolds 52
in [Rum94, pg. 300] and also shown in [JK95, §5]. Let
ζ0(s) := dim Ker∆0 + ζ0(s)
ζ1(s) := dim Ker∆1 + ζ1(s)
denote the zeta functions as defined in [RS08, §3]. For the next proposition we will need
to make the following
3.3.3 Definition. Let (X, κ) be a contact three-manifold and ∆q the contact Laplacian
(See Eq. (3.1.14)).6 Define the contact torsion function KX : C→ C as follows:
KX(s) :=1∑q=0
(−1)q(2− q)ζq(s)
= 2ζ0(s)− ζ1(s).
We then have the following
3.3.4 Proposition. [RS08, Cor. 3.8 (1)] Let (X, κ) be a three-dimensional contact
manifold and let KX denote the contact torsion function as in Def. 3.3.3. Then KX(0) =
0.
3.3.5 Remark. It is still not known whether or not KX(0) vanishes in all dimensions,
[RS08, pg. 17].
The next theorem is important in our definition of a regularized determinant of the
operator −k ? D : Ω1(H)→ Ω1(H) as in Eq. (6.7.3).
3.3.6 Theorem. [BHR07, Theorem 8.8] Let X be a CR-Seifert manifold,7 ∆H :=
dHd∗H + d∗HdH the horizontal Laplacian, R ∈ C∞(X) the Tanaka-Webster scalar cur-
vature of X and η0(X, κ)8 the renormalized eta invariant. Then,
η0(X, κ) = η(?D)(0) + ζ(∆H)(0)
6Note that this definition makes sense for all contact manifolds, but we will not need this definitionhere.
7See Def. 6.2.1. Equivalently, see §2.3 for a definition of a quasi-regular K-contact manifold; suchstructures are equivalent to CR-Seifert structures.
8See Def. 6.10.7.
Chapter 3. Heisenberg Calculus on Contact Manifolds 53
with
ζ(∆H)(0) =1
512
∫X
R2 κ ∧ dκ.
Finally, the last result that we need is an explicit identification of the spaces of
harmonic and contact harmonic forms.
3.3.7 Proposition. [Rum94, Prop. 12] Let (E , dH) be the complex defined in Def. 6.4.4,
∆q, 0 ≤ q ≤ 3 denote the contact Laplacian and ∆DRq = d∗d + dd∗ denote the de Rham
Laplacian. If
Hq(E , dH) := α ∈ Eq |∆qα = 0,
and,
Hq(X, d) := α ∈ Ωq(X) |∆DRq α = 0,
denote the contact and de Rham harmonic forms, respectively, then for q ≤ 1,
Hq(E , dH) = Hq(X, d).
Chapter 4
U(1) Chern-Simons Theory
4.1 Induced Principal Bundles and Induced Connec-
tions
In this section we will show how to construct a canonical SU(2)-bundle P over a manifold
X given a U(1)-bundle P over X. This will allow us to define the Chern-Simons action in
the U(1) case, even when our principal U(1)-bundle P is non-trivial. This construction
is crucial to our analysis and is due to [Man98], which we follow here.
We start by viewing U(1) as an embedded maximal torus in SU(2) via a specific inclusion
homomorphism,
ρ : U(1) → SU(2) (4.1.1)
e2πiϕ 7→
e2πiϕ 0
0 e−2πiϕ
(4.1.2)
The induced map on the Lie algebra is,
ρ∗ : u(1) → su(2) (4.1.3)
α 7→
α 0
0 −α
(4.1.4)
54
Chapter 4. U(1) Chern-Simons Theory 55
where u(1) := Lie(U(1)) ' R, and su(2) := Lie(SU(2)). Let ϑ denote the Maurer-Cartan
form on SU(2). Choose a non-degenerate, Ad-invariant bilinear form on su(2),1
Tr : su(2)× su(2) → R (4.1.5)
(a, b) 7→ 1
8πtr(ab) (4.1.6)
The bilinear form Tr is normalized so that the closed three-form 16Tr(ϑ∧[ϑ, ϑ]) represents
an integral cohomology class in H3(SU(2),R). The form Tr on su(2) then restricts to a
symmetric bilinear form on u(1),
〈·, ·〉 : u(1)× u(1) → R (4.1.7)
(α, β) 7→ 〈α, β〉 = Tr(ρ∗α, ρ∗β) =1
4παβ (4.1.8)
As before, let ϑ denote the Maurer-Cartan form on U(1) and observe that ρ∗(ϑ) = ρ∗(ϑ).
Given a principal U(1)-bundle over X we may now use the inclusion map ρ : U(1) →
SU(2) to construct an SU(2)-bundle P on X. We define the induced SU(2)-bundle P
as follows, [Man98, pg. 14]:
Define a right action of U(1) on P × SU(2) by
(p, g) · λ = (p · λ, (ρ(λ))−1g) (4.1.9)
where λ ∈ U(1) and (p, g) ∈ P × SU(2). The natural right SU(2) action on P × SU(2)
given by (p, g) · h = (p, gh), commutes with the U(1) action and therefore passes to a
free action on the quotient P = P ×U(1) SU(2). Thus,
π : P → X (4.1.10)
with projection map π([p, g]) = π(p) is a principal SU(2)-bundle. Define
ιP : P → P (4.1.11)
p 7→ [p, e] (4.1.12)
1This is standard, and follows from the fact that su(2) is a simple Lie algebra.
Chapter 4. U(1) Chern-Simons Theory 56
which is a natural morphism of bundles covering the identity map on X.
Given a connection A ∈ AP on the U(1)-bundle P , there is an induced connection A ∈ AP
on the SU(2)-bundle P . Define, pr1 : P × SU(2) → P and pr2 : P × SU(2) → SU(2)
as the natural projections. A is obtained from A′, which is the following su(2)-valued
one-form on P × SU(2), [Man98, Eq. 3.1],
A′(p,g) := Adg−1(ρ∗pr∗1Ap) + pr∗2(ϑg), (4.1.13)
by pushing A′ down to an su(2)-valued one-form on P = P ×U(1) SU(2). It is not hard
to see that A′ is invariant under the U(1)-action on P × SU(2). We have the following
relationship between A and A [Man98, pg. 14],
ι∗P A = ρ∗A. (4.1.14)
We also have the following relationship between the curvature forms FA = dA and FA =
dA+ 12[A, A], [Man98, pg. 14],
ι∗PFA = ρ∗FA. (4.1.15)
In the next section we will see how to define the Chern-Simons action naturally using
this construction.
4.2 The U(1) Chern-Simons Action
Let X be a closed oriented 3-manifold. Given a principal U(1)-bundle P over X, we
have previously constructed an induced SU(2)-bundle P = P ×U(1)SU(2)→ X as in Eq.
(4.1.10). Also, for any U(1)-connection A ∈ AP we defined in Eq. (4.1.13) an induced
SU(2)-connection A on P . Since for any 3-manifold X, P is trivializable, let s : X → P
be a global section. In this section we define the U(1) Chern-Simons action and study
some of its properties. We primarily follow [Man98] here.
Chapter 4. U(1) Chern-Simons Theory 57
4.2.1 Definition. [Man98, Eq. 3.2] The Chern-Simons action functional of a U(1)-
connection A ∈ AP is defined by:
SX,P (A) =
∫X
s∗α(A) (mod Z) (4.2.2)
where α(A) ∈ Ω3(P ,R) is the Chern-Simons form of the induced SU(2)-connection
A ∈ AP ,
α(A) = Tr(A ∧ FA)− 1
6Tr(A ∧ [A, A]) (4.2.3)
We claim that the definition of the action SX,P does not depend on the choice of section
s : X → P . To see this, we first establish the following property of the SU(2) Chern-
Simons form α(A):
4.2.4 Proposition. [Man98, Prop. 3.3] If s1, s2 : X → P are two global sections of the
SU(2)-bundle P over X, then∫X
s1∗α(A) =
∫X
s2∗α(A) +
∫X
dTr(Ada−1 s2∗A ∧ a∗ϑ)− 1
6
∫X
a∗Tr(ϑ ∧ [ϑ, ϑ]), (4.2.5)
where a : X → SU(2) is the map defined by s2(x) = s1 · a(x), for any x ∈ X.
One can prove this after straightforward manipulations using the standard relation
s∗2A = Ada−1(s1A) + a∗ϑ. Using this result we can prove:
4.2.6 Proposition. [Man98] The definition of the action SX,P does not depend on the
choice of section s : X → P .
Proof. Looking at Eq. (4.2.5), one can see in the case ∂X = ∅ that the second integral
vanishes by Stokes’ theorem because the form is exact. The last integral in Eq. (4.2.5) is
an integer due to the normalization of the bilinear form Tr on su(2). Thus, for any two
sections s1, s2 : X → P , ∫X
s∗1α(A) =
∫X
s∗2α(A) (mod Z) (4.2.7)
and the Chern-Simons action SX,P is well defined.
Chapter 4. U(1) Chern-Simons Theory 58
4.2.8 Remark. [Man98, Remark 3.4] In the case that the U(1)-bundle P over X is trivi-
alizable, we have the expected result:
SX,P (A) =1
4π2
∫X
s∗(A ∧ FA) (mod Z) (4.2.9)
=1
4π2
∫X
s∗(A ∧ dA) (mod Z) (4.2.10)
where s : X → P is a section of the U(1)-bundle P and s : X → P is taken to be
s = ιP s. ιP was defined in Eq. (4.1.11). One can see this with a straightforward
calculation,
s∗α(A) = s∗ι∗Pα(A)
= s∗Tr(ι∗P A ∧ ι∗PFA)− 1
6s∗Tr(ι∗P A ∧ [ι∗P A, ι
∗P A])
= s∗Tr(ρ∗A ∧ ρ∗FA), by eq.’s (4.1.14) and (4.1.15)
=1
4π2s∗(A ∧ FA), by Eq. (4.1.8).
We now collect several properties of the Chern-Simons action SX,P .
4.2.11 Theorem. [Man98, Theorem 3.6] The Chern-Simons functional SX,P : AP →
R/Z defined for a closed oriented 3-manifold X and a principal U(1)-bundle P over X
has the following properties:
1. Functorality
If φ : P1 → P2 is a morphism of principal U(1)-bundles covering an orientation
preserving diffeomorphism φ : X1 → X2 and if A ∈ AP is a connection on P , then
SX1,P1(φ∗A) = SX2,P2(A) (4.2.12)
2. Orientation
If −X denotes the manifold X with the opposite orientation, then
S−X,P (A) = −SX,P (A) (4.2.13)
Chapter 4. U(1) Chern-Simons Theory 59
3. Disjoint Union
Let X = X1tX2 be a disjoint union and P = P1tP2 a principal U(1)-bundle over
X. If Ai ∈ APi , i = 1, 2 are connections, then
SX1tX2,P1tP2(A1 t A2) = SX1,P1(A1) + SX2,P2(A2) (4.2.14)
Proof. 1. Let P1 and P2 be the induced principal SU(2)-bundles coming from P1 and
P2 respectively, as in Eq. (4.1.10). The morphism of U(1) bundles φ : P1 → P2
induces a morphism of SU(2)-bundles φ : P1 → P2 covering φ : X1 → X2. It is
defined by φ([p1, a]) = [φ(p1), a]. Given a section s1 : X1 → P1 we have a section
s2 = φ s1 φ−1 : X2 → P2. We then have
SX2,P2(A) =
∫X2
s∗2α(A) (mod Z) (4.2.15)
=
∫X2
(φ−1)∗s∗1φ∗α(A) (mod Z) (4.2.16)
=
∫X1
s∗1α(φ∗A) (mod Z) (4.2.17)
= SX1,P1(φ∗A), by definition 4.2.1 (4.2.18)
The last equality follows from the fact that φ∗A is the induced SU(2)-connection
on P2 coming from the U(1)-connection φ∗A on P2:
φ∗A(p,g) = Adg−1ρ∗(φ∗pr∗1Ap) + φ∗pr∗2ϑg (4.2.19)
= Adg−1ρ∗(pr∗1φ∗Ap) + pr∗2ϑg (4.2.20)
2. This follows from the fact that integration over a manifold with a chosen orientation
is equivalent to minus the integral over the opposite orientation.
3. This is also an easy consequence of a property of integration. That is, an integral
over a disjoint union is the sum of the integrals over each component in the disjoint
union. In our case, A = A1 t A2 is the extension of A = A1 t A2 on P = P1 t P2
Chapter 4. U(1) Chern-Simons Theory 60
to an SU(2)-connection on P = P1 t P2. Let si : Xi → Pi and s = s1 t s2 : X =
X1 tX2 → P . Then,
SX,P (A) =
∫X1tX2
s∗α(A) (mod Z) (4.2.21)
=
∫X1
s∗1α(A1) +
∫X2
s∗2α(A2) (mod Z) (4.2.22)
= SX1,P1(A1) + SX2,P2(A2) (4.2.23)
As a consequence of part 1) of Theorem 4.2.11, we can see that the Chern-Simons action
is invariant under the action of the gauge group GP on AP ; where the action is defined
in Proposition 2.5.18. We reiterate this in the following
4.2.24 Proposition. [Man98, Prop 3.7] The Chern-Simons functional SX,P for a closed
three-manifold is invariant under the action of the group of gauge transformations,
SX,P (A · h) = SX,P (A) (4.2.25)
for all h ∈ GX . Hence SX,P is a well defined functional on the quotient space AP/GP .
We will need the following
4.2.26 Proposition. [Man98, Lemma 3.18] Let X be a closed, oriented three-manifold
and P a U(1)-bundle over X. If A1 and A2 are connections on P , then
SX,P (A2)− SX,P (A1) =1
2π2
∫X
s∗(FA1 ∧ ω) +1
4π2
∫X
s∗(ω ∧ dω) (mod Z) (4.2.27)
where ω = A2 − A1.
Proof. Let A1 and A2 be the induced connections on P as usual. It follows from the
definition of the induced connection that the su(2)-valued 1-form ω = A2 − A1 on P is
related to ω by
ω[p,g] = Adg−1(ρ∗pr∗1ωp) (4.2.28)
Chapter 4. U(1) Chern-Simons Theory 61
and ι∗P ω = ρ∗ω. It is straightforward to derive the following relation:
α(A2)−α(A1) = −dTr(A1∧ ω)+2Tr(FA1∧ ω)+Tr(ω∧ (dω+[A1, ω]))+
1
3Tr(ω∧ [ω, ω])
(4.2.29)
It is not hard to see from relation 4.2.28 that
Tr(ω ∧ [ω, ω]) = 0 (4.2.30)
and,
Tr(ω ∧ [A1, ω]) = Tr([ω, ω] ∧ A1) = 0 (4.2.31)
where Eq. (4.2.31) follows from the associativity of the Killing form. Recall that since
su(2) is simple, the inner product Tr is a non-zero scalar multiple of the Killing form.
We then have,
α(A2)− α(A1) = −dTr(A1 ∧ ω) + 2Tr(FA1∧ ω) + Tr(ω ∧ dω) (4.2.32)
Thus, we may write
SX,P (A2)− SX,P (A1) =
∫X
s∗(α(A2)− α(A1))
=
∫X
s∗[−dTr(A1 ∧ ω) + 2Tr(FA1
∧ ω) + Tr(ω ∧ dω)]
=
∫X
−s∗dTr(A1 ∧ ω) +
∫X
2s∗Tr(FA1∧ ω) (4.2.33)
+
∫X
s∗Tr(ω ∧ dω)
=
∫X
2s∗Tr(FA1∧ ω) +
∫X
s∗Tr(ω ∧ dω)
=1
2π2
∫X
s∗(FA1 ∧ ω) +1
4π2
∫X
s∗(ω ∧ dω) (mod Z)
where the second last equality comes from Stokes’ theorem and the fact that the first
quantity in Eq. (4.2.33) is exact. The last equation follows from the fact that Tr is
Ad-invariant, s := pr1 s : X → P defines a global section of P and by the definition of
the symmetric bilinear form 〈·, ·〉 on u(1) defined in Eq. (4.1.8).
Chapter 4. U(1) Chern-Simons Theory 62
We further have the following
4.2.34 Proposition. [Man98, Prop. 3.8] The stationary points of the Chern-Simons
functional SX,P : AP → R/Z are the flat connections. In other words, δSX,P (A) = 0 if
and only if FA = dA = 0.
Proof. We consider the variation of SX,P along lines At = A + tω, for t ∈ R and ω ∈
Ω1(X,R). This is sufficient because AP is an affine space. Observe, by Proposition 4.2.26,
we have
SX,P (At) = SX,P (A) +t
2π2
∫X
s∗(FA1 ∧ ω) +t2
4π2
∫X
s∗(ω ∧ dω) (mod Z) (4.2.35)
and we may calculate the variation as
δSX,P (A) =d
dt
∣∣∣t=0SX,P (At) =
1
2π2
∫X
s∗(FA ∧ ω), (4.2.36)
which proves the assertion.
4.3 The U(1) Partition Function
In this section we recall the definition of the partition function, ZU(1)(X, k), for U(1)
Chern-Simons theory. This partition function is defined and studied in both [MPR93]
and [Man98], for example. We follow §7 of [Man98] and “derive”2 a rigorous definition
of the partition function from a heuristically defined path integral (see eq.’s (4.3.2) and
(4.3.3) below). We modify the rigorous definition of the U(1) Chern-Simons partition
function [Man98, Eq. 7.28] to take into account the dependence that the partition func-
tion has on a choice of metric g on X. We follow [Wit89] and revise the definition
of [Man98] by adding a “counterterm,” the gravitational Chern-Simons term (see Eq.
2Note that it would be interesting to make an explicit study of the choices that one needs to makein order to rigorously define the partition function using the methods of this thesis. For example, ourregularization of the eta-invariant is seemingly not unique. Of course, a rigorous definition of the pathintegral measure would probably answer this question.
Chapter 4. U(1) Chern-Simons Theory 63
(4.3.14)), to the eta-invariant (see Eq. (4.3.13)) that shows up in this calculation. By the
Atiyah-Patodi-Singer theorem [APS75a], [APS75b], [APS76], this counterterm effectively
restores topological invariance for the partition function. We note that the existence of
such a counterterm is expected from a physics perspective. That is, the U(1) Chern-
Simons partition function should be a topological invariant a priori, since this theory is
“generally covariant.”3 In order to actually compute the partition function one needs to
make a gauge choice, which in our case is tantamount to a choice of metric g on X. In
physics terminology, this choice introduces a quantum anomaly that is made manifest in
the dependence of the eta-invariant (Eq. (4.3.13)) on the choice of metric g. Introducing
the counterterm (Eq. (4.3.14)) effectively cancels this anomaly.
Before we define the partition function, we recall some notation and terminology. For
each p ∈ TorsH2(X; Z),4 we choose a U(1)-bundle P on X with c1(P ) = p. AP 5 is
the affine space of connections on P modeled on the vector space Ω1(X; R). GP is the
group of gauge transformations and acts on AP in the standard way.6 SX,P (A) is the
Chern-Simons functional of a U(1)-connection A on P → X and is defined in Eq. (4.2.1).
We then heuristically define the partition function as follows.
4.3.1 Definition. [Man98, Eq.’s 7.13, 7.14] Let k ∈ Z be an (even) integer, and X a
closed, oriented three-manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k),
3See the introduction to [Wit89] for a brief discussion of this.4Recall that an element in a Z-module is said to be torsion if some integral multiple of it is zero.
TorsH2(X; Z) denotes the torsion subgroup of H2(X; Z). We take the sum over only torsion classes inEq. (4.3.2) because these are the bundle classes that contain flat connections, by Prop. 2.5.47. We areinterested in summing over bundle classes with flat connections because we expect that these are theonly classes that will contribute non-trivially to the partition function. This is because the integrandof Eq. (4.3.3) is oscillatory, and we expect to get contributions only from the critical points (i.e. flatconnections) of the action via stationary phase.
5See Eq. (2.5.10) and §2.5 for more background.6See Prop. 2.5.18 and §2.5 for a general discussion.
Chapter 4. U(1) Chern-Simons Theory 64
is the heuristic quantity
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k) (4.3.2)
where,
ZU(1)(X, p, k) =1
Vol(GP )
∫APDAeπikSX,P (A). (4.3.3)
Note that Eq. (4.3.3) is a formal expression, where we heuristically assume the
existence of the measure DA. It is precisely the quantity DAVol(GP )
in Eq. (4.3.3) that is
not well defined. One goal of future work is to find a rigorous definition of this quantity.
For now we follow a different approach to make definition 4.3.1 rigorous. What follows
is mainly a summary of §7 of [Man98], and one should refer to this for more details. To
start, we write
ZU(1)(X, p, k) =1
Vol(GP )
∫APDAeπikSX,P (A) (4.3.4)
=1
VolU(1)
∫AP /GP
DAeπikSX,P (A)[det′(τ ∗AτA)]1/2 (4.3.5)
where the measure DA is formally induced from a choice of metric g on X. That is, g
defines the Hodge-? on the tangent space TAAP ' Ω1(X; R), which in turn induces the
(infinite dimensional) GP -invariant Riemannian metric
〈α, β〉 :=
∫X
α ∧ ?β, (4.3.6)
on AP , for α, β ∈ TAAP ' Ω1(X; R). Clearly, this metric is invariant under the action
of the gauge group GP on AP . Thus, this metric descends to a metric on the quotient
AP/GP , and thereby induces a quotient measure that we denote by DA in Eq. (4.3.5).
The factor of VolU(1) in Eq. (4.3.5) is identified as the volume of the isotropy subgroup
of GP at any A ∈ AP and is given by, [Man98, Eq. 7.16]:
VolU(1) = [VolX]1/2 =
[∫X
?1
]1/2
. (4.3.7)
Chapter 4. U(1) Chern-Simons Theory 65
Eq. (4.3.7) follows from the definition of the invariant metric on the group GP that is
induced by the inner product on LieGP ' Ω0(X; R) that comes from g:
G(θ, φ) = 〈θ, φ〉 :=
∫X
θ ∧ ?φ, (4.3.8)
for θ, φ ∈ LieGP ' Ω0(X; R). The isotropy subgroup of GP at A ∈ AP is the group of
constant maps from X into U(1) since
θ ∈ LieGP : A 7→ A+ dθ; (4.3.9)
I.e. dθ = 0⇒ θ = constant. Observe that G restricted to the space of constant functions
is simply a scalar at each Ψ ∈ GP ' Maps(X,U(1)):
GΨ(θ, φ) =
∫X
θ ∧ ?φ
= θφ
(∫X
?1
),
since θ, φ ∈ R are constant. We may therefore write GΨ =∫X?1. If
√GΨDΨ denotes
the measure on U(1) < GP , then
Vol(U(1)) =
∫U(1)
√GΨDΨ
=√GΨ, setting
∫U(1)
DΨ = 1,
=
[∫X
?1
]1/2
.
This proves Eq. (4.3.7). τA denotes the differential of the map from GP to AP that
defines the GP -action. The map τA sends LieGP ' Ω0(X; R) to TAAP ' Ω1(X; R), and
can be identified with the differential τA = d : Ω0(X; R)→ Ω1(X; R).
Observe that SX,P is basically a quadratic functional, i.e.
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4π
(∫X
A ∧ dA)]
, (4.3.10)
=eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4π〈A, ?dA〉
], (4.3.11)
(4.3.12)
Chapter 4. U(1) Chern-Simons Theory 66
where we use Prop. 4.2.26 to rewrite this partition function after identifying AP =
AP + Ω1(X) for a flat base point AP in AP (so that FAP = 0). The inner product 〈·, ·〉
is as defined in Eq. (4.3.6) with respect to a choice of Riemannian metric g on X. Thus,
we use the stationary phase method7 to obtain an exact result, [Man98, Eq. 7.17′′]:
ZU(1)(X, p, k) =eπikSX,P (AP )
VolU(1)
∫MP
eπi4
sgn(?d) [det′(d∗d)]1/2
[det′(k ? d)]1/2ν,
where ν is the measure induced on the moduli space of flat connections on P , MP (see
Eq. (2.5.57)), and AP ∈ AP is a flat connection on P . This last expression has rigorous
mathematical meaning if the determinants and signatures of the operators therein are
regularized. The signature of the operator ?d on Ω1(X; R) is regularized via the eta-
invariant, so that sgn(?d) = η(?d) + 13
CS(Ag)2π
, where
η(?d) = lims→0
∑λj 6=0
signλj|λj|−s, (4.3.13)
and λj are the eigenvalues of ?d, and
CS(Ag) =1
4π
∫X
Tr(Ag ∧ dAg +2
3Ag ∧ Ag ∧ Ag) (4.3.14)
is the gravitational Chern-Simons term, with Ag the Levi-Civita connection on the spin
bundle of X. The notation det′ refers to regularized determinants and are regularized as
in Remark 7.6 of [Man98].
It is shown in [Man98] that the term inside of the integral
1
VolU(1)
[det′(d∗d)]1/2
[det′(k ? d)]1/2
may be identified, up to a factor depending on k, with the Ray-Singer analytic torsion
of the three-manifold X, T dRS [Man98, Eq. 7.25]. Recall that for a closed three-manifold
X with a given Riemannian metric g, the Ray-Singer analytic torsion can be viewed as
a density on X. Recall the Hodge-de Rham Laplacian:
∆q := d∗d+ dd∗, on Ωq(X; R). (4.3.15)
7See Appendix A.1 for a review of the method of stationary phase.
Chapter 4. U(1) Chern-Simons Theory 67
The analytic Ray-Singer torsion, [RS73],
TRS := exp
(1
2
3∑q=0
(−1)qqζ ′(∆q)(0)
), (4.3.16)
in the case where dim(X) = 3, where ζ denotes the zeta function of ∆q:
ζ(∆q)(s) :=∑
λ∈spec∗(∆q)
λ−s. (4.3.17)
We would like to view TRS as a density T dRS. Consider:
| detH•(Ω(X), d)∗| :=3⊗j=0
(detHj(Ω(X), d))(−1)j (4.3.18)
and,
|| · ||RS := TRS|| · ||L2(Ω(X),d), (4.3.19)
where the L2 norm comes from the identification:
H•(Ω(X), d) ' H•(Ω(X), d). (4.3.20)
Then if δ| detH•((Ω(X),d))∗| = density for || · ||L2(Ω(X),d) on | detH•((Ω(X), d))∗|, then
T dRS := TRS · δ| detH•((Ω(X),d))∗|. (4.3.21)
We then obtain a rigorous definition of the partition function, which is to be contrasted
with [Man98, Eq. 7.27].
4.3.22 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented three-
manifold. The U(1) Chern-Simons partition function, ZU(1)(X, k), is the rigorous quan-
tity
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k) (4.3.23)
where,
ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2 (4.3.24)
where mX = 12(dimH1(X; R)− dimH0(X; R)).
Chapter 4. U(1) Chern-Simons Theory 68
The Atiyah-Patodi-Singer theorem [APS75b] says that the combination
η(?d)
4+
1
12
CS(Ag)
2π(4.3.25)
is a topological invariant depending only on a 2-framing of X. Recall,8 that a 2-framing
is choice of a homotopy equivalence class π of trivializations of TX ⊕ TX, twice the
tangent bundle of X viewed as a Spin(6) bundle. The possible 2-framings correspond to
Z. The identification with Z is given by the signature defect defined by
δ(X, π) = sign(M)− 1
6p1(2TM, π)
where M is a 4-manifold with boundary X and p1(2TM, π) is the relative Pontrjagin
number associated to the framing π of the bundle TX ⊕ TX. The canonical 2-framing
πc corresponds to δ(X, πc) = 0. Either we can choose the canonical framing, and work
with this throughout, or we can observe that if the framing of X is twisted by s units,
then CS(Ag) transforms by
CS(Ag)→ CS(Ag) + 2πs.
The partition function ZU(1)(X, k) is then transformed by
ZU(1)(X, k)→ ZU(1)(X, k) · exp
(2πis
24
). (4.3.26)
Then ZU(1)(X, k) is a topological invariant of framed, oriented three-manifolds, with
a transformation law under change of framing. This is tantamount to a topological
invariant of oriented three-manifolds without a choice of framing.
4.4 Shift Symmetry and the U(1) Partition Function
Our goal in this section is to follow [BW05, §3.1] and obtain a heuristic “shift invariant”
expression for the U(1) Chern-Simons partition function (See Def. 4.4.21 below) by de-
coupling one of the three components of the gauge field A ∈ AP using a shift symmetry
8See [Ati90b].
Chapter 4. U(1) Chern-Simons Theory 69
S. We note that the constructions in this section are largely heuristic and should be
viewed as an initial step in obtaining a rigorous definition for a shift invariant expression
of the U(1) Chern-Simons partition function. For a finite dimensional analogue of the
shift symmetry see Appendix B.1.
First we define S by its variation δS on a field A ∈ AP . Define:
δSA = σκ
where σ ∈ (Ω0(P ) ⊗ u(1))U(1) ' Ω0(X) is an arbitrary form and κ ∈ Ω1(X) is a fixed
contact form on X. Clearly, the Chern-Simons action, SX,P (A),9 does not respect the
shift symmetry. That is,
δSSX,P (A) 6= 0. (4.4.1)
In order to study a shift invariant version of U(1) Chern-Simons theory, we follow [BW05,
§3.1] and introduce a new scalar field Φ ∈ Ω0(X) such that
δSΦ = σ.
We postulate the scaling
Φ→ t−1Φ,
for a non-zero function t ∈ C∞(X), whenever
κ→ tκ
9See §4.2 for a definition and general properties.
Chapter 4. U(1) Chern-Simons Theory 70
so that κΦ ∈ Ω1(X) is invariant under the scaling by t and is a well defined form.
Then for any principal U(1)-bundle P we follow [BW05] and define a new action10
SX,P (A,Φ) := SX,P (A− κΦ)
:=
∫X
α(A− κΦ)
=
∫X
α(A− κΦ) (4.4.2)
= SX,P (A)−∫X
[2κ ∧ Tr(Φ ∧ FA)− κ ∧ dκ Tr(Φ2)] (4.4.3)
where Eq. (4.4.2) follows from the definition of A and Φ (where Φ|[p,g] := Adg−1(ρ∗pr∗1Φ|p))
on P = P ×U(1) SU(2) (See §4.1). It is easy to see that the new action, SX,P (A,Φ), is
invariant under the shift symmetry:
δSSX,P (A,Φ) =δSX,PδA
(A− Φκ) · δS(A− Φκ) (4.4.4)
=δSX,PδA
(A− Φκ) · (δS(A)− δS(Φ)κ) (4.4.5)
=δSX,PδA
(A− Φκ) · (σκ− σκ) (4.4.6)
= 0. (4.4.7)
Now define a “new”11 partition function:
ZU(1)(X, p, k) :=1
Vol(S)
1
Vol(GP )
∫AP
DADΦ eπikSX,P (A,Φ), (4.4.8)
where DΦ is defined by the invariant, positive definite quadratic form, [BW05, Eq. 3.8],
(Φ,Φ) = − 1
4π2
∫X
Φ2 κ ∧ dκ. (4.4.9)
As observed in [BW05], the new partition function of Eq. (4.4.8) should be identically
equal to the original partition function defined for U(1) Chern-Simons theory as in Eq.
(4.3.3)
ZU(1)(X, p, k) =1
Vol(GP )
∫APDAeπikSX,P (A). (4.4.10)
10See §4.1 for a definition of the “hat” notation used here.11The point is that this partition function is really not new, per se, but is expressed in a different form
using an a priori choice of contact structure.
Chapter 4. U(1) Chern-Simons Theory 71
This is seen by fixing Φ = 0 using the shift symmetry, δΦ = σ, which will cancel the
pre-factor Vol(S) from the resulting group integral over S and yield exactly our original
partition function:
ZU(1)(X, p, k) =1
Vol(GP )
∫APDA eπikSX,P (A).
Thus, we obtain the heuristic result,
ZU(1)(X, p, k) = ZU(1)(X, p, k). (4.4.11)
On the other hand, we obtain another description of ZU(1)(X, p, k) by integrating Φ out.
We will briefly review this computation here. Our starting point is the formula for the
shifted partition function (See Eq. (4.4.3))
ZU(1)(X, p, k) =1
Vol(S)
1
Vol(GP )
∫A(P )
DADΦ eπikSX,P (A,Φ) (4.4.12)
where
SX,P (A,Φ) = SX,P (A)−∫X
[2κ ∧ Tr(Φ ∧ FA)− κ ∧ dκ Tr(Φ2)]. (4.4.13)
We formally complete the square with respect to Φ as follows:∫X
[κ ∧ dκ Tr(Φ2) − 2κ ∧ Tr(Φ ∧ FA)]
=
∫X
[Tr(Φ2)− 2κ ∧ Tr(Φ ∧ FA)
κ ∧ dκ
]κ ∧ dκ
=
∫X
Tr
(Φ2 − 2κ ∧ FA
κ ∧ dκΦ
)κ ∧ dκ
=
∫X
Tr
([Φ− κ ∧ FA
κ ∧ dκ
]2
−[κ ∧ FAκ ∧ dκ
]2)κ ∧ dκ
We then only need to compute the Gaussian∫DΦ exp
[πik
∫X
Tr
([Φ− κ ∧ FA
κ ∧ dκ
]2)κ ∧ dκ
]
=
∫DΦ exp
[πik
∫X
Tr(Φ2)κ ∧ dκ]
=
∫DΦ exp
[ik
4π
∫X
Φ2κ ∧ dκ]
=
∫DΦ exp
[−1
2(Φ, TΦ)
]
Chapter 4. U(1) Chern-Simons Theory 72
where we take T = 2πikI acting on the space of fields Φ and the inner product (Φ,Φ) is
defined as in Eq. (4.4.9). We then formally get∫DΦ exp
[−1
2(Φ, TΦ)
]=
√(2π)∆G
detT(4.4.14)
=
(−ik
)∆G/2
(4.4.15)
where the quantity ∆G is formally the dimension of the gauge group G. Note that we
follow [BW05] and abuse notation slightly throughout by writing 1κ∧dκ . We have done
this with the understanding that since κ ∧ dκ is non-vanishing, then κ ∧ FA = φκ ∧ dκ
for some function φ ∈ Ω0(X, su(2)),12 and we identifyκ∧FAκ∧dκ := φ.
Our new description of the partition function is now,
ZU(1)(X, p, k) = C
∫AP
DA exp
[πik
(SX,P (A)−
∫X
Tr[(κ ∧ FA)2]
κ ∧ dκ
)](4.4.16)
where C = 1Vol(S)
1Vol(GP )
(−ik
)∆G/2. Using Prop. 4.2.26, we may rewrite this partition
function after identifying AP = AP + Ω1(X) for a flat base point AP in AP (so that
FAP = 0). We then obtain
ZU(1)(X, p, k) = C1
∫AP
DA exp
[ik
4π
(∫X
A ∧ dA−∫X
(κ ∧ dA)2
κ ∧ dκ
)](4.4.17)
where
C1 =eπikSX,P (AP )
Vol(S) Vol(GP )
(−ik
)∆G/2
.
Note that the critical points of this action, up to the action of the shift symmetry, are
precisely the flat connections, [BW05, Eq. 5.3]. In our notation, A ∈ TAPAP . Let us
define the notation
S(A) :=
∫X
A ∧ dA−∫X
(κ ∧ dA)2
κ ∧ dκ(4.4.18)
12Note that we also abuse notation here again and make the identification of forms on P with formson X via the pullback of some trivializing section of P .
Chapter 4. U(1) Chern-Simons Theory 73
for the new action that appears in the partition function. Also, define
S(A) :=
∫X
(κ ∧ dA)2
κ ∧ dκ(4.4.19)
so that we may write
S(A) = CS(A)− S(A) (4.4.20)
The primary virtue of Eq. (4.4.17) above is that it is heuristically equal to the original
Chern-Simons partition function of Def. 4.3.1 and yet it is expressed in such a way that
the action S(A) is invariant under the shift symmetry. This means that S(A+σκ) = S(A)
for all tangent vectors A ∈ TAP (AP ) ' Ω1(X) and σ ∈ Ω0(X). We may naturally view
A ∈ Ω1(H), the subset of Ω1(X) restricted to the contact distribution H ⊂ TX. If ξ
denotes the Reeb vector field of κ, then Ω1(H) = ω ∈ Ω1(X) | ιξω = 0. The remaining
contributions to the partition function come from the orbits of S in AP , which turn out
to give a contributing factor of Vol(S), [BW05, Eq. 3.32]. We thus reduce our integral
to an integral over AP := AP/S and obtain:
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4π
(∫X
A ∧ dA−∫X
(κ ∧ dA)2
κ ∧ dκ
)]=
eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4πS(A)
]
where DA denotes an appropriate quotient measure on AP , and we can now assume that
A ∈ Ω1(H) ' TAP AP .
We therefore make the following heuristic definition:
4.4.21 Definition. Let k ∈ Z be an (even) integer, and (X, κ) a closed, oriented contact
three-manifold. The shifted U(1) Chern-Simons partition function, ZU(1)(X, k), is the
heuristic quantity
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k) (4.4.22)
Chapter 4. U(1) Chern-Simons Theory 74
where,
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4πS(A)
], (4.4.23)
where S(A) :=∫XA ∧ dA−
∫X
(κ∧dA)2
κ∧dκ is the shifted Chern-Simons action.
Note that we are justified in excluding the factor(−ik
)∆G/2from Eq. (4.4.23) since we
may redefine the heuristic partition function to cancel this factor.
Chapter 5
Non-Abelian Localization for U(1)
Chern-Simons Theory
In [BW05] the authors study the Chern-Simons partition function, [BW05, Eq. 3.1],
Z(k) =1
Vol(G)
(k
4π2
)∆G ∫DA exp
[ik
4π
∫X
Tr
(A ∧ dA+
2
3A ∧ A ∧ A
)], (5.0.1)
where,
• A ∈ AP = A ∈ (Ω1(P )⊗ g)G | A(ξ]) = ξ, ∀ ξ ∈ g is a connection on a principal
G-bundle π : P → X1 over a closed three-manifold X,
• g = LieG and ξ] ∈ Γ(TP ) is the vector field on P generated by the infinitesimal
action of ξ ∈ g on P ,
• k ∈ Z, thought of as an element of H4(BG,Z) that parameterizes the possible
Chern-Simons invariants,
• G := ψ ∈ (Diff(P, P ))G | π ψ = π is the gauge group,
1In fact, [BW05] consider only G compact, connected and simple, and for concreteness one mayassume G = SU(2).
75
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 76
• ∆(G) is formally defined as the dimension of the gauge group.2
In general, the partition function of Eq. (5.0.1) does not admit a general mathematical
interpretation in terms of the cohomology of some classical moduli space of connections, in
contrast to Yang-Mills theory for example, [Wit92]. The main result of [BW05], however,
is that if X is assumed to carry the additional geometric structure of a Seifert manifold,
then the partition function of Eq. (5.0.1) does admit a more conventional interpretation
in terms of the cohomology of some classical moduli space of connections. Using the
additional Seifert structure on X, [BW05] decouple one of the components of a gauge
field A, and introduce a new partition function, [BW05, Eq. 3.7],
Z(k) = K ·∫DADΦ exp
[ik
4π
(CS(A)−
∫X
2κ ∧ Tr(ΦFA) +
∫X
κ ∧ dκ Tr(Φ2)
)],
(5.0.2)
where
• K := 1Vol(G)
1Vol(S)
(k
4π2
)∆G,
• κ ∈ Ω1(X,R) is a contact form associated to the Seifert fibration of X, [BW05,
§3.2],
• Φ ∈ Ω0(X, g) is a Lie algebra-valued zero form on X,
• DΦ is a measure on the space of fields Φ,3
• S is the space of local shift symmetries4 that “acts” on the space of connections
AP and the space of fields Φ, [BW05, §3.1],
2Note that the definition of the Chern-Simons partition function in Eq. (5.0.1) is completely heuristic.The measure DA has not been defined, but only assumed to “exist heuristically,” and the volume anddimension of the gauge group, Vol(G) and ∆(G), respectively, are at best formally defined.
3The measure DΦ is defined independently of any metric on X and is formally defined by the positivedefinite quadratic form
(Φ,Φ) := −∫X
κ ∧ dκ Tr(Φ2),
which is invariant under the choice of representative for the contact structure (X,H) on X, i.e. underthe scaling κ 7→ fκ, Φ 7→ f−1Φ, for some non-zero function f ∈ Ω0(X,R).
4S may be identified with Ω0(X, g), where the “action” on AP is defined as δσ(A) := σκ, and on thespace of fields Φ is defined as δσ(Φ) := σ, for σ ∈ Ω0(X, g). δσ denotes the action associated to σ.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 77
• FA ∈ Ω2(X, g) is the curvature of A, and
• CS(A) :=∫X
Tr(A ∧ dA+ 2
3A ∧ A ∧ A
)is the Chern-Simons action. 5
[BW05] then give a heuristic argument showing that the partition function computed
using the alternative description of Eq. (5.0.2) should be the same as the Chern-Simons
partition function of Eq. (5.0.1). In essence, they show, [BW05, pg.13]:
Z(k) = Z(k), (5.0.3)
by gauge fixing Φ = 0 using the shift symmetry (See §4.4 for a description of the shift
symmetry in the U(1) gauge group case). [BW05] then observe that the Φ dependence
in the integral can be eliminated by simply performing the Gaussian integral over Φ in
Eq. (5.0.2) directly. They obtain the alternative formulation:
Z(k) = Z(k) = K ′ ·∫DA exp
[ik
4π
(CS(A)−
∫X
1
κ ∧ dκTr[(κ ∧ FA)2
])], (5.0.4)
where K ′ := 1Vol(G)
1Vol(S)
(−ik4π2
)∆G/2. Note that we follow [BW05, Eq. 3.9] here and abuse
notation slightly by writing 1κ∧dκ . We have done this with the understanding that since
κ ∧ dκ is non-vanishing (since κ is a contact form), then κ ∧ FA = φκ ∧ dκ for some
function φ ∈ Ω0(X, g), and we identify κ∧FAκ∧dκ := φ.
The original argument of [BW05] was to decouple one of the components of the gauge
field A ∈ AP 6 by introducing a local shift symmetry,7 and then to translate the Chern-
Simons partition function into a “moment map squared” form using this symmetry. The
general “moment map squared” form for the partition function is a symplectic integral
5Note that the partition functions of Eq.’s 5.0.1 and 5.0.2 are defined implicitly with respect thepullback of some trivializing section of the principal G-bundle P . Of course, every principal G-bundleover a three-manifold for G compact, connected and simple is trivializable. It is basic fact that thepartition functions of Eq.’s 5.0.1 and 5.0.2 are independent of the choice of such trivializations.
6See §2.5 for notation and terminology.7See [BW05, §3.1].
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 78
of the canonical form
Z(ε) =1
Vol(H)
(1
2πε
)∆H/2 ∫Y
exp
[Ω− 1
2ε(µ, µ)
]where Y is a symplectic manifold with symplectic form Ω, and H is a Lie group that acts
on Y in a Hamiltonian fashion with moment map µ. ∆H = dim(H) and ε = 2πk
.8 The
technique of non-abelian localization [Wit92] can then be applied to study such integrals.
The goal of this chapter is to develop analogous results for the case of a U(1) gauge group.
5.1 Symplectic Formulation of the U(1) Partition Func-
tion
The goal of this section is to provide a rigorous definition of the U(1) Chern-Simons
partition function using the technique of non-abelian localization following [BW05].
We study the U(1) Chern-Simons partition function using the final results of [BW05],
which a priori are only valid for a simply connected gauge group G (e.g. G = SU(2)). Us-
ing the results of [BW05], we derive an alternative definition for the U(1) Chern-Simons
partition function on a closed, oriented Seifert three-manifold X, called the symplectic
U(1) Chern-Simons partition function, denoted as ZSU(1)(X, k) (See Def. 5.1.13). We note
that the results in this section are intended to serve as motivation for a definition of the
U(1) partition function using the results of [BW05], and do not comprise a rigorous anal-
ysis of the method of non-abelian localization for a U(1) gauge group. It is surprising,
however, that this method yields a definition that is closely9 related with our previous
definitions, although we could not infer this from Beasley and Witten’s calculation be-
cause our situation does not satisfy the hypotheses of [BW05] (i.e. G simply connected).
The results of this section also further support the conclusions of [BW05]. Our starting
8Note that we have chosen to express the quantities in this thesis in terms of the Chern-Simonscoupling constant, k ∈ Z.
9See §5.3 and §5.4 for a detailed study of the precise relations.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 79
point is the following, [BW05, Eq. 5.172]:
ZX := Z(ε)|M0=
1
|Γ|exp
(−iπ
2η0
)∫M0
A(M0)exp
[1
2πεΩ +
1
2c1(TM0) +
in
4π2εrΘ
].
(5.1.1)
We have:
• M0 is a smooth component of the moduli space of irreducible flat connections on
a Seifert manifold X (this formula is derived in [BW05] under the assumption that
X is a principal U(1) bundle of degree n over Σ),
• Γ = Z(G) is the center of G,
• η0, is called the adiabatic eta-invariant, and is recalled in §5.4,10
• A(M0) =∏dimM0
j=1xj/2
sinh(xj/2)where xj(1 ≤ j ≤ n), are the Chern roots of TM0, so
that c(TM0) =∏n
j=1(1 + xj), xj ∈ H2(M0,Z),
• Ω is the symplectic form on M0,
• εr = 2πk+cg
, where cg is the dual Coxeter number of G,
• Θ ∈ H4(M0) is the cohomology class corresponding to the degree 4 element
−(φ, φ)/2 in the equivariant cohomology H4G(pt) (for φ ∈ g using the Cartan model
of equivariant cohomology), Θ can also be described in terms of the universal bundle
U
C // U
Jac(Σ)× Σ
in other words
Θ = −1
2c1(U)2|pt.∈Σ
where Jac(Σ) is the Jacobian of Σ.
10We have chosen the opposite sign convention to [BW05] and [Bea07] in defining η0, and their eta-invariant is the negative of ours.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 80
Our first observation is that the overall constant |Γ| does not really make sense for
G = U(1), if interpreted as the cardinality of U(1). We observe, however, that Eq.
(5.1.1) is derived by localization on a smooth component of the moduli space of flat
connections, and since such a component consists of irreducible connections, the isotropy
group Γ arises solely from the center of G. In U(1) Chern-Simons theory, the isotropy
group is Γ = H ' U(1) and as in [Man98] the correct quantity to replace |Γ| with here is
|Γ| = Vol(H) =
[∫X
?1
]1/2
, (5.1.2)
where ? is the Hodge star relative to some choice of metric g on X (See §4.3). We consider
e(−iπ2η0)∫M0
A(M0)exp
[1
2πεΩ +
1
2c1(TM0) +
in
4π2εrΘ
]. (5.1.3)
By Prop. 2.5.66 and our observations in §5.2, MX ' U(1)2g × HT , where MX is the
moduli space space of flat connections on X and HT ' Tors(H2(X,Z)). We posit that
M0 = MP ' U(1)2g and Eq. (5.1.1) applies to each connected component of MX .
That is, we take each component of the U(1) partition function to be proportional to Eq.
(5.1.3), and weighted by eπikSX,P (AP ) for some flat connection in a bundle P over X, with
p := c1(P ). We do this in complete analogy with the considerations in §4.3. In summary,
we make the initial definition
ZSU(1)(X, k) =
∑p∈TorsH2(X;Z)
ZSU(1)(X, p, k) (5.1.4)
where,
ZSU(1)(X, p, k) =
eπikSX,P (AP )
Vol(H)e(−
iπ2η0)∫MP
A(MP )exp
[1
2πεΩP +
1
2c1(TMP ) +
in
4π2εrΘ
],
(5.1.5)
where MP is the moduli space of flat connections on the principal U(1)-bundle P and
ΩP :=∑
1≤i≤g dθi ∧ dθi, is the standard symplectic form on U(1)2g 'MP .
5.1.6 Remark. We note that ΩP |A is naturally viewed as an element of ∧2H1(X), where
H1(X) is the space of harmonic one-forms and is canonically identified as the tangent
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 81
space to the moduli space of flat connections at A, i.e. H1(X) ' TAMP . In particular,
for α, β ∈ H1(X), differential forms,
ΩP |A(α, β) :=
∫X
κ ∧ α ∧ β, (5.1.7)
where κ ∈ Ω1(X) is our chosen contact form as usual.
This is not our final definition of ZSU(1)(X, k), however. The first thing we observe is
that Θ = 0 in the case that G = U(1). This follows since the universal bundle U for U(1)-
bundles is the classical Poincare line bundle, and the Poincare line bundle is normalized
to have degree d = 0 when restricted to the Jacobian of Σ. Since c1(U) = d[Σ] ∈ H2(Σ),
this implies c1(U) = 0, and hence Θ = 0. Also, since MP ' U(1)2g, we know
c(MP ) := c(TMP ),
=
g∏i=1
c(Li) =
g∏i=1
(1 + xi), where Li = TΣi, and xi = c1(Li) ∈ H2(Σi,Z)
where Σi ' (U(1))2. Then the tangent bundles TΣi are trivial, and hence
xi = c1(TΣi) = 0
Thus
A(MP ) =
dimMP∏j=1
xj/2
sinh(xj/2)= 1 (5.1.8)
Clearly, c1(TMP ) = 0 as well. Recalling that ε = 2πk
, we have∫M0
exp
[k
(ΩP
(2π)2
)]=∫M0
kg(
ΩP(2π)2
)g/g!
= kg∫M0
(ΩP
(2π)2
)g/g!
Define
ωP :=ΩgP
g!(2π)2g Vol(H).
Our revised definition of the U(1) Chern-Simons partition function becomes
ZSU(1)(X, p, k) = kgeπikSX,P (AP )e(−
iπ2η0)∫MP
ωP . (5.1.9)
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 82
Eq. (5.1.15) is still not a final definition of ZSU(1)(X, p, k). We notice in Def. 4.3.22 that
aside from the term eπikSX,P (AP ), the k-dependence of ZSU(1)(X, p, k) should be of the form
kmX where mX = 12(dimH1(X; R) − dimH0(X; R)). Eq. (5.1.15) has a k-dependence
of kg, however. Since we are assuming that X is connected, we have dimH0(X; R) = 1.
We account for this extra factor of k−1/2 by observing that the dimension of the stabi-
lizer of the gauge group action (for U(1) gauge group) is dim(H0(X,R)) = 1. A similar
phenomenon occurs in Yang-Mills theory at the higher non-flat critical points of the
Yang-Mills action. As observed in [BW05, Eq. 4.45], there is a k1/2 dependence on the
partition function coming from the fact that the gauge group G doesn’t act freely on the
locus of non-flat Yang-Mills solutions. This k1/2 dependence comes from the U(1) < G
subgroup generated by the non-flat Yang-Mills solution, which acts as a stabilizer at the
corresponding connection. U(1) Chern-Simons theory also comes with a U(1) stabilizer
subgroup; the subgroup of constant gauge transformations with values in U(1). Here we
get a factor of k−1/2. In the computation of Eq. 5.172 of [BW05] it is assumed that one
is localizing at the locus of irreducible flat connections, and therefore the isotropy group
of A, ΓA = u ∈ G | u(A) = A, is finite. We do not get a factor of k−1/2 here because
the dimension of the stabilizer is zero. We take this difference of the k-dependence into
account when we make our rigorous definition of the partition function below (See Def.
5.1.13).
Finally, as is noted in §5 of [BW05],11 the computation of their partition function is
done implicitly with respect to a choice of the so called Seifert framing on X. This
choice of framing results in a difference of a factor of eiδΨ · e(−iπ2η0) in the partition
function for the canonical framing, where for general gauge group, [BW05, Eq. 5.101]:
eiδΨ = exp
(iπ∆G
4− iπ∆Gcg
12(k + cg)θ0 +
iπ
2η0
). (5.1.10)
11See [BW05, Pages 89-92].
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 83
For the case of a U(1) gauge group, we take ∆G = 1 and we posit that the term iπ∆Gcg
12(k+cg)θ0
vanishes in the exponential. In this thesis, we assume the choice of canonical framing.
Thus, we should replace
e(−iπ2η0), (5.1.11)
in the partition function with,
e(iπ4− iπ
2η0). (5.1.12)
Before we make our rigorous definition, we recall the following:
mX :=1
2(dimH1(X; R)− dimH0(X; R)),
η0 is the adiabatic eta-invariant defined in §5.4, AP ∈ AP is a flat connection on P ,MP ,
is the moduli space of flat connections on P , and ωP :=ΩgP
g!(2π)2g Vol(H)is the symplectic
volume form on MP with
ΩP :=∑
1≤i≤g
dθi ∧ dθi,
the standard symplectic form on U(1)2g 'MP .
We are now ready to make the following
5.1.13 Definition. Let k ∈ Z be an (even) integer, and X a closed, oriented Seifert
three-manifold,
U(1) // X
Σ
,
where Σ = |Σ|,U is an orbifold with underlying space |Σ| a Riemann surface of genus
g. The symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), is the rigorous
quantity
ZSU(1)(X, k) =
∑p∈TorsH2(X;Z)
ZSU(1)(X, p, k) (5.1.14)
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 84
where,
ZSU(1)(X, p, k) = kmXeπikSX,P (AP )eiπ(
14− 1
2η0)∫MP
ωP . (5.1.15)
One of our main objectives in this thesis is to establish the equivalence of Def. 5.1.13
and Def. 4.3.22.
5.2 Orbifolds and Symplectic Volume
In order to be more explicit in our computation of the U(1) Chern-Simons partition
function we find an explicit form for the moduli space of flat connections, MX (See Eq.
(2.5.65)), over a Seifert manifold X (See Def. 2.3.6) in this section. The main result of
this section is contained in Prop. 5.2.4 below.
Recall that the fundamental group of X ([Orl72]), π1(X), is generated by the follow-
ing elements
ap, bp, p = 1, . . . , g
cj, j = 1, . . . , N
h
which satisfy the relations,
[ap, h] = [bp, h] = [cj, h] = 1
cαjj h
βj = 1g∏p=1
[ap, bp]N∏j=1
cj = hn.
The generator h is associated to the generic S1 fiber over Σ, the generators ap, bp come
from the 2g non-contractible cycles on Σ, and the generators cj come from the small one
cycles in Σ around each of the orbifold points pj.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 85
The moduli space of flat connections over X can be realized as the space of homomor-
phisms from π1(X) to U(1). Consider,
ρ : π1(X)→ U(1)
and let, Aj = ρ(cj), B = ρ(h). Then since U(1) is abelian, the generating relations for
π1(X) translate into the following restrictions on ρ:
Aαjj B
βj = 1, j = 1, . . . , N
N∏j=1
Aj ·B−n = 1,
where, Aj = e2πiψj , j = 1, . . . , N , and B = e2πiψ0 for some ψj ∈ R, j = 0, . . . , N .
We condense the above restrictions and write,
N∏j=0
e2πiψlKj,l = 1 (5.2.1)
where Kj,l is the following matrix
K =
−n β1 β2 · · · βN
1 α1 0 · · · 0
... 0. . . 0
...
1... 0 αN−1 0
1 0 · · · 0 αN .
(5.2.2)
Let v = (ψ0, ψ1, . . . , ψN) ∈ RN+1. Then v solves the above equation, 5.2.1, if and only if
K · v = w ∈ ZN+1
Observe that detK = (−∏N
j=1 αj) ·(n+
∑Nj=1
βjαj
)= (−
∏Nj=1 αj) · c1(L) 6= 0, since
αj 6= 0, ∀ 1 ≤ j ≤ N , and the orbifold first chern number c1(L) > 0 by assumption (See
Remark 2.3.7). Thus, K is invertible, and
v = K−1w =(CofK)T
detKw
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 86
where the cofactor matrix CofK ∈MN+1(Z) and detK ∈ Z. So,
v =1
detKw ∈ 1
detKZN+1
where w = (CofK)T ·w ∈ ZN+1.
In general, once we fix the values of ρ(ap), ρ(bp) ∈ U(1) there are (detK)N+1 possibilities
for ρ. Thus, for each point in U(1)2g there is a factor of order |(detK)N+1| in the moduli
space of flat connections of X. It is interesting to note that it has been independently
shown in [Man98] (See Prop. 2.5.66) that
π0(MX) ' Tors(H2(X,Z)). (5.2.3)
Thus, we have HT ' Tors(H2(X,Z)) and in particular |(detK)N+1| = |Tors(H2(X,Z))|.
We have therefore established the following
5.2.4 Proposition. Let X be a closed, oriented Seifert three-manifold, and MX denote
the moduli space of flat connections on X. Then
MX ' U(1)2g × Tors(H2(X,Z)) (5.2.5)
where |Tors(H2(X,Z))| = |(detK)N+1| for the matrix K defined in Eq. (5.2.2). Note
that the special case where N = 0, we recover det K = −n and MX ' U(1)2g × Zn.
5.3 Reidemeister Torsion and Symplectic Volume
One of our main objectives in this thesis is to establish the equivalence of Def. 5.1.13
and Def. 4.3.22. This boils down to proving the equivalence of the quantities
ZSU(1)(X, p, k) = kmXeπikSX,P (AP )eiπ(
14− 1
2η0)∫MP
ωP , (5.3.1)
and,
ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2, (5.3.2)
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 87
where all the requisite notation for Eq. (5.3.1) is defined in §5.1 and similarly the no-
tation for Eq. (5.3.2) is defined in §4.3. In this section we compare the square root of
the Ray-Singer torsion (T dRS)1/2 (see Appendix C.2) to the symplectic volume form ωP .
Superficially, these two quantities are both elements of Ω2g(MP ,R). The goal of this
section is to establish that in fact these two quantities agree up to some undetermined
constant. Ultimately, we would like to show these quantities are identically equal
(T dRS)1/2 = ωP , (5.3.3)
as differential forms on Ω2g(MP ,R).
Our first observation for this section is that it is much more useful for us to work with
the combinatorially defined Reidemeister torsion (R-torsion), τ(X) (see Remark 5.3.12
below), than directly with the Ray-Singer torsion, T dRS. It is a well known fact that
τ(X) = T dRS. This was independently shown by Cheeger, [Che79] and Muller, [Mul78],
and more recently a new proof has been given by Braverman, [Bra02].
We begin by reviewing the Reidemeister torsion, (R-torsion), and provide some relevant
examples. Recall that the R-torsion is an invariant for a CW-complex and a representa-
tion of its fundamental group. Before we define the R-torsion, we recall the definition of
the torsion of a chain complex. We primarily follow [JCW93] and [Yam08].
Let C∗ =(
0→ Cndn−→ Cn−1
dn−1−−−→ · · ·C1d1−→ C0 → 0
)be a chain complex over F (either
R or C). Let Zi denote the cycles of this complex, Bi denote the boundaries, and Hi the
homology. We say that C∗ is acyclic if Hi = 0 for all i.
Let ci be a basis of Ci and c be the collection cii≥0. We call a pair (C∗, c) a based chain
complex, c the preferred basis of C∗ and ci the preferred basis of Ci. Let hi be a basis of
Hi.
We construct another basis as follows. By the definitions of Zi, Bi, and Hi, the following
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 88
two split exact sequences exist
0→ Zi → Cidi−→ Bi−1 → 0,
0→ Bi → Zi → Hi → 0.
Let Bi−1 be a lift of Bi−1 to Ci and Hi a lift of Hi to Zi. Then we can decompose Ci as
follows.
Ci = Zi ⊕ Bi−1
= Bi ⊕ Hi ⊕ Bi−1
= di+1Bi ⊕ Hi ⊕ Bi−1
Choose a basis bi for Bi. We write bi+1 = bi+1j
nij=1 for a lift of bi and hi = hij
mij=1 for
a lift of hi. By construction, the set bi ∪ di+1(bi+1) ∪ hi forms another ordered basis of
Ci. Denote this basis as bidi+1(bi+1)hi. The definition of the R-torsion, Tor(C∗, c), is as
follows:
Tor(C∗, c)h = (−1)|C∗| ·n∏i=1
[bidi+1(bi+1)hi/ci](−1)i+1 ∈ F∗. (5.3.4)
where [bidi+1(bi+1)hi/ci] denotes the determinate of the change of basis matrix from
the basis ci to the basis bidi+1(bi+1)hi; |C∗| =∑
i≥0 αi(C∗) · βi(C∗), where αi(C∗) =∑ik=0 dimCk and βi(C∗) =
∑ik=0 dimHk.
An alternative definition12 that we will also use is to equip our complex C∗ with volumes
µi ∈ (∧maxCi)′, one for each i, and then define
Tor(C∗, µ)h =
∧i even µi [b
i ∧ di+1(bi+1) ∧ hi]∧i odd µi [b
i ∧ di+1(bi+1) ∧ hi]. (5.3.5)
where we take bi = ∧nij=1bij, di+1b
i+1 = ∧nij=1di+1bi+1j , and hi = ∧mij=1h
ij. The torsion is an
element
Tor(C∗, µ) ∈ ⊗2i+1| ∧max H2i+1(C∗)| ⊗2i | ∧max H2i(C∗)′ |,
12See [JCW93].
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 89
where |·| denotes absolute values in the determinant lines. It is well known that the torsion
is independent of the choices of bi, and lifts bi, hi.13 The latter definition specializes to
the former definition when we choose the canonical volumes associated to a choice of
preferred basis c for C. In the case that our complex C∗ is acyclic, we define
Tor(C∗, µ) =
∧i even µi [b
i ∧ di+1(bi+1)]∧i odd µi [b
i ∧ di+1(bi+1)]∈ R∗. (5.3.6)
We will be interested in a specific chain complex C∗. In particular, let N be a cell
complex, and ρ a representation of π1(N) in G. The Lie algebra g is acted on by π1(N)
under the composition of the adjoint action of G and the representation ρ. Let gρ denote
g with the π1(N)-module structure from ρ. Let N denote the universal cover of N . Since
the fundamental group π1(N) acts on N by covering transformations, the chain complex
C∗(N) also has a natural π1(N)-module structure. The chain complex of interest is then
C∗(N, gρ), defined as the quotient of C∗(N)⊗ g under the equivalence:
σ ⊗ Y ∼ σa⊗ Ad(ρ(a))−1Y, (5.3.7)
where a ∈ π1(N), σ ∈ C∗(N), and Y ∈ g. The usual differential on C∗(N) is compatible
with the equivalence relation, and thus descends to a differential δρ on C∗(N, gρ). By
dualizing one obtains the corresponding cochain complex C∗(N, gρ) with differential dρ =
δ∗ρ. We have the following
5.3.8 Lemma. [JCW93] Suppose h ∈ G. If ρ and hρh−1 are conjugate representations of
π1(N) in G, then the map Ad(h) : g→ g induces an isomorphism of the chain complexes
C∗(N, gρ) and C∗(N, ghρh−1). Hence, one obtains a natural isomorphism between the
cohomology groups H i(C∗(N, gρ)) and H i(C∗(N, ghρh−1)).
We will mainly be interested in the zeroth and first cohomology groups of this complex.
We recall the following
13See [RS73].
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 90
5.3.9 Proposition. [JCW93] Let [ρ] ∈ Hom(π1(N), G)/G. The choice of a particular
ρ ∈ Hom(π1(N), G) in the conjugacy class [ρ] identifies the Zariski tangent space to the
space Hom(π1(N), G)/G at ρ with the first cohomolgy group H1(N, gρ).
Furthermore, the Lie alegbra of the isotropy group of ρ(the subgroup of G fixing the
representation ρ under conjugation) is H0(N, gρ).
Using the definition of the R-torsion in equation (5.3.5) above, we may define volumes
on C∗(N, gρ) by using the natural metric on g. We take σij ⊗ Yk to be an orthonormal
basis of C∗(N, gρ), where σij are the i-cells in the universal cover N and the Yk are an
orthonormal basis of g. This volume is well defined since the adjoint representation is
an orthogonal representation of G, and hence compatible with the equivalence relation
in Eq. (5.3.7). We then have
Tor(C∗(N, gρ), µ) ∈ ⊗2i+1| ∧max H2i+1(N, gρ)| ⊗2i | ∧max H2i(N, gρ)′|.
Since H i(N, gρ) ' Hi(N, gρ)′, we have
Tor(C∗(N, gρ), µ) ∈ ⊗2i+1| ∧max H2i+1(N, gρ)′ | ⊗2i | ∧max H2i(N, gρ)|. (5.3.10)
Define the determinant line,
| detH•(N, gρ)∗| :=
⊗j
| detHj(N, gρ)(−1)j |,
where,
detHj(N, gρ) := ∧maxHj(N, gρ),
and H−1 := H ′ denotes the dual space. Then we may also write
Tor(C∗(N, gρ), µ) ∈ | detH•(N, gρ)∗|. (5.3.11)
5.3.12 Remark. The isomorphisms in Lemma 5.3.8 identify the torsion Tor(C∗(N, gρ))
with Tor(C∗(N, ghρh−1), so the torsion descends to an equivalence class τ(N, ρ) depending
only on the conjugacy class [ρ] ∈ Hom(π1(N), G)/G.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 91
It is instructive to consider the case when N is a genus g surface Σg.
5.3.13 Example. From the relation (5.3.10) above, we see that the torsion τ(Σg; ρ) of a
surface Σg takes values in
∧maxH1(Σg, gρ)′ ⊗ ∧maxH2(Σg, gρ)⊗ ∧maxH0(Σg, gρ).
Note that [Wit91] assumes that the flat connection ρ is irreducible, which in particular
implies that H0(Σ, adE) ' H2(Σ, adE) ' 0.14 Thus, we have
τ(Σg; ρ) ∈ ∧maxH1(Σg, gρ)′.
Observe that H1(Σg, gρ) is a symplectic vector space with the symplectic form given
by the cup product H1(Σg, gρ) ⊗ H1(Σg, gρ) → H2(Σg,R) ' R. One can show that in
fact the torsion may be identified with the symplectic volume on a smooth component
of irreducible flat connections in the moduli space Hom(π1(Σ), G)/G. Note that this
identification naturally follows from Prop. 5.3.9 above. A rigorous proof of this is given
in [Wit91].
5.3.14 Remark. The case of interest is when N = X is a closed, oriented Seifert manifold,
and G = U(1). In particular, we observe that H•(N, gρ) ' H•(Ω(X), d) for any flat U(1)
connection ρ, where Hj(Ω(X), d) denotes the usual de Rham cohomology of X. We
henceforth impose these conditions in this section.
As above, we observe that the torsion τ(X; ρ) takes values in
| detH•(Ω(X), d)∗| :=3⊗j=0
| detHj(Ω(X), d)(−1)j |,
where by Poincare duality H3(Ω(X), d)′ is canonically isomorphic to H0(Ω(X), d), and
H1(Ω(X), d)′ is canonically isomorphic to H2(Ω(X), d). Thus,
τ(X; ρ) ∈ | detH1(Ω(X), d)′|⊗2⊗| detH0(Ω(X), d)|⊗2.
14E denotes a principal G bundle over Σ and adE denotes the corresponding adjoint bundle. Notealso that [Wit91] restricts to the case where the gauge group G is compact, semi-simple and connected.
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 92
Choosing the canonical trivialization | detH0(Ω(X), d)| ' R+ with respect to an or-
thonormal basis for H0(Ω(X), d), we may then view the square root of the R-torsion as
a density, √τ(X; ρ) ∈ | detH1(Ω(X), d)′|.
Define,
√τ(X) : MP → | detH1(Ω(X), d)′|, by, (5.3.15)√
τ(X)(ρ) :=√τ(X)|ρ :=
√τ(X; ρ). (5.3.16)
Note that MX ' MP × TorsH2(X; Z) by Prop. 2.5.66, and we are implicitly choosing
only one bundle class [P ] to work with here in this decomposition ofMX , so that c1(P ) ∈
TorsH2(X; Z). Then we may identify√τ(X) ∈ Ω2g(MP ,R) since By Prop. 4.1.2 of
[JM09], H1(X) is identified with the Zariski tangent space to the moduli space MP '
U(1)2g.
5.3.17 Remark. Note that we are assuming that√τ(X) defined above is a smooth density
here with respect to the natural smooth structure on MP and the smooth structure on
H1(X) as identified with the Zariski tangent space to the moduli space MP ' U(1)2g.
We could not find an explicit proof of this fact in the literature, and it is of interest to
the author to show this in the future. For now we simply take this for granted.
We would like to see√τ(X) is proportional up to a constant to the symplectic volume
ωP on the moduli space MP , i.e.
√τ(X) = C · ωP , (5.3.18)
where C ∈ R∗ is some non-zero constant. Recall that ωP :=ΩgP
g!(2π)2g Vol(H)with
ΩP :=∑
1≤i≤g
dθi ∧ dθi,
the standard symplectic form on U(1)2g ' MP . It will be sufficient to identify√τ(X)
and ωP at a single point of the moduli space since√τ(X) and ΩP (and hence ωP ) are
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 93
invariant under left multiplication, i.e. invariant under the action,
Lδ : U(1)2g → U(1)2g
defined by Lδ : γ 7→ δ · γ for γ, δ ∈ U(1)2g. Let us show this presently. First, we ob-
serve that√τ(X) is invariant under this action. This follows directly from the fact that
in the G = U(1) case√τ(X; ρ) '
√τ(X; ρ′) by definition (see (5.3.5)). In particular,√
τ(X; ρ) '√τ(X; δ · ρ) for all δ, ρ ∈ U(1)2g. It is also clear that ΩP :=
∑1≤i≤g dθi∧dθi
is invariant under this action since this action just represents rotations in each copy of
U(1) on U(1)2g.
Thus, let e denote the identity element of U(1)2g. Then at the point e,√τ(X) and
ωP must agree up to a non-zero constant (recall that√τ(X) 6= 0 is a volume form by
definition, and similarly ωP 6= 0):
√τ(X)|e = C · ωP |e
for some C ∈ R∗. By left invariance, we therefore have:
√τ(X)|ρ = C · ωP |ρ, ∀ρ ∈ U(1)2g.
Thus, √τ(X) = C · ωP , for C ∈ R∗.
We have shown the following
5.3.19 Proposition. Let X be a closed, oriented Siefert three-manifold,√τ(X) the
square root of the R-torsion of X, and ωP the symplectic volume form on the moduli
space of flat connections MP ' U(1)2g for some flat bundle P over X. Then, there
exists C ∈ R∗, such that √τ(X) = C · ωP .
We make the following
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 94
5.3.20 Conjecture. C = 1 in Prop. 5.3.19 above.
5.3.21 Remark. We propose to prove Conjecture 5.3.20 by modeling our proof on [Wit91]
for the analogous result in two dimensions (see Ex. 5.3.13). We note that Conjecture
5.3.20 would yield a calculation of the R-torsion for the class of closed, oriented Siefert
three manifolds. Consider,∫MP
√τ(X) =
∫MP
ωP , assuming Conjecture 5.3.20 is true, (5.3.22)
=
∫MP
ΩgP
g!(2π)2g Vol(H), (5.3.23)
=1
Vol(H), (5.3.24)
where (see §6.5),
Vol(H) =
[∫X
κ ∧ dκ]1/2
=
[n+
N∑j=1
βjαj
]1/2
,
for [n; (α1, β1), . . . , (αN , βN)] the Seifert invariants of our Seifert manifold X. Also, recall
that (see §2.3)
c1(L) = n+N∑j=1
βjαj
> 0.
Thus, by our observations in Appendix C.2 and the Cheeger-Muller theorem, [Che79],
[Mul78], we would have
TRS = TC = (c1(L))−1.
5.4 The Adiabatic Eta-Invariant
In this section we continue with our aim of proving the equivalence of Def. 5.1.13 and
Def. 4.3.22, i.e. that
ZSU(1)(X, p, k) = kmXeπikSX,P (AP )eiπ(
14− 1
2η0)∫MP
ωP , (5.4.1)
and,
ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2, (5.4.2)
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 95
are equal. Our focus in this section is to compare the eta-invariants that show up in
Eq.’s 5.4.1 and Def. 5.4.2. We will see that these invariants are closely related, and in
fact differ by the exact amount required to establish the equivalence of the eta-invariant
dependent parts of Def. 5.1.13 and Def. 4.3.22. The regularized eta-invariant of Eq.
(5.4.2)
η(?d)
4+
1
12
CS(Ag)
2π, (5.4.3)
is studied in §6.10 for example, and is computed in Prop. 6.10.27
5.4.4 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then,15
η(?d) +1
3
CS(Ag)
2π= η(?D) +
1
512
∫X
R2 κ ∧ dκ
= 1− d
3+ 4
N∑j=1
s(αj, βj),
where 0 < d = c1(X) = c1(L) = n+∑N
j=1βjαj∈ Q, and
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)∈ Q
is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for gcd(αj, βj) =
1) are the Seifert invariants of X. In particular, we have computed the U(1) Chern-
Simons partition function as:
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dC)1/2,
= kmXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dRS)1/2.
Thus, we turn our attention to the study of η0, which was named the adiabatic eta-
invariant by Nicolaescu in [Nic00]. This invariant shows up in Def. 5.1.13 for the
symplectic U(1) Chern-Simons partition function, ZSU(1)(X, k), precisely due to the con-
siderations in §5.2 of [BW05]. See [Bea07, App. C] for an explicit computation of η0.
15This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 96
The main result that we will need is [Bea07, Eq. C.31]:
η0 =∆G
6
[c1(L)− 12
N∑j=1
s(αj, βj)
], (5.4.5)
where ∆G = dim(G) is the dimension of the gauge group. Note that the adiabatic
eta-invariant η0 is a topological invariant defined and computed on a general Seifert
three-manifold X.
5.4.6 Remark. We have chosen the opposite sign convention to [BW05] and [Bea07] in
defining η0, and their eta-invariant is the negative of ours.
In our case G = U(1), ∆G = 1, and we have
− η0
2= −c1(L)
12+
N∑j=1
s(αj, βj). (5.4.7)
L denotes the line V-bundle over the orbifold Σ associated to the Siefert manifold X (i.e.
X is the boundary of the disc bundle of L). Setting d = c1(L) in Eq. (5.4.7), we obtain
− η0
2= − d
12+
N∑j=1
s(αj, βj). (5.4.8)
Inserting this into Eq. (5.4.1), we have
ZSU(1)(X, p, k) = kmXeπikSX,P (AP )e
iπ4 (1− d
3+4∑Nj=1 s(αj ,βj)))
∫MP
ωP . (5.4.9)
This precisely matches the eta-invariant dependence in Eq. (5.4.2),
ZU(1)(X, p, k) := kmXeπikSX,P (AP )eπi4
(η(?d)+ 1
3CS(Ag)
2π
) ∫MP
(T dRS)1/2, (5.4.10)
by Prop. 5.4.4, i.e.
η(?d) +1
3
CS(Ag)
2π= 1− d
3+ 4
N∑j=1
s(αj, βj).
Thus, we have shown the following
Chapter 5. Non-Abelian Localization for U(1) Chern-Simons Theory 97
5.4.11 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then,16
η(?d)
4+
1
12
CS(Ag)
2π=
1
4− 1
2η0,
=1
4− d
12+
N∑j=1
s(αj, βj).
16This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 6
Eta-Invariants and Anomalies
This chapter studies U(1) Chern-Simons theory and its relation to a construction of
Chris Beasley and Edward Witten, [BW05]. The natural geometric setup here is that
of a three-manifold with a Seifert structure (see Chapter 2). Based on a suggestion of
Edward Witten we are led to study the stationary phase approximation (see A.1) of the
path integral for U(1) Chern-Simons theory after one of the three components of the
gauge field is decoupled. This gives an alternative formulation of the partition function
for U(1) Chern-Simons theory that is conjecturally equivalent to the usual U(1) Chern-
Simons theory (see §4.3). We establish this conjectural equivalence rigorously using
appropriate regularization techniques. This approach leads to some rather surprising
results and opens the door to studying hypoelliptic operators (see Chapter 3) and their
associated eta-invariants in a new light.
6.1 Introduction
The objective in this chapter is to study the partition function for U(1) Chern-Simons
theory using the analogue of Eq. (5.0.4) in this case. Thus, we are also assuming here that
X is a Seifert manifold with a “compatible” contact structure, (X, κ), [BW05, §3.2]. Note
that any compact, oriented three-manifold possesses a contact structure and one aim of
98
Chapter 6. Eta-Invariants and Anomalies 99
future work is to extend our results to all closed, oriented three-manifolds using this fact.
For now, we restrict ourselves to the case of closed three-manifolds that possess contact
compatible Seifert structures (see Definition 6.2.1 for example). We restrict to the gauge
group U(1) so that the action is quadratic and hence the stationary phase approximation
is exact (see Appendix A.1). A salient point is that the group U(1) is not simple, and
therefore may have non-trivial principal bundles associated with it. This makes the U(1)-
theory very different from the SU(2)-theory in that one must now incorporate a sum over
bundle classes in a definition of the U(1)-partition function. Recall (definition 4.3.1) that
the partition function for U(1) Chern-Simons theory is heuristically defined as:
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k), (6.1.1)
where,
ZU(1)(X, p, k) =1
Vol(GP )
∫APDAeπikSX,P (A). (6.1.2)
Recall that the torsion subgroup TorsH2(X; Z) < H2(X; Z)1 enumerates the U(1)-bundle
classes that have flat connections. Note that the bundle P → X in Eq. (6.1.2) is taken to
be any representative of a bundle class with first Chern class c1(P ) = p ∈ TorsH2(X; Z).
Also note that some care must be taken to define the Chern-Simons action, SX,P (A), in
the case that G = U(1). More details of this construction can be found in section 4.3,
where a rigorous version of this definition is also constructed (see definition 4.3.22).
The main results of this chapter may be summarized as follows. First, our main ob-
jective is the rigorous confirmation of the heuristic result of Eq. (5.0.3) in the case where
the gauge group is U(1). This statement is certainly non-trivial and involves some fairly
deep facts about the “contact operator” as studied by Michel Rumin, [Rum94]. Recall
1Recall the definition of the torsion of an abelian group is the collection of those elements which havefinite order.
Chapter 6. Eta-Invariants and Anomalies 100
that this is the second order operator “D” that fits into the complex,
C∞(X)dH−→ Ω1(H)
D−→ Ω2(V )dH−→ Ω3(X), (6.1.3)
and in our case2 is defined by:
Dα = κ ∧ [Lξ + dH ?H dH ]α, α ∈ Ω1(H). (6.1.4)
This operator is elaborated upon in §6.4 below. A somewhat surprising observation is
that this operator shows up quite naturally in U(1) Chern-Simons theory (see Prop.
6.4.14 below), and this leads us to make several conjectures motivated by the rigorous
confirmation of the heuristic result of Eq. (5.0.3). Our main result is the following (see
§6.10 below for the relevant definitions, and Prop. 6.10.24 in particular):
6.1.5 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three mani-
fold. If,
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (η(?D)+ 1
512
∫X R2 κ∧dκ)
∫MP
(T dC)1/2, (6.1.6)
and (see §4.3),
ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2 (6.1.7)
then,3
ZU(1)(X, k) = ZU(1)(X, k)
as topological invariants.
Following [Man98], we rigorously define ZU(1)(X, k) in §6.6 using the fact that the
stationary phase approximation for our path integral should be exact. This necessitates
the introduction of the regularized determinant of D in Eq. (6.7.3), which in turn nat-
urally involves the hypoelliptic Laplacian of Eq. (6.7.5). The rigorous quantity that
2The case where X is taken to be a closed, oriented quasi-regular K-contact three-manifold (see §2.3).3This result follows after a particular choice of Vielbein for the gravitational Chern-Simons term
CS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 6. Eta-Invariants and Anomalies 101
we obtain for the integrand of Eq. (6.6.2) in §6.6 is derived in Prop. 6.7.15. Using an
observation from §6.5 that identifies the volume of the isotropy subgroup of the gauge
group GP , we identify the integrand of Eq. (6.6.2) with the contact analytic torsion T dC
defined in Def. 6.9.4. After formally identifying the signature of the contact operator D
with the η-invariant of D in §6.8, we obtain our fully rigorous definition of ZU(1)(X, k)
in Eq. (6.9.17) below, which is repeated in Eq. (6.1.6) above.
On the other hand, [Man98] provides a rigorous definition of the partition function
ZU(1)(X, k) that does not involve an a priori choice of a contact structure on X. The
formula for this is recalled in Eq. (6.9.18) below, and is the term ZU(1)(X, p, k) in Eq.
(6.1.7) of Prop. 6.1.5 above.
Our first main step in the proof of Prop. 6.1.5 is confirmation of the fact that the
Ray-Singer analytic torsion (See §4.3, and Eq.’s 4.3.16, 4.3.21) of X, T dRS, is identically
equal to the contact analytic torsion T dC (See Def. 6.9.4).4 We observe that this result
follows directly from [RS08, Theorem 4.2].
We also observe in Remark 6.7.20 that the quantities mX and nX that occur in Prop.
6.1.5 are equal. This leaves us with the main final step in the confirmation of Prop.
6.1.5, which involves a study of the η-invariants, η(?d), η(?D), that naturally show up
in ZU(1)(X, k), ZU(1)(X, k), respectively. This analysis is carried out in §6.10, where we
observe that the work of Biquard, Herzlich, and Rumin, [BHR07] is our most pertinent
reference. Our main observation here is that the quantum anomalies that occur in the
computation of ZU(1)(X, k) and ZU(1)(X, k) should, in an appropriate sense, be com-
pletely equivalent. In our case, these quantum anomalies are made manifest precisely in
4We consider the square roots thereof, viewed as densities on the moduli space of flat connectionsMP (see Eq. (2.5.57) for a definition of MP ).
Chapter 6. Eta-Invariants and Anomalies 102
the failure of the η-invariants to represent topological invariants. As observed by Wit-
ten in [Wit89], this is deeply connected with the fact that in order to actually compute
the partition function, one needs to make a choice that is tantamount to either a valid
gauge choice for representatives of gauge classes of connections, or in some other way by
breaking the symmetry of our problem. Such a choice for us is equivalent to a choice
of metric, which is encoded in the choice of a quasi-regular K-contact structure on our
manifold X. Witten observes in [Wit89] that the quantum anomaly that is introduced by
our choice of metric may be canceled precisely by adding an appropriate “counterterm”
to the η-invariant, η(?d). This recovers topological invariance and effectively cancels the
anomaly.5 This counterterm is found by appealing to the Atiyah-Patodi-Singer theorem
[APS75b], and is in fact identified as the gravitational Chern-Simons term
CS(Ag) :=1
4π
∫X
Tr(Ag ∧ dAg +2
3Ag ∧ Ag ∧ Ag), (6.1.8)
where Ag is the Levi-Civita connection on the spin bundle of X for the metric,
g = κ⊗ κ+ dκ(·, J ·), (6.1.9)
on our quasi-regular K-contact three manifold, (X,φ, ξ, κ, g). In particular, we use the
fact that,
η(?d)
4+
1
12
CS(Ag)
2π, (6.1.10)
is a topological invariant of X, after choosing the canonical framing. As is discussed in
§6.10, this leads us to conjecture that there exists an appropriate counterterm for the η-
invariant associated to the contact operator D that yields the same topological invariant
as in Eq. (6.1.10). More precisely, we conjecture that there exists a counterterm, CT ,
such that
eπi[η(?d)
4+ 1
12CS(Ag)
2π
]= e
πi4
[η(?D)+CT ], (6.1.11)
as topological invariants. We establish the following in Proposition 6.10.23,
5In this case, topological invariance is recovered only up to a choice of two-framing for X. Of course,there is a canonical choice of such framing as observed in [Ati90b].
Chapter 6. Eta-Invariants and Anomalies 103
6.1.12 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then6 there exists a counterterm, CT , such that eπi4
[η(?D)+CT ] is a topological invariant
that is identically equal to the topological invariant eπi[η(?d)
4+ 1
12CS(Ag)
2π
]. In fact, we have
CT =1
512
∫X
R2 κ ∧ dκ,
where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X.
This proposition is proven in §6.10 by appealing to the following result, which is
established using a “Kaluza-Klein” dimensional reduction technique for the gravitational
Chern-Simons term. This result is modeled after the paper [GIJP03], and is listed as
Proposition 6.10.20 (see Chapter 7 for a detailed study of this result).
6.1.13 Proposition. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact
three-manifold,
U(1) // X
Σ
.
Let gε := ε−1 κ⊗ κ+ π∗h. After a particular choice of Vielbein,7 then,
CS(Agε) =
(ε−1
2
)∫Σ
r ω +
(ε−2
2
)∫Σ
f 2 ω (6.1.14)
where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)
Hodge form of (Σ, h), and f ∈ Ω0orb(Σ) is the invariant field strength on (Σ, h).8 In
particular, the adiabatic limit of CS(Agε) vanishes:
limε→∞
CS(Agε) = 0. (6.1.15)
6This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
7See Equations (7.3.20) and (7.3.21).8In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx
0 + ϕ1dx1 + dx2, and dκ =
(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,
and fαβ =√h εαβ f, where f ∈ Ω0
orb(Σ) is called the invariant field strength on (X,h). See Chapter 7and §2.4 for more background.
Chapter 6. Eta-Invariants and Anomalies 104
Finally, as a consequence of these investigations, we are able to compute in Proposition
6.10.27 the U(1) Chern-Simons partition function fairly explicitly.
6.1.16 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then,9
η(?d) +1
3
CS(Ag)
2π= η(?D) +
1
512
∫X
R2 κ ∧ dκ
= 1− d
3+ 4
N∑j=1
s(αj, βj),
where 0 < d = c1(X) = c1(L) = n+∑N
j=1βjαj∈ Q and
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)∈ Q
is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for gcd(αj, βj) =
1) are the Seifert invariants of X. In particular, we have computed the U(1) Chern-
Simons partition function as:
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dC)1/2,
= kmXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dRS)1/2.
6.2 Structure Operators
At this point, we restrict the structure on our three-manifold and assume that the Seifert
structure is compatible with a contact metric structure (φ, ξ, κ, g) on X. In particular,
we restrict to the case of a quasi-regular K-contact manifold (see §2.3). Note that the as-
sumption that the Seifert structure on X comes from a quasi-regular K-contact structure
(φ, ξ, κ, g) is equivalent to assuming that X is a CR-Seifert manifold (see Prop. 2.3.5).
Recall the following
9This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 6. Eta-Invariants and Anomalies 105
6.2.1 Definition. A CR-Seifert manifold is a three-dimensional compact manifold en-
dowed with both a strictly pseudoconvex CR structure (H, J) and a Seifert structure,
that are compatible in the sense that the circle action ψ : U(1)→ Diff(X) preserves the
CR structure and is generated by a Reeb field ξ. In particular, given a choice of contact
form κ, the Reeb field is Killing for the associated metric g = κ⊗ κ+ dκ(·, J ·).
The assumption that X is CR-Seifert (hence quasi-regular K-contact) is sufficient to
ensure that the assumption in [BW05, Eq. 3.27], which states that the U(1)-action on
X, ψ : U(1)→ Diff(X), acts by isometries, is satisfied.
We now employ the natural Hodge star operator ?, induced by the metric g on X,
that acts on Ω•(X) taking k forms to 3 − k forms. We have chosen a normalization
convention such that ?1 = κ ∧ dκ and ?κ = dκ.10 Now define
6.2.2 Definition. Define the horizontal Hodge star operator to be the operator:
?H : Ωq(X)→ Ω2−q(H) q = 0, 1, 2,
defined for β ∈ Ωq(X) by
?H β = ?(κ ∧ β), (6.2.3)
where ? is the usual Hodge star operator on forms for the metric g = κ⊗ κ+ π∗h on X.
We have
6.2.4 Proposition. (See Prop. C.1.21) ?Hα = (−1)qιξ(?α) = ?(κ∧α) for all α ∈ Ωq(X),
0 ≤ q ≤ 2.11
We also have the following
10We choose h to be any (orbifold) Kahler metric on Σ which is normalized so that the corresponding(orbifold) Kahler form, ω ∈ Ω2
orb(Σ,R), pulls back to dκ. See §2.3 for more details.11Note that the statement is true for q = 3, but only trivially so.
Chapter 6. Eta-Invariants and Anomalies 106
6.2.5 Proposition. (See C.1.22) The following equalities hold
?H κ = 0 (6.2.6)
?H(κ ∧ dκ) = 0 (6.2.7)
?H1 = dκ. (6.2.8)
Lastly, we have the following
6.2.9 Proposition. (See Prop. C.1.19) ?2H = −1 on Ω1(H).
6.3 The Action
Our starting point in this section is the analogue of Eq. (5.0.4) for the U(1) Chern-Simons
partition function:
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4πS(A)
](6.3.1)
where SX,P (AP ) is the Chern-Simons invariant associated to P for AP a flat connection
on P . The derivation of Eq. (6.3.1) can be found in §4.4. It is obtained by expanding
the U(1) analogue of Eq. (5.0.4) around a critical point AP of the action. Our key
observation is that the action S(A) =∫XA ∧ dA −
∫X
(κ∧dA)2
κ∧dκ may now be expressed in
terms of the horizontal quantities of §6.2. Define the notation
S(A) :=
∫X
(κ ∧ dA)2
κ ∧ dκ(6.3.2)
so that we may write
S(A) = CS(A)− S(A), (6.3.3)
where CS(A) :=∫XA ∧ dA. Let us start with the term S(A). First, the term κ ∧ dA
in S(A) is equivalent to κ ∧ dHA since the vertical part of dA is annihilated by κ in the
wedge product. The term κ∧dAκ∧dκ is equivalent to ?(κ∧ dHA) by the properties of ? above.
Chapter 6. Eta-Invariants and Anomalies 107
By the definition of ?H , ?(κ∧ dHA) = ?HdHA (See Prop. C.1.26 for a proof of this). We
then have,
S(A) =
∫X
(κ ∧ dA)2
κ ∧ dκ
=
∫X
?H(dHA) ∧ κ ∧ dHA
=
∫X
κ ∧ [dHA ∧ ?H(dHA)]
We claim that S(A) is now expressed in terms of an inner product on Ω2H. More
generally, we define an inner product on Ωl(H) for 0 ≤ l ≤ 2:
6.3.4 Definition. Define the pairing 〈·, ·〉lκ : ΩlH × ΩlH → R as
〈α, β〉lκ := (−1)l∫X
κ ∧ [α ∧ ?Hβ] (6.3.5)
for any α, β ∈ ΩlH, 0 ≤ l ≤ 2.
6.3.6 Proposition. The pairing 〈·, ·〉lκ is an inner product on ΩlH.
Proof. It can be easily checked that this pairing is just the restriction of the usual L2-inner
product, 〈·, ·〉 : ΩlX × ΩlX → R,
〈α, β〉 :=
∫X
α ∧ ?β (6.3.7)
restricted to horizontal forms. i.e. for any β ∈ ΩlH, 0 ≤ l ≤ 2, we have ?β = κ ∧ ?Hβ.
We then have α ∧ ?β = (−1)lκ ∧ [α ∧ ?Hβ] for any α, β ∈ ΩlH, 0 ≤ l ≤ 2. Thus,
〈·, ·〉lκ = 〈·, ·〉 on ΩlH and therefore defines an inner product.
By our definition, we may now write S(A) = 〈dHA, dHA〉2κ. We make the following
6.3.8 Definition. Define the formal adjoint of dH , denoted d∗H , via:
〈d∗Hγ, φ〉l−1κ = 〈γ, dHφ〉lκ
for γ ∈ Ωl(H), φ ∈ Ωl−1(H) where l = 1, 2 and d∗Hγ = 0 for γ ∈ Ω0(H).
Chapter 6. Eta-Invariants and Anomalies 108
6.3.9 Proposition. d∗H = (−1)l ?H dH?H : Ωl(H) → Ωl−1(H), 0 ≤ l ≤ 2, where
Ω−1(H) := 0.
Proof. This just follows from the definition of d∗ relative to the ordinary inner product
〈·, ·〉, and the facts that 〈·, ·〉l−1κ is just this ordinary inner product restricted to horizontal
forms and d∗ = (−1)l ? d?.
Thus, we may now write S(A) = 〈A, d∗HdHA〉1κ and identify this piece of the action with
the second order operator d∗HdH on horizontal forms.
Now we turn our attention to the Chern-Simons part of the action CS(A) =∫XA ∧ dA.
We would like to reformulate this in terms of horizontal quantities as well. This is
straightforward to do; simply observe that dA = κ∧LξA+ dHA (See Prop. C.1.27 for a
proof). Thus, we have:
CS(A) =
∫X
A ∧ dA (6.3.10)
=
∫X
A ∧ [κ ∧ LξA+ dHA] (6.3.11)
=
∫X
A ∧ [κ ∧ LξA] +
∫X
A ∧ dHA (6.3.12)
=
∫X
A ∧ [κ ∧ LξA] (6.3.13)
where the last line follows from the fact that A∧dHA = 0 since both forms are horizontal.
Putting this all together, we may now express the total action S(A) in terms of horizontal
quantities as follows:
S(A) = CS(A)− S(A)
=
∫X
A ∧ [κ ∧ LξA] +
∫X
A ∧ [κ ∧ dH ?H dHA]
=
∫X
A ∧ [κ ∧ (Lξ + dH ?H dH)A]
Chapter 6. Eta-Invariants and Anomalies 109
6.4 The Contact Operator D
A surprising observation is that κ ∧ (Lξ + dH ?H dH) turns out to be well known. It is
the second order operator “D” that fits into the complex,
C∞(X)dH−→ Ω1(H)
D−→ Ω2(V )dH−→ Ω3(X) (6.4.1)
where,
Ω2(V ) := κ ∧ α | α ∈ Ω1(H) = κ ∧ Ω1(H), (6.4.2)
and for f ∈ C∞(X), dHf ∈ Ω1(H) stands for the restriction of df to H as usual, while
dH : Ω2(V )→ Ω3(X) (6.4.3)
is just de Rham’s differential restricted to Ω2(V ) in Ω2(X).
6.4.4 Definition. Denote the complex in Eq. (6.4.1) as (E , dH). Thus, we denote
D := dH in the middle degree.12
D is defined as follows:13 Since d induces an isomorphism
d0 : Ω1(V )→ Ω2(H), with d0(fκ) = fdκ|Λ2(H) (6.4.5)
then any α ∈ Ω1(H) admits a unique extension l(α) in Ω1(X) such that dl(α) belongs to
Ω2(V ); i.e. given any initial extension α of α, one has
l(α) = α− d−10 (dα)|Λ2(H) (6.4.6)
We then define
Dα := dl(α) (6.4.7)
12Note that we have abused notation in a few places in this thesis regarding the operator dH in middledegree. We will only view D = dH in terms of the complex (E , dH), and otherwise dH : Ω1(X)→ Ω2(H)is meant to denote the operator d : Ω1(X) → Ω2(X) followed by the projection of Ω2(X) onto Ω2(H).That is, dH : Ω1(X) → Ω2(H) is defined as dH := π d where π : Ω2(X) → Ω2(H) is the projectiondefined in Eq. (C.1.13). The notation should also be clear from the context.
13See [BHR07, §6].
Chapter 6. Eta-Invariants and Anomalies 110
We then have, [BHR07, Eq. 39]:
Dα = κ ∧ [Lξ + dH ?H dH ]α (6.4.8)
for any α ∈ Ω1(H). Thus,
S(A) =
∫X
A ∧ [κ ∧ (Lξ + dH ?H dH)A] (6.4.9)
=
∫X
A ∧DA (6.4.10)
= 〈A, ?DA〉 (6.4.11)
where 〈·, ·〉 is the usual L2 inner product on Ω1(X).
Alternatively, we make the following
6.4.12 Definition. Let D1 : Ω1(H)→ Ω1(X) denote the operator
D1 := Lξ + dH ?H dH (6.4.13)
and observe that we can also write S(A) = 〈A, ?HD1A〉1κ, identifying S(A) with the
operator ?HD1 on Ω1(H). Thus, we have proven the following
6.4.14 Proposition. The new action, S(A), as defined in Eq. (4.4.18), for the “shifted”
partition function of Eq. (6.3.1) can be expressed as a quadratic form on the space of
horizontal forms Ω1(H) as follows:
S(A) = 〈A, ?DA〉 (6.4.15)
or equivalently as,
S(A) = 〈A, ?HD1A〉1κ (6.4.16)
where D and D1 are the second order operators defined in Eq.’s 6.4.8 and 6.4.13, re-
spectively. 〈·, ·〉 is the usual L2 inner product on Ω1(X), and 〈·, ·〉1κ is defined in Eq.
(6.3.5).
Chapter 6. Eta-Invariants and Anomalies 111
6.5 Gauge Group and the Isotropy Subgroup
In order to extract anything mathematically meaningful out of this construction we will
need to divide out the action of the gauge group GP on AP . At this point we observe
that the gauge group GP ' Maps(X → U(1)) naturally descends to a “horizontal” action
on AP , which infinitesimally can be written as:
θ ∈ Lie(GP ) : A 7→ A+ dHθ (6.5.1)
Following [Sch79b], we let HA denote the isotropy subgroup of GP at a point A ∈ AP .
Note that HA can be canonically identified for every A ∈ AP , and so we simply write
H for the isotropy group. The condition for an element of the gauge group h(x) = eiθ(x)
to be in the isotropy group is that dHθ = 0, given definition 6.5.1 above. By [Rum94,
Prop. 12], we see that Lξθ = 0 since θ is harmonic.14 Therefore we have dθ = 0 since
d = dH + κ ∧ Lξ. Thus, the group H can be identified with the group of constant maps
from X into U(1); hence, is isomorphic to U(1). We let Vol(H) denote the volume of the
isotropy subgroup, computed with respect to the metric induced from GP , so that
Vol(H) =
[∫X
κ ∧ dκ]1/2
=
[n+
N∑j=1
βjαj
]1/2
, (6.5.2)
where [n; (α1, β1), . . . , (αN , βN)] are the Seifert invariants of our Seifert manifold X (See
§2.3). The last equality in Eq. (6.5.2) above follows from Eq. 3.22 of [BW05]. Let us
briefly derive Eq. (6.5.2). Let u, v ∈ Lie(GP ) ' Ω0(X) and recall that the metric on GP
is given by:
G(u, v) :=
∫X
u ∧ ?v. (6.5.3)
To calculate the volume of H ' U(1) < GP we let dΘ denote the standard measure on
U(1) such that∫U(1)
dΘ = 1 and obtain the induced measure√GΘdΘ on U(1) < GP .
14Note that θ is harmonic since dHθ = 0. We leave the details of this fact to the reader, who shouldconsult the actual proof of [Rum94, Prop. 12].
Chapter 6. Eta-Invariants and Anomalies 112
Since u, v ∈ H are viewed as constant functions in Lie(GP ) ' Ω0(X), we have
G(u, v) :=
∫X
u ∧ ?v = uv
∫X
?1 = uv
∫X
κ ∧ dκ. (6.5.4)
So√GΘ is constant on TH and may be identified as
√GΘ =
[∫Xκ ∧ dκ
]1/2. Then,
Vol(H) :=
∫U(1)
√GΘdΘ,
=√GΘ
∫U(1)
dΘ,
=√GΘ, since
∫U(1)
dΘ = 1,
=
[∫X
κ ∧ dκ]1/2
,
and we have justified Eq. (6.5.2).
6.6 The Partition Function
We now have
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA e[ik4πS(A)]
=Vol(GP )
Vol(H)
eπikSX,P (AP )
Vol(GP )
∫AP /GP
e[ik4πS(A)] [det′(d∗HdH)]
1/2µ
=eπikSX,P (AP )
Vol(H)
∫AP /GP
e[ik4πS(A)] [det′(d∗HdH)]
1/2µ (6.6.1)
where µ is the induced measure on the quotient space AP/GP and det′ denotes a regu-
larized determinant to be defined later. Since S(A) = 〈A, ?DA〉 is quadratic in A, we
may apply the method of stationary phase15 to evaluate the oscillatory integral (6.6.1)
exactly. We obtain,
(6.6.2)
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(H)
∫MP
eπi4
sgn(?D) [det′(d∗HdH)]1/2
[det′(k ? D)]1/2
ν
15Our main references for the method of stationary phase are [GS77], [Sch79a] and [Sch79b]. SeeAppendix A.1 for more details.
Chapter 6. Eta-Invariants and Anomalies 113
whereMP denotes the moduli space of flat connections modulo the gauge group16 and ν
denotes the induced measure on this space. Note that we have included a factor of k in
our regularized determinant since this factor occurs in the exponent multiplying S(A).
6.7 Zeta Function Determinants
We will use the following to define the regularized determinant of k ? D
6.7.1 Proposition. [Sch79b] Let H0, H1 be Hilbert spaces, and S : H1 → H1 and
T : H0 → H1 such that S2 and TT ∗ have well defined zeta functions with discrete spectra
and meromorphic extensions to C that are regular at 0 (with at most simple poles on
some discrete subset). If ST = 0, and S2 is self-adjoint, then
det′(S2 + TT ∗) = det′(S2) det′(TT ∗) (6.7.2)
Proof. This equality follows from the facts that S2TT ∗ = 0 and TT ∗S2 = 0 (i.e. these
operators commute), which both follow from ST = 0 and the fact that S2 and TT ∗ are
both self-adjoint.
Following the notation of Eq.’s (3)-(6) in section 2 of [Sch79b], we set the operators
S = k ?D and T = kdHd∗H on Ω1(H) and observe that ST = 0 since (6.4.1) is a complex.
With Prop. 6.7.1 as motivation, we make the formal definition
det′(k ? D) := C(k, J) · [det′(S2 + TT ∗)]1/2
[det′(TT ∗)]1/2(6.7.3)
where S2 + TT ∗ = k2(D∗D + (dHd∗H)2), TT ∗ = k2(dHd
∗H)2 and
C(k, J) := k(− 11024
∫X R2 κ∧dκ) (6.7.4)
is a function of R ∈ C∞(X), the Tanaka-Webster scalar curvature of X, which in turn
depends only on a choice of a compatible complex structure J ∈ End(H). That is, given
16See Def. 2.5.57 for a definition of MP .
Chapter 6. Eta-Invariants and Anomalies 114
a choice of contact form κ ∈ Ω1(X), the choice of complex structure J ∈ End(H) de-
termines uniquely an associated metric (See Prop. 2.2.15). We have defined det′(k ? D)
in this way to eliminate the metric dependence that would otherwise occur in the k-
dependence of this determinant. The motivation for the definition of the factor C(k, J)
comes explicitly from Prop. 6.7.13 below.
The operator
∆ := D∗D + (dHd∗H)2 (6.7.5)
is actually equal to the middle degree Laplacian defined in Eq. (10) of [RS08] and has
some nice analytic properties.17 In particular, it is maximally hypoelliptic and invertible
in the Heisenberg symbolic calculus.18 We define the regularized determinant of ∆ via
its zeta function19
ζ(∆)(s) :=∑
λ∈spec∗(∆)
λ−s. (6.7.6)
Note that our definition agrees with [RS08] up to a finite constant term, dimH1(E , dH).20
This is a consequence of the following
6.7.7 Proposition. [RS08, Prop. 2.2] Let (X, κ) be a contact three-manifold. The con-
tact complex (E , dH), defined in Def. 6.4.4 above, forms a resolution of the constant sheaf
R and its cohomology therefore coincides with the de Rham cohomology of X. Moreover,
the natural projection π : Ωk(X) → Ek,21 for k ≤ 1, and inclusion i : Ek → Ωk(X), for
k ≥ 2, induce an isomorphism between the two cohomologies.
Also, ζ(∆)(s) admits a meromorphic extension to C that is regular at s = 0 (See
17See Eq. (6.9.3) below for the full definition, and also Chapter 3 for some more background.18See Prop. 3.3.1 and also §3.2 for the relevant definitions.19We follow [RS08, Pg. 10] here.20Recall Def. 6.4.4 for a definition of the complex (E , dH). Also, it is noted in [RS08, Pg. 11] that
dimH1(E , dH) is finite by hypoellipticity.21Define π(α) := α− ιξ(α)κ, for α ∈ Ωk(X), for k ≤ 1.
Chapter 6. Eta-Invariants and Anomalies 115
Prop. 3.3.2). Thus, we define the regularized determinant of ∆ as
det′(∆) := e−ζ′(∆)(0) (6.7.8)
Let ∆0 := (d∗HdH)2 on Ω0(X), ∆1 := ∆ on Ω1(H) and define ζi(s) := ζ(∆i)(s). We claim
the following
6.7.9 Proposition. For any real number 0 < c ∈ R,
det′(c∆i) := cζi(0) det′(∆i) (6.7.10)
for i = 0, 1.
Proof. To prove this claim, recall that ζi(s) = ζ(∆i)(s) for i = 0, 1, scale as follows:
ζ(c∆i)(s) = c−sζ(∆i)(s). (6.7.11)
From here we simply calculate the scaling of the regularized determinants using the
definition
det′(∆i) := e−ζ′(∆i)(0) (6.7.12)
and the claim is proven.
The following will be useful.
6.7.13 Proposition. For ∆0 := (d∗HdH)2 on Ω0(X), ∆1 := ∆ on Ω1(H) (See. Eq.
(6.7.5)) and ζi(s) := ζ(∆i)(s), we have
ζ0(0)− ζ1(0) =
(− 1
512
∫X
R2 κ ∧ dκ)
+ dim Ker∆1 − dim Ker∆0 (6.7.14)
=
(− 1
512
∫X
R2 κ ∧ dκ)
+ dimH1(E , dH)− dimH0(E , dH).
where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X and κ ∈ Ω1(X) is our
chosen contact form as usual.
Chapter 6. Eta-Invariants and Anomalies 116
Proof. Let
ζ0(s) := dim Ker∆0 + ζ0(s)
ζ1(s) := dim Ker∆1 + ζ1(s)
denote the zeta functions as defined in [RS08]. From Prop. 3.3.4, one has that
ζ1(0) = 2ζ0(0)
for all 3-dimensional contact manifolds. By Theorem 3.3.6, one knows that on CR-Seifert
manifolds that
ζ0(0) = ζ(∆0)(0) = ζ(∆H)(0) =1
512
∫X
R2 κ ∧ dκ
Thus,
ζ1(0) =1
256
∫X
R2 κ ∧ dκ
By our definition of the zeta functions, which differ from that of [RS08] by constant
dimensional terms, we therefore have
ζ0(0) =1
512
∫X
R2 κ ∧ dκ− dim Ker∆0
ζ1(0) =1
256
∫X
R2 κ ∧ dκ− dim Ker∆1
Hence,
ζ0(0)− ζ1(0) =
[1
512
∫X
R2 κ ∧ dκ− dim Ker∆0
]−[
1
256
∫X
R2 κ ∧ dκ− dim Ker∆1
]=
(− 1
512
∫X
R2 κ ∧ dκ)
+ dim Ker∆1 − dim Ker∆0
=
(− 1
512
∫X
R2 κ ∧ dκ)
+ dimH1(E , dH)− dimH0(E , dH).
and the result is proven.
We now have the following
Chapter 6. Eta-Invariants and Anomalies 117
6.7.15 Proposition. The term inside of the integral of Eq. (6.6.2) has the following
expression in terms of the hypoelliptic Laplacians, ∆0 and ∆1, as defined in Prop. 6.7.13:
[det′(d∗HdH)]1/2
[det′(k ? D)]1/2
= knX[det′(∆0)]1/2
[det′(∆1)]1/4
(6.7.16)
where
nX :=1
2(dimH1(E , dH)− dimH0(E , dH)). (6.7.17)
Proof.
(6.7.18)
[det′(d∗HdH)]1/2
[det′(−k ?H D1κ)]
1/2= C(k, J)−1 · [det′(d∗HdH)2]
1/4 · [det′ k2(dHd∗H)2]
1/4
[det′(k2∆)]1/4
= C(k, J)−1 · kζ0(0)/2 [det′(∆0)]
1/4 · [det′(∆0)]1/4
kζ1(0)/2 [det′(∆1)]1/4
(6.7.19)
= C(k, J)−1 · k12
(ζ0(0)−ζ1(0)) [det′(∆0)]1/2
[det′(∆1)]1/4
= C(k, J)−1 · C(k, J) · knX [det′(∆0)]1/2
[det′(∆1)]1/4, Prop. 6.7.13,
= knX[det′(∆0)]1/2
[det′(∆1)]1/4
where the second last line comes from Eq. (6.7.14). Also note that d∗HdH and dHd∗H
have the same eigenvalues (by standard arguments), which allows us to proceed to Eq.
(6.7.19) from Eq. (6.7.18).
6.7.20 Remark. Note that by Prop. 6.7.7, the definition of nX (see Eq. (6.7.17)) here
is exactly equal to the quantity mX := 12(dimH1(X, d) − dimH0(X, d)) of [Man98, Eq.
5.18]. This shows that our partition function has the same k-dependence as that in
[Man98].
Chapter 6. Eta-Invariants and Anomalies 118
6.8 The Eta-Invariant
Next we regularize the signature sgn(?D) via the eta-invariant and set sgn(?D) =
η(?D)(0) := η(?D) where
η(?D)(s) :=∑
λ∈spec∗(?D)
(sgnλ)|λ|−s (6.8.1)
Finally, we may now write the result for our partition function
(6.8.2)
ZU(1)(X, p, k) = knXeπikSX,P (A0)eπi4η(?D)
∫MP
1
Vol(H)
[det′(∆0)]1/2
[det′(∆1)]1/4
ν
where nX := 12(dimH1(E , dH) − dimH0(E , dH)). Note that ν is a measure on MP
22
relative to the horizontal structure on the tangent space of MP .
6.9 Torsion
Now we will study the quantity 1Vol(H)
[det′(∆0)]1/2
[det′(∆1)]1/4ν inside of the integral in Eq. (6.8.2),
and in particular how it is related to the analytic contact torsion TC . First, recall that,
[RS08, Eq. 16]:
TC := exp
(1
4
3∑q=0
(−1)qw(q)ζ ′(∆q)(0)
)(6.9.1)
where
w(q) =
q if q ≤ 1,
q + 1 if q > 1.
(6.9.2)
22MP denotes the moduli space of flat connections modulo the gauge group. See Def. 2.5.57.
Chapter 6. Eta-Invariants and Anomalies 119
in the case where dim(X) = 3. Note that we have chosen a sign convention that leads to
the inverse of the definition of TC in [RS08]. Recall, [RS08, Eq. 10]:
∆q =
(d∗HdH + dHd
∗H)2 if q = 0, 3,
D∗D + (dHd∗H)2 if q = 1.
DD∗ + (d∗HdH)2 if q = 2.
(6.9.3)
We would, however, like to work with torsion when viewed as a density on the determinant
line
| detH•(E , dH)∗| := | detH0(E , dH)| ⊗ | detH1(E , dH)∗|
⊗ | detH2(E , dH)| ⊗ | detH3(E , dH)∗|
We follow [RS73] and [Man98] and make the analogous definition.
6.9.4 Definition. Define the analytic torsion as a density as follows
T dC := TC · δ| detH•(E,dH)|
where TC is as defined in Eq. (6.9.1), and
δ|detH•(E,dH)| := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν
qbq|(−1)q
where νq1 , · · · , νqbq is an orthonormal basis for the space of harmonic contact forms
Hq(E , dH) with the inner product defined in Eq. (6.3.5). Note that Hq(E , dH) is canon-
ically identified with the cohomology space Hq(E , dH), and bq := dim(Hq(E , dH)) is the
qth contact Betti number.
Let
ν(q) := νq1 ∧ · · · ∧ νqbq
and write the analytic torsion of a compact connected Seifert 3-manifold X as
T dC = TC × |ν(0)| ⊗ |ν(1)|−1 ⊗ |ν(2)| ⊗ |ν(3)|−1. (6.9.5)
Chapter 6. Eta-Invariants and Anomalies 120
In terms of regularized determinants, we have
TC =[(det′(∆0))0 · (det′(∆1))1 · (det′(∆2))−3 · (det′(∆3))4
]1/4(6.9.6)
where ∆q, 0 ≤ q ≤ 3, denotes the Laplacians on the contact complex as defined in [RS08,
Eq. 10] and recalled in Eq. (6.9.3) above. This notation agrees with our notation for
∆0, ∆1 as in Eq. (6.7.10). The Hodge ?-operator induces the equivalences ∆q ' ∆3−q
and allows us to write
TC =[(det′(∆0))0 · (det′(∆1))1 · (det′(∆2))−3 · (det′(∆3))4
]1/4(6.9.7)
=det′(∆0)
(det′(∆1))1/2(6.9.8)
Also, from the isomorphisms Hq(X,R) ' Hq(E , dH) of Prop. 6.7.7, we have Poincare
duality Hq(E , dH) ' H3−q(E , dH)∗, and therefore
T dC = TC × |ν0|⊗2 ⊗ (|ν1|−1)⊗2 (6.9.9)
Moreover, by Prop. 3.3.7, Hq(E , dH) = Hq(X,R) ⊂ Ωq(X), and thus any orthonormal
basis ν(0) of H0(E , dH) ' R is a constant such that
|ν(0)| =[∫
X
κ ∧ dκ]−1/2
. (6.9.10)
Let us briefly justify Eq. (6.9.10). This is a direct consequence of the definition of the
L2 metric || · ||L2 on H0(E , dH) and the fact that we are taking ||ν(0)||L2 = 1. That is,
consider
1 = ||ν(0)||2L2 ,
:=
∫X
ν(0) ∧ ?ν(0),
= |ν(0)|2∫X
?1,
= |ν(0)|2∫X
κ ∧ dκ.
Thus, |ν(0)| =[∫Xκ ∧ dκ
]−1/2and we have justified Eq. (6.9.10). Also, recall that the
tangent space TAMP ' H1(E , dH) ' H1(X,R), at any point A ∈MP . The measure ν on
Chapter 6. Eta-Invariants and Anomalies 121
MP that occurs in Eq. (6.8.2) is defined relative to the metric on H1(E , dH) ' H1(E , dH),
which can be identified with the usual L2-metric on forms. Thus the measure ν may
be identified with the inverse of the density |ν(1)| by dualizing the orthogonal basis
ν11 , . . . , ν
1b1 for H1(E , dH); i.e.
ν = |ν(1)|−1 = |ν11 ∧ · · · ∧ ν1
b1|−1 (6.9.11)
Putting together equations 6.9.8, 6.9.10, 6.9.11 into Eq. (6.9.9), we have
T dC = TC × |ν0|⊗2 ⊗ (|ν1|−1)⊗2 (6.9.12)
=det′(∆0)
(det′(∆1))1/2·[∫
X
κ ∧ dκ]−1
ν⊗2 (6.9.13)
= Vol(H)−2 det′(∆0)
(det′(∆1))1/2· ν⊗2 (6.9.14)
We have thus proven the following,
6.9.15 Proposition. The contact analytic torsion, when viewed as a density T dC as in
definition 6.9.4, can be identified as follows:
(T dC)1/2 =1
Vol(H)
[det′(∆0)]1/2
[det′(∆1)]1/4
ν (6.9.16)
Our partition function is now
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4η(?D)
∫MP
(T dC)1/2 (6.9.17)
This partition function should be completely equivalent to the partition function defined
in [Man98, Eq. 7.27]:
ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi4η(?d)
∫MP
(T dRS)1/2. (6.9.18)
Our goal in the remainder is to show that this is indeed the case. Our first observation is
that (T dC)1/2 is equal to the Ray-Singer torsion (T dRS)1/2 that occurs in [Man98, Eq. 7.27].
This follows directly from [RS08, Theorem 4.2]; note that their sign convention makes
TC the inverse of our definition (See Prop. C.2.21 for a proof and §C.2 in general).
Chapter 6. Eta-Invariants and Anomalies 122
6.10 Regularizing the Eta-Invariants
Since we have seen that our k-dependence matches that in [Man98] (i.e. mX = nX ; cf.
Remark 6.7.20), the only thing left to do is to reconcile the eta invariants, η(?D) and
η(?d). As observed in [Wit89], the correct quantity to consider is
η(?d)
4+
1
12
CS(Ag)
2π. (6.10.1)
where,
CS(Ag) =1
4π
∫X
Tr(Ag ∧ dAg +2
3Ag ∧ Ag ∧ Ag) (6.10.2)
is the gravitational Chern-Simons term, with Ag the Levi-Civita connection on the spin
bundle of X for a given metric g on X. See §4.3 for an exposition on the regularization
of η(?d) in Eq. (6.10.1).23 It was noticed in [Wit89] that in the quasi-classical limit,
quantum anomalies can occur that can break topological invariance. Invariance may be
restored in this case only after adding a counterterm to the eta invariant. Our job then
is to perform a similar analysis for the eta invariant η(?D), which depends on a choice
of metric. Of course, our choice of metric is natural in this setting and is adapted to the
contact structure. One possible approach is to consider variations over the space of such
natural metrics and calculate the corresponding variation of the eta invariant, giving us
a local formula for the counterterm that needs to be added. Such a program has already
been initiated in [BHR07].
Our starting point is the conjectured equivalence that results from the identification
of Eq.’s 6.9.17 and 6.9.18:
eπi[η(?d)
4+ 1
12CS(Ag)
2π
]“=”e
πi4
[η(?D)+CT ] (6.10.3)
where CT is some appropriate counterterm that yields an invariant comparable to the
left hand of this equation. As noted in §4.3, the left hand side of this equation depends
23See Eq. (4.3.25) in particular.
Chapter 6. Eta-Invariants and Anomalies 123
on a choice of 2-framing on X, and since we have a rule (See Eq. (4.3.26)) for how the
partition function transforms when the framing is twisted, we basically have a topological
invariant. Alternatively, as also noted in §4.3, one can use the main result of [Ati90b]
and fix the canonical 2-framing on TX ⊕ TX. We therefore expect the same type of
phenomenon for the right hand side of this equation, having at most a Z-dependence on
the regularization of our eta invariant, along with a rule that tells us how the partition
function changes when our discrete invariants are “twisted,” once again yielding a topo-
logical invariant.
Let us first make the statement of the conjecture of Eq. (6.10.3) more precise. We
should have the following
6.10.4 Conjecture. (X,φ, ξ, κ, g) a closed quasi-regular K-contact three-manifold. Then
there exists a counterterm, CT , such that
eπi4
[η(?D)+CT ]
is a topological invariant that is identically equal to the topological invariant
eπi[η(?d)
4+ 1
12CS(Ag)
2π
],
where CS(Ag) and all relevant operators are defined with respect to the metric g on X
and we use the canonical 2-framing [Ati90b].
Our regularization procedure for η(?D) will be quite different than that used for
η(?d). Since we are restricted to a class of metrics that are compatible with our contact
structure, we are really only concerned with finding appropriate counterterms for η(?D)
that will eliminate our dependence on the choice of contact form κ and complex structure
J ∈ End(H). In the case of interest, we observe that our regularization may be obtained
in one stroke by introducing the renormalized η-invariant, η0(X, κ), of X that is discussed
in [BHR07, §3]. Before giving the definition of η0(X, κ), we require the following
Chapter 6. Eta-Invariants and Anomalies 124
6.10.5 Lemma. [BHR07, Lemma 3.1] Let (X, J, κ) be a strictly pseudoconvex pseudo-
hermitian 3-manifold. The η-invariants of the family of metrics gε := ε−1κ⊗κ+dκ(·, J ·)
have a decomposition in homogeneous terms:
η(gε) =2∑
i=−2
ηi(X, κ)εi. (6.10.6)
The terms ηi for i 6= 0 are integrals of local pseudohermitian invariants of (X, κ), and
the ηi for i > 0 vanish when the Tanaka-Webster torsion, τ , vanishes.
We then make the following
6.10.7 Definition. Let (X, κ) be a compact strictly pseudoconvex pseudohermitian 3-
dimensional manifold. The renormalized η-invariant η0(X, κ) of (X, κ) is the constant
term in the expansion of Eq. (6.10.6) for the η-invariants of the family of metrics gε :=
ε−1κ⊗ κ+ dκ(·, J ·).
Our assumption that X is K-contact ensures that the Reeb flow preserves the metric.
In this situation, it is known that the Tanaka-Webster torsion necessarily vanishes.24 In
the case where the torsion of (X, κ) vanishes, the terms ηi(X, κ) in Eq. (6.10.6) vanish
for i > 0, so that when ε→∞, one has
η0(X, κ) = limε→∞
η(gε) := ηad (6.10.8)
The limit ηad is known as the adiabatic limit and has been studied in [BC89] and [Dai91],
for example. The adiabatic limit is the case where the limit is taken as ε goes to infinity,
ηad := limε→∞
η(gε), (6.10.9)
while the renormalized η-invariant, η0(X, κ), is naturally interpreted as the constant term
in the asymptotic expansion for (η(gε)) in powers of ε, when ε goes to 0. This reverse
process of taking ε to 0 is also known as the diabatic limit. When torsion vanishes (i.e.
24See [BHR07, §3].
Chapter 6. Eta-Invariants and Anomalies 125
when the Reeb flow preserves the metric), Eq. (6.10.8) is the statement that the diabatic
and adiabatic limits agree. One of the main challenges for our future work will be to
extend beyond the case where torsion vanishes. This will naturally involve the study of
the diabatic limit. For now, we are restricted to the case of vanishing torsion. In this
case, the main result that we will use is the following
6.10.10 Theorem. [BHR07, Theorem 1.4] Let X be a compact CR-Seifert 3-manifold,
with U(1)-action generated by the Reeb field of an U(1)-invariant contact form κ. If R
is the Tanaka-Webster curvature of (X, κ) and D is the middle degree operator of the
contact complex (cf. Eq. (6.4.1) and 6.4.13), then
η0(X, κ) = η(?D) +1
512
∫X
R2 κ ∧ dκ. (6.10.11)
Theorem 6.10.10 compels us to conjecture that CT = 1512
∫XR2 κ∧dκ. Our motivation
for this comes from the fact that η0(X, κ) is a topological invariant in our case. We have
the following,
6.10.12 Theorem. [BHR07, Remark 9.6 and Eq. 27] If X is a CR-Seifert manifold,
then η0(X, κ) is a topological invariant and
η0(X, κ) = 1− d
3+ 4
N∑j=1
s(αj, βj), (6.10.13)
where 0 < d ∈ Q is the degree of X as a compact U(1)-orbifold bundle and
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)(6.10.14)
is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for (αi, βi) =
1 relatively prime) are the Seifert invariants of X.
6.10.15 Remark. Note that we have chosen the opposite sign convention of [BHR07] for
the orbifold first Chern number d = c1(X) in this thesis. Thus, d appears with a plus
sign in Theorem 6.10.12 in [BHR07].
Chapter 6. Eta-Invariants and Anomalies 126
Thus, we are led to consider the natural topological invariant eπi4
[η0(X,κ)] and how it
compares with the topological invariant eπi[η(?d)
4+ 1
12CS(Ag)
2π
]. We consider the limit
limε→∞
eπi[η(?εd)
4+ 1
12CS(Agε )
2π
](6.10.16)
where gε = ε−1κ⊗ κ+ dκ(·, J ·) is the natural metric associated to X. On the one hand,
since this is a topological invariant, and is independent of the metric, we must have
limε→∞
eπi[η(?εd)
4+ 1
12CS(Agε )
2π
]= e
πi[η(?d)
4+ 1
12CS(Ag)
2π
]. (6.10.17)
where we take g1 := g so that ?g1 := ?.
On the other hand, since η(gε) = η(?εd) by definition, and we know that its limit
exists as ε→∞ (in fact η0(X, κ) = limε→∞ η(gε)), we have
limε→∞
eπi[η(?εd)
4+ 1
12CS(Agε )
2π
]= e
πi[η0(X,κ)
4+
limε→∞112
CS(Agε )2π
]. (6.10.18)
Thus, we have
eπi[η(?d)
4+ 1
12CS(Ag)
2π
]= e
πi[η0(X,κ)
4
]eπi
limε→∞112
CS(Agε )2π
. (6.10.19)
We therefore see that if we can understand the limit limε→∞112
CS(Agε )2π
, we will obtain
crucial information for our problem. The following has been established using a “Kaluza-
Klein” dimensional reduction technique modeled after the paper [GIJP03] (see Chapter
7 and Prop. 7.3.40),
6.10.20 Proposition. [McL10] Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact
three-manifold,
U(1) // X
Σ
.
Let gε := ε−1 κ⊗ κ+ π∗h. After a particular choice of Vielbein,25 then,
CS(Agε) =
(ε−1
2
)∫Σ
r ω +
(ε−2
2
)∫Σ
f 2 ω (6.10.21)
25See Equations (7.3.20) and (7.3.21).
Chapter 6. Eta-Invariants and Anomalies 127
where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)
Hodge form of (Σ, h), and f ∈ Ω0orb(Σ) is the invariant field strength on (Σ, h).26 In
particular, the adiabatic limit of CS(Agε) vanishes:
limε→∞
CS(Agε) = 0. (6.10.22)
Proposition 6.10.20 combined with Eq. (6.10.19) and Theorem 6.10.10 gives us the
following,
6.10.23 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then27 there exists a counterterm, CT , such that eπi4
[η(?D)+CT ] is a topological invariant
that is identically equal to the topological invariant eπi[η(?d)
4+ 1
12CS(Ag)
2π
]. In fact, we have
CT =1
512
∫X
R2 κ ∧ dκ,
where R ∈ C∞(X) is the Tanaka-Webster scalar curvature of X.
Prop. 6.10.23 is the final equivalence that was needed to establish our main equiva-
lence of the partition functions Z and Z. We thus have the following
6.10.24 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
If,
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (η(?D)+ 1
512
∫X R2 κ∧dκ)
∫MP
(T dC)1/2 (6.10.25)
and (see §4.3),
ZU(1)(X, p, k) = kmXeπikSX,P (AP )eπi(η(?d)
4+ 1
12CS(Ag)
2π
) ∫MP
(T dRS)1/2 (6.10.26)
26In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx0 + ϕ1dx
1 + dx2, and dκ =(∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx
0 ∧ dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor,and fαβ =
√h εαβ f, where f ∈ Ω0
orb(Σ) is called the invariant field strength on (X,h). See Chapter 7and §2.4 for more background.
27This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 6. Eta-Invariants and Anomalies 128
then,28
ZU(1)(X, k) = ZU(1)(X, k)
as topological invariants.
Given Proposition 6.10.23 and Theorem 6.10.12, we conclude the following as an
immediate consequence,
6.10.27 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold.
Then,29
η(?d) +1
3
CS(Ag)
2π= η(?D) +
1
512
∫X
R2 κ ∧ dκ
= 1− d
3+ 4
N∑j=1
s(αj, βj),
where 0 < d = c1(X) = c1(L) = n+∑N
j=1βjαj∈ Q and
s(α, β) :=1
4α
α−1∑k=1
cot
(πk
α
)cot
(πkβ
α
)∈ Q
is the classical Rademacher-Dedekind sum, where [n; (α1, β1), . . . , (αN , βN)] (for gcd(αj, βj) =
1) are the Seifert invariants of X. In particular, we have computed the U(1) Chern-
Simons partition function as:
ZU(1)(X, p, k) = knXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dC)1/2,
= kmXeπikSX,P (AP )eπi4 (1− d
3+4∑Nj=1 s(αj ,βj))
∫MP
(T dRS)1/2.
Finally, we are justified in making the following
6.10.28 Definition. Let k ∈ Z be an (even) integer, and (X,φ, ξ, κ, g) a closed, quasi-
regular K-contact three-manifold. The shifted U(1) Chern-Simons partition function,
28This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
29This result follows after a particular choice of Vielbein for the gravitational Chern-Simons termCS(Ag). See Equations (7.3.20) and (7.3.21).
Chapter 6. Eta-Invariants and Anomalies 129
ZU(1)(X, k), is the rigorous quantity
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k) (6.10.29)
where,
ZU(1)(X, p, k) := knXeπikSX,P (AP )eπi4 (η(?D)+ 1
512
∫X R2 κ∧dκ)
∫MP
(T dC)1/2 (6.10.30)
where nX = 12(dimH1(E , dH)− dimH0(E , dH)).
Of course, Prop. 6.10.24 is the statement that Def.’s 4.3.22 and 6.10.28 agree.
Chapter 7
Gravitational Chern-Simons and the
Adiabatic Limit
In this chapter we compute the gravitational Chern-Simons term explicitly for an adi-
abatic family of metrics using standard methods in general relativity. We use the fact
that our base three-manifold is a quasi-regular K-contact manifold heavily in this com-
putation. Our key observation is that this geometric assumption corresponds exactly to
a Kaluza-Klein Ansatz for the metric tensor on our three manifold, which allows us to
translate our problem into the language of general relativity. Similar computations have
been performed in [GIJP03], although not in the adiabatic context.
7.1 Introduction
The primary goal of this chapter is to explicitly compute the adiabatic limit
limε→0
CS(Agε)
2π, (7.1.1)
where,
CS(Agε) :=1
4π
∫X
Tr(Agε ∧ dAgε +2
3Agε ∧ Agε ∧ Agε), (7.1.2)
130
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 131
is the gravitational Chern-Simons term with Agε the Levi-Civita connection on the spin
bundle of X for the family of metrics,
gε = ε κ⊗ κ+ dκ(·, J ·), (7.1.3)
on a quasi-regular K-contact three manifold, (X,φ, ξ, κ, g).
The main motivation for the considerations of this chapter come directly from our work
in Chapter 6. In particular, this chapter provides one possible proof of Prop. 6.10.20,
which is the key to establishing Prop.’s 6.10.23, 6.10.24 and 6.10.27. In §6.10, we are
naturally led to compute the adiabatic limit of the regularized eta-invariant,
η(?εd)
4+
1
12
CS(Agε)
2π, (7.1.4)
by considering the limits of each term in Eq. (7.1.4) separately. By the Atiyah-Patodi-
Singer theorem Eq. (7.1.4) is a topological invariant,1 and one of our objectives in
Chapter 6 was to compute this topological invariant explicitly. It turns out that the
computation of adiabatic limit of Eq. (7.1.4) is all that is required to obtain this explicit
identification. This fact led us to study the limit of Eq. (7.1.1).
7.1.5 Remark. In this chapter we consider ε as opposed to ε−1 in our computations (as
in Eq.’s (7.1.1), (7.1.2) and (7.1.3)). This is simply a matter of notational convenience.
The gravitational Chern-Simons term was first introduced in the physics literature
by S. Deser, R. Jackiw, G. ’t Hooft and S. Templeton (cf. [DJT82], for example). The
reduction of the gravitational Chern-Simons term from three to two dimensions was sub-
sequently investigated in [GIJP03] for an abstract three-dimensional space-time. A key
observation in [GIJP03] is that setting a Kaluza-Klein Ansatz for the metric tensor ef-
fects a reduction from three to two dimensions for the gravitational Chern-Simons term.
1Note that Eq. (7.1.4) actually depends on a choice of 2-framing for X, and by a result of Atiyah[Ati90b] there exists a canonical choice of such framing. In our case, we explicitly choose a framing viaa particular choice of Vielbein (see Eq. (7.3.20) below), and work with this throughout this chapter. Inparticular, we may view Eq. (7.1.4) as a topological invariant without ambiguity.
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 132
Our first observation is that the natural associated metric for a quasi-regular K-contact
three-manifold satisfies this Ansatz. Although the work in [GIJP03] necessarily computes
the gravitational Chern-Simons term for our class of three-manifolds, we must extend the
results of [GIJP03] to perform this calculation for the adiabatic family of metrics in Eq.
(7.1.3).
We note that the terminology adiabatic limit in this chapter is meant to describe the
limit in Eq. (7.1.1) with respect to the adiabatic family of metrics in Eq. (7.1.3). We
borrow this terminology from [BHR07], where one considers the adiabatic limit for the
family of eta-invariants,
η(?εd), (7.1.6)
where d : Ω1(X)→ Ω2(X) is the standard exterior derivative on forms, ?ε are the Hodge
star operators for the family of metrics in Eq. (7.1.3), and the eta-invariant is defined as
usual,
η(?d)(s) :=∑
λ∈spec∗(?d)
(sgnλ)|λ|−s, (7.1.7)
so that,
η(?d) := η(?d)(0). (7.1.8)
It is shown in [BHR07], for example, that the adiabatic limit,
limε→0
η(?εd), (7.1.9)
exists, and in fact, for the case of quasi-regular K-contact three-manifolds, is a topologi-
cal invariant of X. As is noted in [BHR07], the adiabatic limit has been known for some
time, and has been studied in [BC89] and [Dai91], for example.
We note that our computations are modeled on [GIJP03]. The novelty here is the
introduction of the adiabatic family of metrics in Eq. (7.1.3), for which one must be
careful to keep track of the ε dependence in the explicit computation of the gravitational
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 133
Chern-Simons term of Eq. (7.1.2).
The first thing we do is find an explicit formula for the family of metrics in Eq. (7.1.3)
relative to a special coordinate system that is adapted to our geometric situation. We
then observe that the result we obtain is precisely the Kaluza-Klein Ansatz of [GIJP03].
This allows us to carry the rest of our analysis out in parallel with [GIJP03]. We then
compute the Christoffel symbols for the Levi-Civita connection for the family of metrics
in Eq. (7.1.3). Using the formulae for the Christoffel symbols and Vielbein, we compute
the components of the spin connection Agε and directly evaluate CS(Agε). Our compu-
tation yields a formula for CS(Agε) in “reduced” terms as in [GIJP03] and also provides
an explicit identification of the ε-dependence for this quantity. We note that the class of
quasi-regular K-contact three-manifolds are necessarily U(1)-bundles that fiber over an
orbifold surface Σ = Σ (See §2.3). Our main result is the following:
7.1.10 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold,
U(1) // X
Σ
.
Let gε := ε κ⊗ κ+ π∗h. After a particular choice of Vielbein,2 then,
CS(Agε) =( ε
2
)∫Σ
r ω +
(ε2
2
)∫Σ
f 2 ω (7.1.11)
where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)
Hodge form of (Σ, h), and f ∈ Ω0orb(X) is the invariant field strength on (Σ, h).3 In
particular, the adiabatic limit of CS(Agε) vanishes:
limε→0
CS(Agε) = 0. (7.1.12)
2See Equations (7.3.20) and (7.3.21).3In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx
0 + ϕ1dx1 + dx2, and dκ =
(∂0ϕ1−∂1ϕ0)dx0∧dx1 = f01dx0∧dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor, and
fαβ =√h εαβ f, where f ∈ Ω0
orb(X) is called the invariant field strength on (X,h). See §2.4 for morebackground.
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 134
As outlined in chapter 6, this allows us to compute the regularized eta-invariant
η(?εd)
4+
1
12
CS(Agε)
2π, (7.1.13)
explicitly, and in fact is an important step in the computation of the U(1) Chern-Simons
partition function as a topological invariant for the class of quasi-regular K-contact three
manifolds.
7.2 Local Formulation of Gravitational Chern-Simons
Term
In order to perform the computation of the gravitational Chern-Simons term in Eq.
(7.1.2) explicitly, we will adopt the conventions of [GIJP03]; indexing everything in sight
and working in local coordinates. To this end, we express the gravitational Chern-Simons
term in a coordinate system x0, x1, x2 for a local chart U ⊂ X as follows:
CS(Agε) :=1
4π
∫X
d3x εµνλ Tr
((Agε)µ∂ν(A
gε)λ +2
3(Agε)µ(Agε)ν(A
gε)λ
), (7.2.1)
where µ, ν, λ ∈ 0, 1, 2, and εµνλ is the three-dimensional Levi-Civita symbol, normalized
so that ε012 = 1, and defined for any σ ∈ S3 = permutations of 0, 1, 2 by:
εσ(012) := (−1)|σ|, (7.2.2)
where |σ| denotes the sign of σ as a permutation, so that
|σ| :=
0 if σ is even,
1 if σ is odd.
(7.2.3)
Note that repeated indices are allowed, and our definition of εµνλ implies that εµνλ = 0
whenever any of µ, ν, λ ∈ 0, 1, 2 are equal.
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 135
Let ∇G : Γ(TX) → Γ(T ∗X ⊗ TX) denote the standard Levi-Civita connection asso-
ciated to the metric G on X. Let
Γλµν :=1
2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) (7.2.4)
be the Christoffel symbols relative to the coordinate basis x0, x1, x2 for the Levi-Civita
connection, i.e.
(∇G)∂µ∂ν = Γλµν∂λ. (7.2.5)
Our computation is facilitated by the basic relationship between the spin connection AG,4
and the Levi-Civita connection ∇G, [GIJP03, Eq. 2.17]:
[(AG)µ]AB = EAν E
λBΓνµλ − Eλ
B∂µEAλ . (7.2.6)
where EA ∈ Γ|U(TX) and EA ∈ Γ|U(T ∗X) are the Vielbein defined in §2.4. Our goal
then is to compute the Christoffel symbols for the family of metrics G = gε, which will
give us an explicit formula for the spin connections [(AG)µ]AB. We note that in three
dimensions, the spin connection is anti-symmetric in A,B,
[(AG)µ]AB := ηAC [(AG)µ]CB = −[(AG)µ]BA, (7.2.7)
and we may use this fact to write
[(AG)µ]AB := ηAC [(AG)µ]CB = εABCACµ , (7.2.8)
where ACµ is a vector-valued one-form defined by this relation. We then obtain a slightly
simpler expression for CS(AG), [GIJP03, Eq. 2.22]:
CS(Agε) = − 1
4π
∫X
d3x εµνλ(
2ηABAAµ∂νA
Bλ −
2
3εABCA
AµA
Bν A
Cλ
)(7.2.9)
= − 1
2π
∫X
d3x εµνλ(ηABA
Aµ∂νA
Bλ
)+
1
π
∫X
d3x detAAµ (7.2.10)
Note that we have suppressed the Agε notation in our integrals to just A. Eq. (7.2.8)
will be the result that we use to perform our computation directly.
4Note that we will generally write the notation G for our metric interchangeably with gε. We willneed to explicitly make the identification G = gε later, but for now this simplfies notation.
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 136
7.3 Computation of Gravitational Chern-Simons Term
The first quantities that we wish to compute are the Christoffel symbols. Before we
can do this, however, we will need to find a useful expression for our family of metrics,
G = gε, in an “arbitrary” coordinate system x0, x1, x2 on X that reflects our geometric
situation. We follow [Ber47, pg. 265], and introduce a “special coordinate system.” Such
a coordinate system is adapted to our geometric situation in the following sense: We
define our coordinates via the local decomposition of X given by a local trivialization
from its bundle structure
π−1(U) ' U × S1, (7.3.1)
where U ⊂ Σ is any open subset of Σ.
7.3.2 Remark. Note that Σ is an orbifold. All of our considerations are completely valid
for the orbifold case by the results of Ichiro Satake [Sat57]. In particular, the notions
of (co-)tangent bundles, (co-)tangent vectors, forms, curvature, integration, Riemannian
metrics, orthogonal frames, etc... all have rigorously defined orbifold counterparts. See
§2.1 for more details on orbifolds. For example, the decomposition π−1(U) ' U × S1
assumes that U ⊂ R2 is an orbifold coordinate chart on Σ.
If ξ denotes the Reeb vector field on our quasi-regular K-contact manifold, (X,φ, ξ, κ, g),
then our coordinate system is chosen such that ξp = [0, 0, 1], for any p ∈ U , and the
first two coordinates x0, x1 coincide with the coordinates on our base orbifold Σ. We
should note that such a choice of coordinates does not necessarily respect the contact
structure, TX ' H⊕Rξ, of our K-contact manifold (X,φ, ξ, κ, g); i.e. for the coordinates
x0, x1, the associated vector fields ∂∂x0 ,
∂∂x1 are not necessarily horizontal vector fields.
The vector fields ∂∂x0 ,
∂∂x1 may have components in the Reeb direction:
∂
∂xα= hα + ϕα
∂
∂x2, α ∈ 0, 1, (7.3.3)
where hα is the horizontal component of ∂∂xα
, and ϕα∂∂x2 is its vertical component in the
direction of the Reeb field. Clearly, we have chosen our coordinates so that the local
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 137
vector field ∂∂x2 coincides with the Reeb direction. We now wish to express our family of
metrics in this coordinate system. By definition:
gε = ε κ⊗ κ+ π∗h. (7.3.4)
Evaluating this in our coordinate system yields:
Gµν = gε =
hαβ + εϕαϕβ εϕα
εϕβ ε
. (7.3.5)
Our matrix is indexed with the understanding that α, β ∈ 0, 1 and µ, ν ∈ 0, 1, 2
index the entire matrix.5
7.3.6 Remark. Note that the K-contact condition (i.e. the Reeb field is Killing for the
metric G = gε) is crucial for our analysis and ensures that the quantities, hαβ, ϕα, are
independent of the third local coordinate x2, [Ber47, Eq. 17.60]. It is precisely this
condition that makes our computation of the gravitational Chern-Simons term feasible.
This result is easy to see in our special coordinate system x0, x1, x2 described above.
Recall the definition of the Lie derivative,
Lξg :=d
dt
∣∣∣t=0φ∗tg, (7.3.7)
where φt : X → X is the flow of ξ. In our coordinate system, φt(x0, x1, x2) = (x0, x1, x2 +
t) and the condition ddt
∣∣∣t=0φ∗tg = 0 is equivalent to d
dt
∣∣∣t=0gφt = d
dt
∣∣∣t=0g(x0, x1, x2 +t) = 0.
Using our explicit expression for the matrix g in Eq. (7.3.5) above we have
d
dt
∣∣∣t=0ϕα(x0, x1, x2 + t) = ∂2ϕα(x0, x1, x2) = 0, (7.3.8)
5We follow [GIJP03] in our notation; letters from the middle Greek alphabet (λ, µ, ν, . . .) will de-note spacetime components on our three-manifold, while beginning Greek letters (α, β, γ, . . .) will de-note spacetime components on our reduced two-manifold. Tangent space components are generallydescribed by Latin letters, upper case (A,B,C, . . .) for three-dimensions and lower case (a, b, c, . . .) fortwo-dimensions.
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 138
and,
d
dt
∣∣∣t=0
(hαβ + ϕαϕβ)(x0, x1, x2 + t) = ∂2hαβ + ∂2(ϕαϕβ)
= ∂2hαβ, since ∂2ϕα(x0, x1, x2) = 0 by Eq. (7.3.8),
= 0.
Thus, ∂2ϕα = 0 and ∂2hαβ = 0.
Thus, we can now see that the Kaluza-Klein Ansatz of [GIJP03, Eq. 3.28] is implied
by our geometric situation. Note that our sign conventions differ, and we follow [Ber47,
Eq. 17.53]. Also, it is not difficult to show that our inverse metric is given by:
Gµν =
hαβ −hαδϕδ
−hβδϕδ ε−1 + hδζϕδϕζ
, (7.3.9)
where hαβ denotes the inverse of the (orbifold) Kahler metric h on Σ. After some calcu-
lation, we find that the Christoffel symbols
Γλµν =1
2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) , (7.3.10)
for the metric G = gε may be computed as (see B.2.1):
Γδαβ = γδαβ −ε
2hδζ(ϕβfζα + ϕαfζβ) (7.3.11)
Γ2αβ =
1
2(Dαϕβ +Dβϕα) +
ε
2ϕζ(ϕβfζα + ϕαfζβ), (7.3.12)
Γδ2β =ε
2hδζfβζ (7.3.13)
Γ22β =
ε
2ϕζfζβ (7.3.14)
Γδ22 = Γ222 = 0. (7.3.15)
Some explanation of notation is in order. First, α, β, δ, ζ ∈ 0, 1 index the coordinates
on Σ. The γδαβ are the Christoffel symbols for the Levi-Civita connection of the metric h
on Σ:
γδαβ :=1
2hδζ (∂βhζα + ∂αhζβ − ∂ζhαβ) . (7.3.16)
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 139
D is the covariant derivative for the Levi-Civita connection of the metric h on Σ:
Dαϕβ := ∂αϕβ − γζαβϕζ . (7.3.17)
fαβ is the “abelian field strength” tensor:
fαβ := ∂αϕβ − ∂βϕα, (7.3.18)
and finally, all two-dimensional indices are raised and lowered with the metric h; i.e.
ϕα := hαζϕζ . (7.3.19)
In order to compute our spin connection using Eq. (7.2.6), we need an explicit formula
for the Vielbein. We choose these as follows:
Eaα = eaα, E
22 =√ε, E2
α =√εϕα, E
a2 = 0 (7.3.20)
Eαa = eαa , E
22 =
1√ε, E2
a = −ϕζ eζa, Eα2 = 0, (7.3.21)
where eaα, eαa are the Vielbein (i.e. Zweibein) for the two-dimensional metric tensor h on
Σ. Note that a, α, ζ ∈ 0, 1 are indices in two-dimensions, and a ∈ 0, 1 denotes the
“tangent space coordinates” that index the Zweibein, as usual. We leave the straight-
forward confirmation that these formulae define a local orthogonal trivialization to the
reader.
Thus, using Eq. (7.2.6),
[Aµ]AB = EAν E
λBΓνµλ − Eλ
B∂µEAλ , (7.3.22)
and our formulae for the Vielbein and the connection components, we may compute the
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 140
spin connections [(AG)µ]AB := [Aµ]AB (see B.2.2):
[Aα]ab = eζb
[−Dαe
aζ −
ε
2eaδh
δρϕαfρζ
](7.3.23)
[Aα]a 2 = −[Aα]2 a = ηa b
√ε
2ebδh
δζfαζ (7.3.24)
[A2]ab =ε
2eζbe
aδh
δρfζρ (7.3.25)
[A2]2 a = [A2]a 2 = 0 (7.3.26)
[Aα]22 = [A2]22 = 0 (7.3.27)
where we have lowered the two dimensional index on [Aα]a 2 := ηab[Aα]b2 above. ηab is
the two-dimensional Kronecker pairing. Then using Eq. (7.2.8) we may compute the
quantities ACµ as follows (see B.2.3):
A2α = −ωα −
ε
2fϕα , A2
2 = − ε2f (7.3.28)
Aaα =
√ε
2eaαf , Aa2 = 0 (7.3.29)
where ωα is defined by the relation ηac(ωα)cb =: ωα,ab = εabωα, and (ωα)ab is the spin
connection on Σ:
(ωα)ab := eζb∂αeaζ − eaδ e
ζbγ
δαζ (7.3.30)
= eζbDαeaζ (7.3.31)
Also, f ∈ C∞orb(Σ) is the invariant field strength defined by the relation:
fαβ =√h εαβ f. (7.3.32)
Thus, using Eq.’s (7.3.28) and (7.3.29) and the formula for CS(Agε) given by Eq. (7.2.10),
we find that (see B.2.4):
CS(Agε) = − 1
4π
∫S1
dx2
∫Σ
dx0 ∧ dx1√h(εfr + ε2f 3) (7.3.33)
= −1
2
∫Σ
dx0 ∧ dx1√h(εfr + ε2f 3) (7.3.34)
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 141
where we take the volume of S1 to be 2π, r ∈ Ω0orb(Σ) is the (orbifold) scalar curvature,
and ω ∈ Ω2orb(Σ) is the (orbifold) Kahler form of Σ. Recall that the Reeb vector field is
dual to the one-form κ under our metric gε when ε = 1:
κ(·) = g1(ξ, ·). (7.3.35)
In our coordinate system, this means:
κ = ϕ0dx0 + ϕ1dx
1 + dx2. (7.3.36)
We then have:
dκ = (∂0ϕ1 − ∂1ϕ0)dx0 ∧ dx1 = f01dx0 ∧ dx1, (7.3.37)
=√hfdx0 ∧ dx1 (7.3.38)
We have implicitly identified the (orbifold) Kahler form ω on Σ with its pullback under
π : X → Σ here in this coordinate system. Strictly speaking we have:
dκ = π∗ω. (7.3.39)
By reversing the orientation of Σ, and substituting ω in for√hfdx0∧dx1 in Eq. (7.3.34),
we obtain:
7.3.40 Proposition. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold,
U(1) // X
Σ
.
Let gε := ε κ⊗ κ+ π∗h. After a particular choice of Vielbein,6 then,
CS(Agε) =( ε
2
)∫Σ
r ω +
(ε2
2
)∫Σ
f 2 ω (7.3.41)
6See Equations (7.3.20) and (7.3.21).
Chapter 7. Gravitational Chern-Simons and the Adiabatic Limit 142
where r ∈ C∞orb(Σ) is the (orbifold) scalar curvature of (Σ, h), ω ∈ Ω2orb(Σ) is the (orbifold)
Hodge form of (Σ, h), and f ∈ Ω0orb(Σ) is the invariant field strength on (Σ, h).7 In
particular, the adiabatic limit of CS(Agε) vanishes:
limε→0
CS(Agε) = 0. (7.3.42)
7In a “special coordinate system,” x0, x1, x2 on U ⊂ X, κ = ϕ0dx0 + ϕ1dx
1 + dx2, and dκ =(∂0ϕ1−∂1ϕ0)dx0∧dx1 = f01dx
0∧dx1. So dκ = fαβ . fαβ is called the abelian field strength tensor, andfαβ =
√h εαβ f, where f ∈ Ω0
orb(Σ) is called the invariant field strength on (X,h). See §2.4 for morebackground.
Appendix A
Localization
A.1 Stationary Phase
In this section we review a finite dimensional model for the method of stationary phase.
First, we recall the classical case where our functional is non-degenerate and then follow-
ing [GS77], [Sch79a], [Sch79b], and [Man98], we briefly review the case of a degenerate
functional.
Take k ∈ R>0, M a manifold of dimension n = 2l, f ∈ C∞(M), and dx a density
on M . The primary reason why the method of stationary phase is crucial for this thesis
is that it gives us a way to study integrals of the form:
F (k) :=
∫M
eikfdx. (A.1.1)
Of course, the study of such integrals is rigorously understood in the case that M is finite
dimensional.1 In this thesis, the main quantity of interest is the U(1) Chern-Simons path
integral (See §4.3 and §4.4), given heuristically as:
ZU(1)(X, k) =∑
p∈TorsH2(X;Z)
ZU(1)(X, p, k), (A.1.2)
1See [BGV92, §7.4].
143
Appendix A. Localization 144
where,
ZU(1)(X, p, k) =1
Vol(GP )
∫APDAeπikSX,P (A). (A.1.3)
The similarities between Eq.’s (A.1.1) and (A.1.3) are clear. In fact, this analogy is so
strong that we formally use the method of stationary phase to obtain a rigorous definition
of the U(1) Chern-Simons partition function for this thesis (See §4.3 and §6.6).
As a warm up, let us first assume that f ∈ C∞(M) is non-degenerate, meaning that
the set M0 where the differential of f vanishes is finite and the Hessian Hp := ∇pdf of
f is non-degenerate for all p ∈ M0. Let sgn(Hp) be the signature of the quadratic form
Hp. Let T+p , T−p denote the subspaces of TpM on which Hp is positive/negative definite,
respectively. Then sgn(Hp) = dimT+p − dimT−p . The following is standard, [BGV92,
§7.4]:
F (k) =
∫M
eikfdx =∑p∈M0
(2π
k
)leikf(p)eπi sgn(Hp)/4 1
| detHp|1/2+O(k−(l+1)). (A.1.4)
A.1.5 Example. The best finite dimensional analogue for us is when M = Rn, n = 2l and
f(x) = 〈x, Tx〉 is quadratic, for some symmetric, non-degenerate matrix T . This is a
nice analogue because AP is an affine space and the U(1) Chern-Simons action SX,P (A)
may be viewed as a quadratic functional (See Eq. (4.3.10)):
ZU(1)(X, p, k) =eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4π
(∫X
A ∧ dA)]
, (A.1.6)
=eπikSX,P (AP )
Vol(GP )
∫AP
DA exp
[ik
4π〈A, ?dA〉
], (A.1.7)
(A.1.8)
so that basically,
SX,P (A) =
(∫X
A ∧ dA)
= 〈A, ?dA〉. (A.1.9)
In this case, the stationary phase approximation of Eq. (A.1.16) is exact (i.e. the error
term O(k−(l+1)) vanishes). This can be seen by evaluating the following Gaussian integral
Appendix A. Localization 145
directly: ∫Rneik〈x,Tx〉dnx. (A.1.10)
Since T is symmetric, we change variables and let x = Py for an orthogonal matrix P
such that D = P TTP is diagonal. Let λj, 1 ≤ j ≤ n denote the eigenvalues of T . We
have ∫Rneik〈x,Tx〉dnx =
∫Rneik〈y,P
TTPy〉dny (A.1.11)
=
∫Rneik
∑nj=1 λjy
2j dny (A.1.12)
=n∏j=1
∫Reikλjy
2j dy. (A.1.13)
Individually, we have ∫Reikλjy
2j dy =
√2π
k
1
|√λj|
eiπ4
sgn(λj). (A.1.14)
Thus, ∫Rneik〈x,Tx〉dnx =
(2π
k
)leπi sgn(T )/4 1
| detT |1/2. (A.1.15)
Clearly, this matches the stationary phase formula Eq. (A.1.16) when f(x) = 〈x, Tx〉,
since the only critical point of f is when x = 0 and the Hessian is H0 = T .
Example A.1.5 demonstrates our main motivation for making the rigorous definitions,
Def.’s 4.3.22 and 6.10.28, and why we make these definitions with the expectation that
the partition functions are exact. In order to be more complete in our finite dimensional
analogue we must consider the case where our functional is allowed to be non-degenerate.
This is the case for the U(1) Chern-Simons action (See Eq. (A.1.9)). This case was first
considered in [Sch79a], [Sch79b]. We will thence consider a slightly more general class of
integrals:
F (k) :=1
VolG
∫M
eikfdµ. (A.1.16)
where:
• (M, g) is a Riemannian manifold with a compact Lie group G acting as a group of
isometries.
Appendix A. Localization 146
• G is assumed to be endowed with an invariant inner product that defines an invari-
ant Riemannian metric on G. Vol(G) denotes the volume of G with respect to this
metric. Let g := LieG denote the Lie algebra of G as usual.
• The action G M generates a map τp : g→ TpM for each p ∈M via:
τp(Y ) :=d
dt
∣∣∣t=0
exp(tY ) · p, for Y ∈ g. (A.1.17)
• Hp := g ∈ G | g · p = p is the isotropy subgroup of G at p. Assume that Hp is
conjugate to a fixed subgroup H < G ∀p ∈M .2 So Vol(Hp) = Vol(H).
• Let gM/G denote the Riemannian metric on the quotient space M/G induced from
the metric g on M . Then µ, µM/G denote the associated measures for the metrics
on M and M/G, respectively.
• f is assumed to be a G-invariant function on M such that the stationary points
of f (i.e. p ∈ M 3: dfp = 0) form a G-invariant submanifold F of M , and
f(p) = Q ∀p ∈ F , say.
• Assume that TpF = Ker(Hessf)p for all p ∈M.
• Assume thatM := F/G is a manifold and we let µM denote the naturally induced
measure on M.
Eq. (A.1.16) is the true analogue of Eq. (A.1.3) that we use to make our rigorous
definitions of the partition functions. All of the ingredients match up as in the following
dictionary:
2This is the case as in §6.5.
Appendix A. Localization 147
Chern-Simons theory, Eq. (A.1.3) Finite dimensional analogue, Eq.
(A.1.16)
AP = A ∈ (Ω1(P ) ⊗ g)G | A(ξ]) =
ξ, ∀ ξ ∈ g, and g = 〈·, ·〉 =∫X· ∧ ?·
(M, g).
GP = ψ ∈ (Diff(P, P ))G | π ψ =
π ' Maps(X,U(1)).
G.
LieGP ' Ω0(X), TAAP ' Ω1(X), d :
Ω0(X)→ Ω1(X).
τp : g→ TpM .
SX,P (A) : AP → R/Z. f : M → R.
MP ⊂ AP/GP . M⊂M/G.
The main result that we use in §4.3 and §6.6 is the following stationary phase approxi-
mation, [Sch79a], [Sch79b]:
1
VolG
∫M
eikfµ = C0 ·∫Meπi4
Sgn(Hessf)p| det′(τ ∗p τp)|1/2
| det′(Hessf)p|1/2µM +O(k−r) (A.1.18)
for some r ∈ R>0, and where,
C0 :=1
k(dimM−dimF )/2
eπiQk
Vol(H), (A.1.19)
and det′ denotes a regularized determinant, which is intended to represent a product
of non-zero eigenvalues. Our basic assumption is that O(k−r) vanishes since SX,P (A) :
AP → R/Z is quadratic.
Appendix B
Constructions and Computations
B.1 Finite Dimensional Analogue of the Shift Sym-
metry
In this section we present a finite dimensional analogue of the shift symmetry construc-
tion of [BW05, §3.1]. We exhibit a specific example that should help to clarify the basic
elements of the heuristic infinite dimensional construction of [BW05]. In order to depict
this analogue more clearly, we will set up a dictionary between our finite dimensional
example and the infinite dimensional Chern-Simons theory.
148
Appendix B. Constructions and Computations 149
Chern-Simons theory Finite dimensional analogue
AP = A ∈ (Ω1(P ) ⊗ g)G | A(ξ]) =
ξ, ∀ ξ ∈ g.
R3.
G = ψ ∈ (Diff(P, P ))G | π ψ = π. e = trivial group.
Contact form κ ∈ Ω1(X,R). Fixed vector 0 6= v0 ∈ R3.
S(A) =∫X
Tr(A ∧ dA+ 2
3A ∧ A ∧ A
),
for A ∈ AP .
S(w) = ‖w‖2 + ‖w‖2 · ‖w × v0‖, for
w ∈ R3 and fixed 0 6= v0 ∈ R3.
Lie algebra-valued zero form on X, Φ ∈
Ω0(X, g).
Φ ∈ R scalar coordinate.
The space of local shift symmetries S
that “acts” on the space of connections
AP and the space of fields Φ.
S = R; For any r ∈ S, r ·w := w+rv0,
w ∈ R3 and r · Φ := Φ + r, Φ ∈ R.
Vol(S) =∫S DΦ :=∫
S exp(−∫Xκ ∧ dκ Tr(σ2)
)DS.
Vol(S) =∫
RDΦ :=∫
R exp (−σ2) dσ =
√π, where σ ∈ S = R.
Our finite dimensional analogue of the partition function of Eq. (5.0.1) is defined as
follows:
Z :=
∫R3
dw exp
[−1
2S(w)
]. (B.1.1)
where S(w) = ‖w‖2 + ‖w‖2 · ‖w× v0‖ is as defined above. Note that we will work with
a “Wick rotated” Euclidean action in our finite dimensional analogue. This just means
that the factor of i =√−1 is replaced by −1 in the exponential. These two viewpoints
are formally equivalent, and it suffices to consider such an action our purposes.
The first thing that we notice is that the action S(w) = ‖w‖2 + ‖w‖2 · ‖w × v0‖ is
not invariant under the action of the shift symmetry S = R on R3; i.e. S(r · w) =
S(w + rv0) 6= S(w) for general r ∈ S = R and w ∈ R3. This is exactly analogous to
Appendix B. Constructions and Computations 150
the fact that S(A) =∫X
Tr(A ∧ dA+ 2
3A ∧ A ∧ A
)is not invariant under the local shift
symmetry action, δσ(A) := σκ, for σ ∈ S. In order to recast their partition function,
Z(k), into a form that is invariant under the shift symmetry, [BW05] introduce a new
scalar field Φ ∈ Ω0(X, g) which also has a transformation rule under the shift symmetry;
δσ(Φ) := σ, for σ ∈ S. The shifted partition Z(k) with shifted action:
S(A,Φ) := S(A− Φκ) (B.1.2)
is then considered since it is the only shift invariant action that incorporates both Φ
and the shift symmetry. To see that S(A − Φκ) is invariant under the shift symmetry,
observe:
δσS(A− Φκ) =δS
δA(A− Φκ) · δσ(A− Φκ) (B.1.3)
=δS
δA(A− Φκ) · (δσ(A)− δσ(Φ)κ) (B.1.4)
=δS
δA(A− Φκ) · (σκ− σκ) (B.1.5)
= 0, (B.1.6)
After introducing some normalization factors and expanding the action S(A−Φκ) in their
partition function, [BW05] obtain Eq. (5.0.2) and establish the equivalence Z(k) = Z(k)
heuristically. [BW05] then observe that S(A−Φκ) is quadratic in Φ, which allows them
to evaluate the Gaussian integral over Φ easily, and obtain the completely equivalent
formulation for their partition function as in Eq. (5.0.4). The action in Eq. (5.0.4)
CS(A)−∫X
1
κ ∧ dκTr[(κ ∧ FA)2
],
is now manifestly invariant under the shift symmetry, allowing one to quotient out by
the shift symmetry, and reduce the problem from an integral over AP to an integral over
AP/S. It turns out that AP/S is a symplectic space and [BW05] are able to show that
their problem can now be solved using the technique of non-abelian localization as in
SU(2)-Yang-Mills theory (cf. [Wit92]).
Appendix B. Constructions and Computations 151
We analogously define:
S(Φ,w) := S(w− Φv0), (B.1.7)
and consider,
Z :=1
Vol (S)
∫R×R3
DΦ dw exp
[−1
2S(Φ,w)
]. (B.1.8)
where the action of S = R is extended to R× R3 via:
r · (Φ,w) := (Φ + r,w + rv0).
S(Φ,w) = S(w− Φv0) : R× R3 → R is clearly invariant under this action:
S[r · (Φ,w)] = S(r · Φ, r ·w)
= S[(r ·w)− (r · Φ)v0)]
= S[(w + rv0)− (Φ + r)v0)]
= S(Φ,w).
Also, notice that we can trivially fix Φ = 0 in our action by simply changing the inde-
pendent coordinate w to w′ = w− Φv0. Our partition function is:
Z =1
Vol (S)
∫R×R3
DΦ dw′ exp
[−1
2S(w′)
]=
[1
Vol (S)
∫RDΦ
] ∫R3
dw exp
[−1
2S(w)
]=
∫R3
dw exp
[−1
2S(w)
]= Z
where the third line comes from the fact that Vol(S) =∫
RDΦ :=∫
R exp (−σ2) dσ =√π,
as noted in our table above. Hence, our new description of the partition, Z, is completely
equivalent to our old description, Z; i.e. Z = Z.
B.1.9 Remark. Our primary motivation for choosing an action of the form:
S(w) = ‖w‖2 + ‖w‖2 · ‖w× v0‖,
Appendix B. Constructions and Computations 152
comes directly from the analogue of our finite dimensional construction with [BW05].
We choose this action so that it is cubic in w, and S(w − Φv0) is quadratic in Φ. The
fact that S(w− Φv0) is quadratic in Φ simply follows from the observation that
(w− Φv0)× v0 = w× v0,
since the cross product of v0 with itself vanishes.
At this point, we will continue with our analogy and evaluate the Gaussian integral
over Φ directly. For simplicity, we assume that ‖v0‖ = 1. We compute:
S(Φ,w) = S(w− Φv0)
= ‖w− Φv0‖2 + ‖w− Φv0‖2 · ‖(w− Φv0)× v0‖
= ‖w− Φv0‖2 · [1 + ‖w× v0‖]
=[Φ2 − (2w · v0)Φ + ‖w‖2
]· [1 + ‖w× v0‖]
=[(Φ− (w · v0))2 + (‖w‖2 − (w · v0)2)
]· [1 + ‖w× v0‖] ,
where we have completed the square in the last line. We then make the change of
variables:
Φ 7→ Φ′ = Φ− (w · v0),
in the partition function, and obtain:
Z = Z =1
Vol (S)
∫R×R3
DΦ dw exp
[−1
2S(Φ,w)
]=
1√π
∫R×R3
DΦ dw exp
[−1
2([(Φ− (w · v0))2 + (‖w‖2 − (w · v0)2)
]· [1 + ‖w× v0‖])
]=
1√π
∫R×R3
dw DΦ′ exp
[−1
2[1 + ‖w× v0‖] Φ′2
]×
× exp
[−1
2([‖w‖2 − (w · v0)2
]· [1 + ‖w× v0‖])
].
There are a couple of natural and equivalent ways to proceed from here. Since it is
our intention to elaborate on the way a physicist performs these kinds of computations,
we will first perform a “gauge fixing” for the shift symmetry to evaluate the partition
Appendix B. Constructions and Computations 153
function Z. Recall that a gauge choice for the shift symmetry is defined by choosing a
function
c : R3 → R (B.1.10)
such that c is one-to-one when restricted to the gauge orbits and is normalized so that
0 ∈ Im(c|Orbit) on each gauge orbit. Clearly the gauge orbits of the shift symmetry are
just lines of the form x + r · v0 ∈ R3 | r ∈ R, and we can choose a representative x for
each gauge orbit in the plane defined by
Pv0 := x ∈ R3 |x · v0 = 0. (B.1.11)
Our natural choice for the gauge function is
c(w) := w · v0; (B.1.12)
i.e. c(w) is just the distance from w ∈ R3 to the plane Pv0 (recall that we set ‖v0‖ = 1).
We may then evaluate our partition function by inserting the delta function
δ(c(w)) (B.1.13)
and the measure fixing “Faddeev-Popov” determinant
∆c(w) :=
∣∣∣∣ ∂∂rc(r ·w)
∣∣∣∣r=r0
, (B.1.14)
into the integrand of the partition function. Note that r0 ∈ S = R is defined by the
relation c(r0 ·w) = 0. It is then usually assumed in the physics literature that one may
simply write:
Z =
∫R3/S
dw δ(c(w))∆c(w)
[2π
1 + ‖w× v0‖
]1/2
× (B.1.15)
× exp
[−1
2([‖w‖2 − (w · v0)2
]· [1 + ‖w× v0‖])
].
after computing the Guassian integral with respect to Φ′ and the group integral over S,
which produces a factor of Vol(S) =√π that cancels the factor out front of the integral.
Appendix B. Constructions and Computations 154
∆c(w)dw is interpreted in the integral as a measure on the quotient space R3/S, dx.
After applying the delta function, δ(c(w)), observing that the Jacobian ∆c(w) = 1 is
trivial, and identifying x = r0w (where r0 is defined by the relation c(r0 · w) = 0), we
obtain:
Z =
∫Pv0'R2
dx
[2π
1 + ‖x‖
]1/2
exp
[−1
2(‖x‖2 · [1 + ‖x‖])
]We further observe that this integral may be computed by making a change of variables
to polar coordinates
x = (rcos(θ), rsin(θ)).
We obtain:
Z =√
2π
∫ 2π
0
dθ
∫ ∞0
drr√
1 + rexp
[−1
2(r2 · (1 + r))
]= (2π)3/2
∫ ∞0
drr√
1 + rexp
[−1
2(r2 · (1 + r))
]Although the gauge fixing procedure that we have just demonstrated yields the correct
answer, it is instructive to consider an alternative method of computing Z that will shed
some light on some of the steps that are being skipped by the physicists. Our starting
point is the integral:
Z =1√π
∫R×R3
dw DΦ′ exp
[−1
2[1 + ‖w× v0‖] Φ′2
]×
× exp
[−1
2([‖w‖2 − (w · v0)2
]· [1 + ‖w× v0‖])
].
A point that was missed in our computation is that one must be careful to observe that
formally
DΦ = exp[−σ2]dσ (B.1.16)
where σ is viewed as a variable on S = R, and that the change of variables Φ 7→ Φ′ =
Φ− (w · v0) gives
DΦ′ = exp[−(σ′ + (w · v0))2]dσ′ (B.1.17)
Appendix B. Constructions and Computations 155
where σ′ is a new variable on S. We cannot simply “perform the Gaussian integral
over Φ′” here, as there are mixed terms in the measure DΦ′. In order to carry out the
computation, we should change coordinates from w = (x, y, z) to (r,x), where r = w ·v0
and x = w− (w · v0)v0 = ProjPv0(w), so that
w = x + rv0. (B.1.18)
Then our measure becomes, dw = |J(w)|drdx = drdx, where the Jacobian |J(w)| = 1
since J(w) ∈ SO(3,R) can be taken to be a rotation in R3. We are then essentially
performing a gauge fixing where x = w−(w·v0)v0 becomes our gauge orbit representative
for each w ∈ R3 and the measure fixing Faddeev-Popov determinant ∆c(w) is identified
as the Jacobian |J(w)|. Making this substitution yields:
Z =1√π
∫R×R2×
dr dx dσ′ exp[−(σ′ + r)2] ×
× exp
[−1
2[1 + ‖x‖]σ′2
]exp
[−1
2(‖x‖2 · [1 + ‖x‖])
].
Now we make the independent substitution, r 7→ r′ = σ′ + r, which gives:
Z =1√π
∫Rdr′exp[−(r′)2]
∫R2×R
dx dσ′ exp
[−1
2[1 + ‖x‖]σ′2
]× exp
[−1
2(‖x‖2 · [1 + ‖x‖])
].
This last step justifies the cancelation of Vol(S) from the group integral over S; i.e.∫Rdr′exp[−(r′)2] =
√π. (B.1.19)
Performing the Gaussian integral over σ′ then yields our result:
Z =
∫Pv0'R2
dx
[2π
1 + ‖x‖
]1/2
exp
[−1
2(‖x‖2 · [1 + ‖x‖])
].
The main reason why this analysis is possible is because we have expressed our partition
function in the shift invariant form∫R3
dw
[2π
1 + ‖w× v0‖
]1/2
exp
[−1
2([‖w‖2 − (w · v0)2
]· [1 + ‖w× v0‖])
]. (B.1.20)
Appendix B. Constructions and Computations 156
i.e. if,
f(w) :=
[2π
1 + ‖w× v0‖
]1/2
exp
[−1
2([‖w‖2 − (w · v0)2
]· [1 + ‖w× v0‖])
], (B.1.21)
then f(r ·w) = f(w) for all r ∈ S and w ∈ R3. Beasley and Witten obtain the analogue
of this result as in Eq. (5.0.4) by “evaluating the Gaussian integral with respect to Φ” in
Eq. (5.0.2). The shift invariant partition function that they obtain in Eq. (5.0.4) then
allows them to reduce their integral modulo the shift symmetry as in Eq. (B.1.15).
B.2 Gravitational Chern-Simons Calculations
B.2.1 Levi-Civita Connection
In this section we explicitly compute the Christoffel symbols
Γλµν =1
2Gλρ (∂νGρµ + ∂µGρν − ∂ρGµν) , (B.2.1)
for the Levi-Civita connection ∇G for the family of metrics
Gµν := gε =
hαβ + εϕαϕβ εϕα
εϕβ ε
. (B.2.2)
with inverse metric
Gµν =
hαβ −ϕα
−ϕβ ε−1 + ϕζϕζ
. (B.2.3)
B.2.4 Remark. We will use the comma notation to denote partial derivatives; i.e. Gρβ,α :=
∂αGρβ, and the semi-colon notation to denote covariant derivatives, so ωαβ;ρ := ∂ρωαβ −
Γζαρωζβ − Γζρβωαζ for some (0, 2) tensor ωαβ, for example.
We break this computation down into cases.
I. Γδαβ, δ, α, β ∈ 0, 1. Reading off the components of the metric from Eq.’s (B.2.2)
and (B.2.3):
Appendix B. Constructions and Computations 157
• ρ ∈ 0, 1:
1
2Gδρ (Gρβ,α +Gρα,β −Gαβ,ρ)
=1
2hδρ ([hρβ,α + ε(ϕρϕβ),α] + [hρα,β + ε(ϕρϕα),β]− [hαβ,ρ + ε(ϕαϕβ),ρ])
=1
2hδρ ([hρβ,α + ε(ϕρ,αϕβ + ϕρϕβ,α)] + [hρα,β + ε(ϕρ,βϕα + ϕρϕα,β)]) +
− 1
2hδρ ([hαβ,ρ + ε(ϕα,ρϕβ + ϕαϕβ,ρ)])
= γδαβ +ε
2hδρ ((ϕρ,αϕβ + ϕρϕβ,α) + (ϕρ,βϕα + ϕρϕα,β)− (ϕα,ρϕβ + ϕαϕβ,ρ))
= γδαβ +ε
2
(hδρ[ϕβ(ϕρ,α − ϕα,ρ) + ϕα(ϕρ,β − ϕβ,ρ)] +
ϕδϕα,β + ϕδϕβ,α
)where
γδαβ :=1
2hδρ (hρβ,α + hρα,β − hαβ,ρ) (B.2.5)
are the Christoffel symbols for the two-dimensional metric tensor h.
• ρ = 2:
1
2Gδ2 (G2β,α +G2α,β −Gαβ,2)
=ε
2(−ϕδ)
(ϕα,β + ϕβ,α − [ε−1hαβ,2 + (ϕαϕβ),2]
)=−ε2
ϕδϕα,β + ϕδϕβ,α
where the last line follows from the fact that
ε−1hαβ,2 + (ϕαϕβ),2 = 0, (B.2.6)
since the ∂2 derivatives of hαβ, and ϕα vanish.
Clearly, the terms in curly brackets cancel in the sum of the ρ ∈ 0, 1 and
ρ = 2 cases above, and we have for δ, α, β ∈ 0, 1:
Γδαβ = γδαβ −ε
2hδζ (ϕβfζα + ϕαfζβ) , (B.2.7)
where,
fαβ := ϕβ,α − ϕα,β. (B.2.8)
Appendix B. Constructions and Computations 158
II. Γ2αβ, α, β ∈ 0, 1.
• ρ ∈ 0, 1:
1
2G2ρ (Gρβ,α +Gρα,β −Gαβ,ρ)
=1
2(−ϕρ) ([hρβ,α + ε(ϕρϕβ),α] + [hρα,β + ε(ϕρϕα),β]− [hαβ,ρ + ε(ϕαϕβ),ρ])
=1
2(−ϕρ) ([hρβ,α + ε(ϕρ,αϕβ + ϕρϕβ,α)] + [hρα,β + ε(ϕρ,βϕα + ϕρϕα,β)]) +
− 1
2(−ϕρ) ([hαβ,ρ + ε(ϕα,ρϕβ + ϕαϕβ,ρ)])
= −ϕργραβ −ε
2ϕρ ((ϕρ,αϕβ + ϕρϕβ,α) + (ϕρ,βϕα + ϕρϕα,β)− (ϕα,ρϕβ + ϕαϕβ,ρ))
= −ϕργραβ +ε
2(ϕρ[ϕβfρα + ϕαfρβ)]− ϕρϕρϕα,β + ϕρϕρϕβ,α)
• ρ = 2:
1
2G22 (G2β,α +G2α,β −Gαβ,2)
=ε
2(ε−1 + ϕζϕζ) (ϕα,β + ϕβ,α) , since Gαβ,2 = 0,
=1
2(ϕα,β + ϕβ,α) +
ε
2
ϕζϕζϕα,β + ϕζϕζϕβ,α
Clearly, the terms in curly brackets cancel in the sum of the ρ ∈ 0, 1 and
ρ = 2 cases, and we have:
Γ2αβ =
1
2(ϕα,β + ϕβ,α)− ϕζγζαβ +
ε
2ϕζ [ϕβfζα + ϕαfζβ]
=1
2(Dαϕβ +Dβϕα) +
ε
2ϕζ(ϕβfζα + ϕαfζβ),
where the last line follows from the fact that the Levi-Civita connection is
symmetric (i.e. γζαβ = γζβα), and
Dαϕβ := ϕβ,α − ϕζγζαβ. (B.2.9)
III. Γδ2β, δ, β ∈ 0, 1.
Appendix B. Constructions and Computations 159
• ρ ∈ 0, 1:
1
2Gδρ (Gρβ,2 +Gρ2,β −G2β,ρ)
=ε
2hδρ (ϕρ,β − ϕβ,ρ) , since Gρβ,2 = 0,
=ε
2hδζfβζ .
• ρ = 2: It is easy to see that
1
2Gδ2 (G2β,2 +G22,β −G2β,2) = 0. (B.2.10)
This follows from observing that G22,β = ∂βε = 0, and the ∂2 derivative G2β,2
vanishes. Thus,
Γδ2β =ε
2hδζfβζ . (B.2.11)
IV. Γ22β, β ∈ 0, 1.
• ρ ∈ 0, 1:
1
2Gδρ (Gρβ,2 +Gρ2,β −G2β,ρ)
=ε
2(−ϕρ) (ϕρ,β − ϕβ,ρ) , since Gρβ,2 = 0,
=ε
2ϕζfζβ.
• ρ = 2: As above, it is not difficult to see that
1
2Gδ2 (G2β,2 +G22,β −G2β,2) = 0. (B.2.12)
Thus,
Γ22β =
ε
2ϕζfζβ. (B.2.13)
V. Γλ22, λ ∈ 0, 1, 2. It is also easy to see that
1
2Gλρ (Gρ2,2 +Gρ2,2 −G22,ρ) = 0. (B.2.14)
Appendix B. Constructions and Computations 160
B.2.2 Spin Connection
In this section we use our formulae for the Vielbeins,
Eaα = eaα, E
22 =√ε, E2
α =√εϕα, E
a2 = 0 (B.2.15)
Eαa = eαa , E
22 =
1√ε, E2
a = −ϕζ eζa, Eα2 = 0, (B.2.16)
and our formulae for the Christoffel symbols for the Levi-Civita connection associated to
our family of metrics G := gε to compute the spin connections [(AG)µ]AB := [Aµ]AB using
Eq. (7.2.6),
[Aµ]AB = EAν E
λBΓνµλ − Eλ
B∂µEAλ . (B.2.17)
We break this down into cases.
I. [Aα]ab , a, b, α ∈ 0, 1. Plug in the appropriate quantities from Eq.’s (B.2.15) and
(B.2.16) and for the Christoffel symbols.
• λ, ν ∈ 0, 1:
Eaν E
λb Γναλ − Eλ
b ∂αEaλ = eζbe
aδ
(γδαζ −
ε
2hδρ (ϕζfρα + ϕαfρζ)
)− eζb∂αe
aζ
= eζb
[(eaδγ
δαζ − ∂αeaζ)−
ε
2eaδh
δρϕαfρζ
]− ε
2eζbe
aδh
δρϕζfρα
= eζb
[−Dαe
aζ −
ε
2eaδh
δρϕαfρζ
]− ε
2eζbe
aδh
δρϕζfρα
• λ = 2, ν ∈ 0, 1:
Eaν E
λb Γναλ − Eλ
b ∂αEaλ = eaδ(−ϕζ e
ζb)( ε
2
)hδρfαρ
= ε
2eζbe
aδh
δρϕζfρα
These two cases, λ, ν ∈ 0, 1 and λ = 2, ν ∈ 0, 1, are the only cases for which we
get a non-zero contribution to [Aα]ab , since the term Eaν E
λb Γνµλ always vanishes for
ν = 2 when a, b ∈ 0, 1 by our formulae for the Vielbeins, and the term Eλb ∂µE
aλ
vanishes for λ = 2 for the same reason. After observing that the terms in the curly
Appendix B. Constructions and Computations 161
brackets from the cases, λ, ν ∈ 0, 1 and λ = 2, ν ∈ 0, 1, cancel in our sum, we
obtain:
[Aα]ab = eζb
[−Dαe
aζ −
ε
2eaδh
δρϕαfρζ
]. (B.2.18)
II. [Aα]a2, a, α ∈ 0, 1. The only terms to contribute to the sum over λ, ν ∈ 0, 1, 2 in
the sum of Eq. (B.2.17) are when λ = 2 and ν ∈ 0, 1, since Eλ2 = 0 for λ ∈ 0, 1.
Even when λ = 2, the sum Eλ2 ∂αE
aλ = 0 in Eq. (B.2.17) since Ea
2 = 0 for a ∈ 0, 1.
Thus,
[Aα]a2 =1√εeaδΓ
δα2
=1√εeaδε
2hδζfαζ
=
√ε
2eaδh
δζfαζ
By lowering the two dimensional index [Aα]a 2 := ηab[Aα]b2, with ηab the two-
dimensional Kronecker pairing, we obtain
[Aα]a 2 = −[Aα]2 a = ηa b
√ε
2ebδh
δζfαζ . (B.2.19)
III. [A2]ab , a, b ∈ 0, 1. The only terms to contribute to the sum over λ, ν ∈ 0, 1, 2
in the sum of Eq. (B.2.17) are when λ, ν ∈ 0, 1. All other terms vanish by
our formulae for the Vielbeins and the Christoffel symbols. Note that the sum
Eλb ∂2E
aλ = 0 in Eq. (B.2.17) since all ∂2 derivatives vanish for the Vielbeins. Thus,
[A2]ab = eζbeaδΓ
δζ2
= eζbeaδ
ε
2hδρfζρ
=ε
2eζbe
aδh
δρfζρ
IV. [A2]a2, a, α ∈ 0, 1. First, the sum Eλ2 ∂2E
aλ = 0 in Eq. (B.2.17) since all ∂2
derivatives vanish for the Vielbeins. The only term to contribute to the sum over
Appendix B. Constructions and Computations 162
λ, ν ∈ 0, 1, 2 in the sum of Eq. (B.2.17) is ν ∈ 0, 1 and λ = 2. Thus,
[A2]a2 = E22eaδΓ
δ22
= 0
since Γδ22 = 0. By lowering the two dimensional index [A2]a 2 := ηab[Aα]b2, and using
our anti-symmetry properties, we see
[A2]2 a = [A2]a 2 = 0. (B.2.20)
V. [Aµ]22, µ ∈ 0, 1, 2. Lastly, since [Aµ]a b is anti-symmetric in a, b, we see trivially
that
[Aµ]22 = 0. (B.2.21)
for all µ ∈ 0, 1, 2.
B.2.3 Reduced Spin Connection
In this section we compute the corresponding vector valued one-forms ACµ for our spin
connections [(A)µ]AB defined by the relation
[(A)µ]AB := ηAC [Aµ]CB = εABCACµ . (B.2.22)
Contracting with the Levi-Civita symbol εABC , we have
ACµ =1
2εABCηAD[Aµ]DB . (B.2.23)
Eq. (B.2.23) combined with our formulae for the spin connections [Aµ]DB are the main
relations that we use to compute ACµ throughout this section. As usual, we do this
computation in cases.
Appendix B. Constructions and Computations 163
I. A2α, α ∈ 0, 1. Let ηab denote the two-dimensional Kronecker pairing as usual.
Eq. (B.2.23) gives us
A2α =
1
2εAB2ηAD[Aα]DB
=1
2[η0a[Aα]a1 − η1a[Aα]a0]
=1
2
[η0ae
ζ1
[−Dαe
aζ −
ε
2eaδh
δρϕαfρζ
]− η1ae
ζ0
[−Dαe
aζ −
ε
2eaδh
δρϕαfρζ
]]=
−1
2(η0ae
ζ1 − η1ae
ζ0)Dαe
aζ
+− ε
4eaδh
δρϕαfρζ(η0aeζ1 − η1ae
ζ0)
(B.2.24)
We compute the quantities in the curly brackets of Eq. (B.2.24) separately. First
we recall that the spin connection (ωα)ab on Σ is defined by:
(ωα)ab := eζb∂αeaζ − eaδ e
ζbγ
δαζ (B.2.25)
= eζbDαeaζ (B.2.26)
Then ωα is defined by the relation
ωα,ab = εabωα = ηac(ωα)cb (B.2.27)
Thus, we compute the first term in Eq. (B.2.24):−1
2(η0ae
ζ1 − η1ae
ζ0)Dαe
aζ
= −1
2(ωα,01 − ωα,10), by Eq.’s (B.2.26) and (B.2.27),
= −1
2(2ωα,01), by anti-symmetry of ωα,ab,
= −ωα, by Eq. (B.2.27).
Before computing the second term in Eq. (B.2.24), we recall the following relations:
eζq = hζληqbebλ, (B.2.28)
fρζ =√hfερζ , (B.2.29)
εab = η0aη1b − η1aη0b, (B.2.30)
εδλεabeaδebλ = 2
√h. (B.2.31)
Appendix B. Constructions and Computations 164
where
εδλ = |h|hρδhζλερζ (B.2.32)
in Eq. (B.2.31). Using Eq.’s (B.2.28) and (B.2.29), we compute the second term
in curly brackets from Eq. (B.2.24):
− ε
4eaδh
δρϕαfρζ(η0aeζ1 − η1ae
ζ0)
= − ε4
√hfϕα(eaδe
bλ)(hρδhζλερζ
)(η0aη1b − η1aη0b)
= − ε4
(√h)−1fϕα
(eaδe
bλεδλεab
), by (B.2.30), (B.2.32),
= − ε2
(√h)−1fϕα
√h, by Eq. (B.2.31),
= − ε2fϕα
Thus, we have:
A2α = −ωα −
ε
2fϕα. (B.2.33)
II. A22. As before, we compute:
A22 =
1
2εAB2ηAD[A2]DB
=1
2[η0a[A2]a1 − η1a[A2]a0]
=1
2
[η0a
[ ε2eζ1e
aδh
δρfζρ
]− η1a
[ ε2eζ0e
aδh
δρfζρ
]]= − ε
4
√hfebλe
aδ(h
δρhζλερζ) (η0aη1b − η1aη0b) , by Eq.’s (B.2.28) and (B.2.29),
= − ε4
(√h)−1f(eaδe
bλεδλεab), by Eq.’s (B.2.30) and (B.2.32),
= − ε2
(√h)−1f
√h, by Eq. (B.2.31),
= − ε2f
Thus, we have:
A22 = − ε
2f. (B.2.34)
III. Aaα, a, α ∈ 0, 1. Before we perform this computation we recall:
eδa = (√h)−1ελδεabe
bλ (B.2.35)
Appendix B. Constructions and Computations 165
Let a ∈ 0, 1\a be the element of 0, 1 that represents the compliment of
a ∈ 0, 1. We then compute:
Aaα =1
2εABaηAD[Aα]DB
=1
2εaa[ηaD[Aα]D2 − η2D[Aα]Da
]=
1
2εaa [[Aα]a2 − [Aα]2a]
= εaa[Aα]a2, by anti-symmetry of [Aα]ab,
= εaaηa b
√ε
2ebδh
δζfαζ
= εaa
√ε
2
√h f eζa εαζ , by Eq.’s (B.2.28) and (B.2.29),
= εaa
√ε
2
√h f [(
√h)−1ελζεabe
bλ] εαζ , by Eq. (B.2.35),
=
√ε
2f [ελζ εαζεaaεabe
bλ]
=
√ε
2f [δλαe
aλ]
=
√ε
2f eaα.
Thus, we have:
Aaα =
√ε
2f eaα. (B.2.36)
IV. Aa2, a ∈ 0, 1.
Aa2 =1
2εABaηAD[A2]DB
=1
2εaa[ηaD[A2]D2 − η2D[A2]Da
]=
1
2εaa [[A2]a2 − [A2]2a]
= εaa[A2]a2, by anti-symmetry of [Aα]ab,
= 0, since [A2]a2 = 0 in general.
Thus,
Aa2 = 0. (B.2.37)
Appendix B. Constructions and Computations 166
B.2.4 Reduced Gravitational Chern-Simons
In this section we compute the gravitational Chern-Simons term CS(Agε) in terms of
reduced quantities using Eq. (7.2.10):
CS(AG) = − 1
2π
∫X
d3x εµνλ(ηABA
Aµ∂νA
Bλ
)+
1
π
∫X
d3x detACµ . (B.2.38)
I. We first compute εµνληABAAµ∂νA
Bλ from the first integral term. First observe that
Aaµ∂νAaλ = 0 for any a ∈ 0, 1 and any permutation σ(012) = µνλ, since if ν = 2,
then all ∂2 derivatives vanish, and if ν 6= 2 then Aa2 = 0 by our previous results.
Thus, we only need compute the term εµνλA2µ∂νA
2λ, where ν 6= 2. We do this in
cases.
• (µ, ν, λ) = (2, 0, 1):
ε201A22∂0A
21 = [− ε
2f ] · ∂0[−ω1 −
ε
2fϕ1]
= [ε
2f ] · ∂0[ω1 +
ε
2fϕ1]
• (µ, ν, λ) = (2, 1, 0):
ε210A22∂1A
20 = −[− ε
2f ] · ∂1[−ω0 −
ε
2fϕ0]
= −[ε
2f ] · ∂1[ω0 +
ε
2fϕ0]
• (µ, ν, λ) = (0, 1, 2):
ε012A20∂1A
22 = [−ω0 −
ε
2fϕ0] · ∂1[− ε
2f ]
= [ω0 +ε
2fϕ0] · ∂1[
ε
2f ]
• (µ, ν, λ) = (1, 0, 2):
ε102A21∂0A
22 = −[−ω1 −
ε
2fϕ1] · ∂0[− ε
2f ]
= −[ω1 +ε
2fϕ1] · ∂0[
ε
2f ]
Appendix B. Constructions and Computations 167
Adding these four cases together and grouping terms by powers of ε we obtain:
εµνλA2µ∂νA
2λ =
( ε2
)[f(∂0ω1 − ∂0ω1) + (ω0∂1f − ω1∂0f)] +
+( ε
2
)2
[(f∂0(fϕ1)− f∂1(fϕ0)) + (fϕ0∂1f − fϕ1∂0f)]
=( ε
2
)[2f(∂0ω1 − ∂1ω0) + ∂1(ω0f)− ∂0(ω1f)] +
+( ε
2
)2
[f 2(∂0ϕ1 − ∂1ϕ0)]
The term in curly brackets in the second last line above yields a global exact form
on Σ, and since ∂Σ = ∅, Stokes’ theorem implies that this term vanishes in the
integral of Eq. (B.2.38). It is also well known that the term in the second last line
above, ∂0ω1− ∂1ω0 = 12
√hr, where r ∈ Ω0
orb(Σ) is the (orbifold) scalar curvature of
(Σ, h). Also, the term ∂0ϕ1 − ∂1ϕ0 = f01 =√hf . Thus, we may write:∫
X
d3x εµνλ(ηABA
Aµ∂νA
Bλ
)=
∫S1
dx2
∫Σ
dx0 ∧ dx1√h
[( ε2
)fr +
( ε2
)2
f 3
](B.2.39)
This completes our computation of the first integral term in Eq. (B.2.38).
II. We now compute the second integral term detACµ from Eq. (B.2.38). For this we
note that detACµ may be computed directly, since:
ACµ =
√ε
2eaαf 0
−ωα − ε2fϕα − ε
2f
. (B.2.40)
Thus,
detACµ = det
(√ε
2eaαf
)·(− ε
2f)
=( ε
4f 2 det(eaα)
)·(− ε
2f)
= −1
2
( ε2
)2√hf 3, since det(eaα) =
√h.
Thus, ∫X
d3x detACµ =
∫S1
dx2
∫Σ
dx0 ∧ dx1√h
[−1
2
( ε2
)2
f 3
]. (B.2.41)
Appendix B. Constructions and Computations 168
Adding our main results from Eq.’s (B.2.39) and (B.2.41), we obtain:
CS(Agε) = − 1
4π
∫S1
dx2
∫Σ
dx0 ∧ dx1√h(εfr + ε2f 3). (B.2.42)
Appendix C
Miscellaneous Results
C.1 Horizontal Operators
In this section we collect some results used in this thesis. First we recall some notation
and terminology. Let (X,φ, ξ, κ, g) be a closed, quasi-regular K-contact three-manifold
(See §2.3) and
Ωq(H) := α ∈ Ωq(X) | ιξα = 0,
for q = 1, 2, be the space of horizontal forms on X (for q = 0 we take Ω0(H) = C∞(X)).
Recall that ξ ∈ Γ(TX) is the Reeb vector field for κ. Let H ⊂ TX denote the contact
distribution on X and H∗ ⊂ T ∗X denote the dual bundle of H. Also let H0 ⊂ T ∗ denote
the annihilator subbundle of H in TX. The cotangent bundle has the natural splitting
induced by the choice of κ:
T ∗X ' H0 ⊕H∗. (C.1.1)
Let Ω1(V ) := Γ(H0). In this way, sections of T ∗X are identified as
Γ(T ∗X) = Ω1(X) ' Ω1(V )⊕ Ω1(H). (C.1.2)
Clearly,
Ω1(V ) := fκ ∈ Ω1(X) | f ∈ C∞(X). (C.1.3)
169
Appendix C. Miscellaneous Results 170
Let
Ω2(V ) := κ ∧ α | α ∈ Ω1(H) = κ ∧ Ω1(H). (C.1.4)
From Eq. (C.1.1) we also have the decomposition
Γ(∧2T ∗X) = Ω2(X) ' Ω2(V )⊕ Ω2(H). (C.1.5)
Let ? : Ωq(X) → Ω3−q(X) be the Hodge star operator associated to the natural metric
g = κ ⊗ κ + π∗h (See Eq. (2.3.11)) on X. We recall the definition and some properties
of the Hodge star operator presently. Let (U, φ) be some local coordinate chart on X
such that x0, x1, x2 are the coordinates about a given point x ∈ X. We choose this
coordinate system so that ξx = ∂∂x0 |x and
∂
∂x0,∂
∂x1,∂
∂x2
,
forms a positively oriented orthonormal basis for our metric g at the point x. Let
dx0, dx1, dx2 denote the dual basis. Let
µx := dx0 ∧ dx1 ∧ dx2,
denote the volume form associated to g at x. Recall that the Hodge star operator may
be defined pointwise on Ωq(X). That is, given β ∈ Ωq(X), we may define ?β ∈ Ω3−q(X)
uniquely by specifying (?β)|x pointwise at x via:
α|x ∧ (?β)|x = µx,
for all α ∈ Ωq(X). Note that κ|x = dx0|x and dκ|x = dx1 ∧ dx2 follows easily from the
fact that our coordinate system can be taken such that κ = dx0 + x1dx2 locally about x
(and x is centered about the origin in our coordinate system). Thus, µx = κ ∧ dκ|x and
µ = ?(1) = κ ∧ dκ. (C.1.6)
Recall that on a three-manifold X
? ?β = β, ∀ β ∈ Ωq(X). (C.1.7)
We will find it convenient to define the Hodge star at x via:
Appendix C. Miscellaneous Results 171
i. ?dx0 = dx1 ∧ dx2,
ii. ?dx1 = −dx0 ∧ dx2,
iii. ?dx2 = dx0 ∧ dx1.
Note that Eq. (i) is the global statement
? κ = dκ. (C.1.8)
Note that Eq. (C.1.7) combined with Eq.’s (C.1.6), (i), (ii), and (iii) together define the
Hodge star operator for any β ∈ Ω3(X). Let us collect some further useful observations.
First, since Ω1(H) = Γ(H∗), we have by definition
β ∈ Ω1(H) ⇐⇒ β|x ∈ spandx1, dx2 ∀ x ∈ X, (C.1.9)
in our choice of coordinate system about a given point x. Similarly,
β ∈ Ω2(H) ⇐⇒ β|x ∈ spandx1 ∧ dx2 ∀ x ∈ X. (C.1.10)
β ∈ Ω1(V ) ⇐⇒ β|x ∈ spandx0 ∀ x ∈ X. (C.1.11)
β ∈ Ω2(V ) ⇐⇒ β|x ∈ spandx0 ∧ dx1, dx0 ∧ dx2 ∀ x ∈ X. (C.1.12)
We will find it useful to give a different characterization of horizontal vectors β ∈ Ωq(H),
for q = 1, 2, using the projection operator
π : Ωq(X) → Ωq(H) (C.1.13)
β 7→ β − κ ∧ ιξ(β). (C.1.14)
It is not difficult to see, for q = 1, 2, that
β ∈ Ωq(H) ⇐⇒ π(β) = β. (C.1.15)
Finally, we recall the following
Appendix C. Miscellaneous Results 172
C.1.16 Definition. Define the horizontal Hodge star operator to be the operator:
?H : Ωq(X)→ Ω2−q(H) q = 0, 1, 2,
defined for β ∈ Ωq(X) by
?H β = ?(κ ∧ β), (C.1.17)
where ? is the usual Hodge star operator on forms for the metric g = κ⊗ κ+ π∗h on X.
It is easy to verify by the properties of ? that ?H maps strictly to Ω2−q(H) q = 0, 1, 2.
We are now ready to prove the following
C.1.18 Proposition. ? : Ω1(H)→ κ ∧ Ω1(H) =: Ω2(V ) and ? : Ω2(H)→ Ω1(V ).
Proof. To prove the first part, let β ∈ Ω1(H). By C.1.9, for any x ∈ X ∃ ax, bx ∈ R 3:
β|x = axdx1 + bxdx
2.
By Eq.’s (ii) and (iii) we have
?β|x = dx0 ∧ (−axdx2 + bxdx1),
which by C.1.9, C.1.11 and C.1.3 says that ?β = κ∧β′ for some β′ ∈ Ω1(H). This proves
? : Ω1(H)→ κ ∧ Ω1(H).
To see the second part, let α ∈ Ω2(H). Then by C.1.10 ∃ cx ∈ R 3:
α|x = cxdx1 ∧ dx2.
By C.1.7 and i,
?α|x = cxdx0,
and by C.1.11 this says that ?α ∈ Ω1(V ). This proves ? : Ω2(H) → Ω1(V ), and we are
done.
C.1.19 Proposition. ?2H = −1 on Ω1(H).
Appendix C. Miscellaneous Results 173
Proof. To see this, let β ∈ Ω1(H). Consider
?2H β = ?(κ ∧ (?(κ ∧ β)), (C.1.20)
by definition C.1.16. By C.1.9 and C.1.11, at a given x ∈ X,
κ ∧ β|x = dx0 ∧ (axdx1 + bxdx
2),
for some ax, bx ∈ R. By ii, iii and C.1.7 we have
?(κ ∧ β)|x = axdx2 − bxdx1.
Similarly,
κ ∧ ?(κ ∧ β)|x = dx0 ∧ (axdx2 − bxdx1),
and so,
?(κ ∧ ?(κ ∧ β))|x = −axdx1 − bxdx2 = −β|x.
Thus,
?(κ ∧ (?(κ ∧ β)) = ?2Hβ = −β,
and ?2H = −1 as claimed.
C.1.21 Proposition. ?Hα = (−1)qιξ(?α) = ?(κ ∧ α) for all α ∈ Ωq(X), 0 ≤ q ≤ 2.1
Proof. We prove this in cases.
i. Let us first prove this for α ∈ C∞(X) = Ω0(X).
On the one hand,
?(κ ∧ α) = αdκ,
since α ∈ C∞(X) and by C.1.8. On the other hand,
ιξ(?α) = αιξ(κ ∧ dκ), by C.1.6,
= αdκ, by property of the contraction ιξ.
1Note that the statement is true for q = 3, but only trivially so.
Appendix C. Miscellaneous Results 174
Thus, ιξ(?α) = ?(κ ∧ α) for α ∈ C∞(X) = Ω0(X).
ii. Assume α ∈ Ω1(X) ' Ω1(V )⊕ Ω1(H).
Write
α = αV + αH ,
uniquely for αV ∈ Ω1(V ) and αH ∈ Ω1(H). Let
α := ?(κ ∧ α) = ?(κ ∧ αH) ∈ Ω1(H).
We want to show that
α = −ιξ(?α).
To see this, observe that by Prop. C.1.19,
?(κ ∧ α) = ?(κ ∧ (?(κ ∧ αH)), by def. of α.
= ?2HαH , by def. of ?2
H ,
= −αH , by Prop. C.1.19.
Thus, if we operate with −ιξ? on both sides of the equation
− ? (κ ∧ α) = αH ,
we obtain
−ιξ(?α) = −ιξ(?αH), since −ιξ(?αV ) = 0 by Prop. C.1.18 and Eq. (C.1.7),
= ιξ(?(?(κ ∧ α)),
= ιξ(κ ∧ α)), by C.1.7,
= α, by property of the contraction ιξ and α ∈ Ω1(H).
Thus, we have shown
α = −ιξ(?α),
Appendix C. Miscellaneous Results 175
and therefore −ιξ(?α) = ?(κ ∧ α) for α ∈ Ω1(X).
iii. Assume α ∈ Ω2(X) ' Ω2(V )⊕ Ω2(H).
Write
α = αV + αH ,
uniquely for αV ∈ Ω2(V ) and αH ∈ Ω2(H). First observe that we can write
αH = φdκ,
for some φ ∈ C∞(X). This follows from observing that κ∧ αH = φκ∧ dκ for some
φ ∈ C∞(X) since κ ∧ dκ is non-vanishing. Then contract the equation κ ∧ αH =
φκ ∧ dκ by ιξ to obtain ιξ(κ ∧ αH) = αH on the one hand, and ιξ(φκ ∧ dκ) = φdκ
on the other hand. Thus, αH = φdκ as claimed. Now,
ιξ(?α) = ιξ(?(αV + αH))
= ιξ(αH + ?(φdκ)), αH := ?αV ∈ Ω1(H), by Prop. C.1.18,
= ιξ(φκ), since ιξαH = 0 and ?dκ = κ by C.1.7 and C.1.8,
= φ, since ιξκ = 1.
Also,
?(κ ∧ α) = ?(κ ∧ (αV + αH)
= ?(κ ∧ αH), since αV ∈ Ω2(V ) = κ ∧ Ω1(H),
= ?(φκ ∧ dκ),
= φ, since ?(κ ∧ dκ) = 1 by C.1.6 and C.1.7.
Thus, ιξ(?α) = ?(κ ∧ α) for α ∈ Ω2(X), as claimed.
This completes the proof.
Appendix C. Miscellaneous Results 176
C.1.22 Proposition. The following equalities hold
?H κ = 0 (C.1.23)
?H(κ ∧ dκ) = 0 (C.1.24)
?H1 = dκ. (C.1.25)
Proof. These are just an easy consequence of C.1.6 and C.1.8 and from Prop. C.1.21
?Hα = (−1)qιξ(?α) = ?(κ∧α). First, ?Hκ = −ιξ(?κ) = −ιξ(dκ), by C.1.8, and ιξ(dκ) = 0
since ξ is the Reeb field for κ. Thus, ?Hκ = 0. Second, ?H(κ∧dκ) = ?(κ∧κ∧dκ) = ?(0) =
0. Thus, ?H(κ ∧ dκ) = 0. Last, ?H1 = ιξ(?1) = ιξ(κ ∧ dκ) by C.1.6, and ιξ(κ ∧ dκ) = dκ
by property of ιξ. Thus, ?H1 = dκ.
C.1.26 Proposition. κ∧dAκ∧dκ = ?HdHA for A ∈ Ω1(X).
Proof. For this proof we first recall what is meant by the notation κ∧dAκ∧dκ . By definition,
κ ∧ dAκ ∧ dκ
:= φ ∈ C∞(X),
where κ∧dA = φκ∧dκ. Since dA = dVA+dHA for some dVA ∈ Ω2(V ) and dHA ∈ Ω2(H)
(note that we define dH : Ω1(X) → Ω2(H) as the restriction of d to the horizontal
distribution H), we see that κ ∧ dA = κ ∧ dHA. We have
κ ∧ dAκ ∧ dκ
= φ,
= ?(φκ ∧ dκ), by C.1.6 and C.1.7,
= ?(κ ∧ dA),
= ?(κ ∧ dHA),
= ?HdHA, by Def. C.1.16 of ?H .
Thus, κ∧dAκ∧dκ = ?HdHA as claimed.
C.1.27 Proposition. dA = κ ∧ LξA+ dHA for A ∈ Ω1(H).
Appendix C. Miscellaneous Results 177
Proof. To see this we recall that the definition of dH : Ω1(X)→ Ω2(H) is the restriction
of d to the horizontal distribution H. In other words,
dHA = π(dA),
where π : Ω2(X)→ Ω2(H) is defined in Eq. (C.1.13) above. We have
dHA = π(dA),
= dA− κ ∧ ιξ(dA).
Observe that LξA = ιξ(dA) + d(ιξA) = ιξ(dA) since ιξA = 0 for A ∈ Ω1(H). Thus,
dHA = dA− κ ∧ ιξ(dA),
= dA− κ ∧ LξA.
Rearranging this last equation we obtain our result, dA = κ ∧ LξA+ dHA.
C.2 T dRS = T dC
Next we would like to prove T dRS = T dC . As we will see, this follows directly from [RS08,
Theorem 4.2]. First we recall the definition of the contact Laplacian,:
∆q =
(d∗HdH + dHd
∗H)2 if q = 0, 3,
D∗D + (dHd∗H)2 if q = 1,
DD∗ + (d∗HdH)2 if q = 2.
(C.2.1)
on (E , dH). The analytic contact torsion is given by,
TC := exp
(1
4
3∑q=0
(−1)qw(q)ζ ′(∆q)(0)
)(C.2.2)
where,
w(q) =
q if q ≤ 1,
q + 1 if q > 1.
(C.2.3)
Appendix C. Miscellaneous Results 178
in the case where dim(X) = 3. Also recall the Hodge-de Rham Laplacian:
∆DRq := d∗d+ dd∗, on Ωq(X; R). (C.2.4)
The analytic Ray-Singer torsion is given by,
TRS := exp
(1
2
3∑q=0
(−1)qqζ ′(∆DRq )(0)
)(C.2.5)
in the case where dim(X) = 3. Want to see T dRS = T dC as densities. Consider:
detH•(Ω(X), d)∗ :=3⊗j=0
(detHj(Ω(X), d))(−1)j (C.2.6)
detH•((E , dH))∗ :=3⊗j=0
(detHj(E , dH))(−1)j (C.2.7)
and,
|| · ||RS := TRS|| · ||L2(Ω(X),d), (C.2.8)
|| · ||C := TC || · ||L2(E,dH), (C.2.9)
where the L2 norms come from the identifications:
H•(Ω(X), d) ' H•(Ω(X), d), (C.2.10)
H•(E , dH) ' H•(E , dH). (C.2.11)
We have the following
C.2.12 Proposition. [RS08, Prop. 2.3]
detH•(Ω(X), d)∗ ' detH•((E , dH))∗. (C.2.13)
We also have the following
C.2.14 Theorem. [RS08, Theorem 4.2] On any quasi-regular K-contact three manifold
(X,φ,R, κ, g):
|| · ||RS ' || · ||C . (C.2.15)
Appendix C. Miscellaneous Results 179
Let us make the following
C.2.16 Definition. Let
δ| detH•(E,dH)|∗ := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν
qbq|(−1)q
be the density for ||·||L2(E,dH) on | detH•((E , dH))∗| where νq1 , · · · , νqbq is an orthonormal
basis for the space of harmonic contact forms Hq(E , dH). Similarly, let
δ| detH•(Ω(X),d)|∗ := ⊗dimXq=0 |νq1 ∧ · · · ∧ ν
qbq|(−1)q
be the density for || · ||L2(Ω(X),d) on | detH•(Ω(X), d)∗| where νq1 , · · · , νqbq is an orthonor-
mal basis for the space of harmonic forms Hq(Ω(X), d). Then we define
T dC := TC · δ| detH•((E,dH))∗|, (C.2.17)
and,
T dRS := TRS · δ| detH•(Ω(X),d)|∗ . (C.2.18)
Recall that the absolute value of the determinant lines are defined as
| detH•(Ω(X), d)∗| :=3⊗j=0
| detHj(Ω(X), d)(−1)j | (C.2.19)
| detH•((E , dH))∗| :=3⊗j=0
| detHj(E , dH)(−1)j |, (C.2.20)
so that T dC ∈ | detH•((E , dH))∗| and T dRS ∈ | detH•(Ω(X), d)∗|. Our desired result is the
following
C.2.21 Proposition. T dRS ' T dC.
Proof. This follows precisely from the fact that the isomorphism in C.2.12 comes directly
from an isomorphism Ψ : (Ω(X), d) → (E , dH), which we can take to preserve the bases
νq1 , · · · , νqbq, νq1 , · · · , ν
qbq. That is, we take Ψ(νql ) = νql . Thus, Ψ preserves the densities
δ|detH•(Ω(X),d)|∗ , and δ| detH•(E,dH)|∗ and also induces the equivalence Ψ : || · ||RS ' || ·
||C by Theorem C.2.14. Therefore, by these observations, we see that T dRS = TRS ·
δ|detH•(Ω(X),d)|∗ ' TC · δ| detH•((E,dH))∗| = T dC .
Appendix C. Miscellaneous Results 180
C.3 A Standard Result in Cohomology
The result that we would like to prove in this section is the following
C.3.1 Proposition. [FS92], [Nic00, Theorem 1.3] Let X be a Seifert manifold over an
orbifold Σ = (|Σ|,U) (See §2.1). Then,
dimH1(X,R) =
2g n ≥ 1
2g + 1 n = 0
where n is the degree of X over Σ,
U(1) // X
Σ
,
and g is the genus of the base space |Σ|.
Proof. We provide a proof of this fact in the case that X is a principal U(1) bundle
and leave the general case to the references [FS92], and [Nic00, Theorem 1.3]. By the
Universal Coefficient Theorem (UCT),
H1(X,R) ' Hom(H1(X,Z),R) (C.3.2)
i.e. the UCT implies that:
0→ Ext(H0(X,Z),R)→ H1(X,R)→ Hom(H1(X,Z),R)→ 0
is exact. Also
Ext(H0(X,Z),R) ' Ext(Z,R) ' 0
since Z is free. Thus we compute Hom(H1(X,Z),R). By Hurewicz,
H1(X,Z) ' π1(X)
[π1(X), π1(X)].
Hence we have the following presentation of π1(X), [Orl72]:
π1(X) ' 〈ap, bp, h|[ap, h] = [bp, h] = 1,
g∏p=1
[ap, bp] = hn〉
Appendix C. Miscellaneous Results 181
where g is the genus of the base space Σ of our X,
U(1) // X
Σ
and n = c1(X) is the Chern number of the U(1)-bundle X. The generator h arises
from the generic fibre over Σ. Observe that in the abelianization of π1(X) the following
relation is satisfied:g∏p=1
[ap, bp] = hn (C.3.3)
We have
[π1(X), π1(X)] = 〈[ap, bp]|g∏p=1
[ap, bp] = hn〉, (C.3.4)
and therefore,
π1(X)
[π1(X), π1(X)]= 〈ap, bp, h | [ap, bp] = hn = 1〉 =
g⊕p=1
〈ap〉g⊕p=1
〈bp〉⊕(
〈h〉〈hn〉
), (C.3.5)
where ap, bp, h now represent equivalence classes in the abelianization and 〈ap〉 ' Z,
〈bp〉 ' Z, 〈h〉〈hn〉 ' Z/nZ ' Zn. Thus,
π1(X)
[π1(X), π1(X)]'
Z2g × Zn n ≥ 1
Z2g+1 n = 0.
(C.3.6)
Finally we have,
H1(X,R) ' Hom(H1(X,Z),R) ' Hom
(π1(X)
[π1(X), π1(X)],R)'
Hom(Z2g × Zn,R) n ≥ 1
Hom(Z2g+1,R) n = 0
'
R2g n ≥ 1
R2g+1 n = 0
In conclusion,
dimH1(X,R) =
2g n ≥ 1
2g + 1 n = 0
(C.3.7)
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