YANG-MILLS TO SEIBERG-WITTEN VIA TQFT

THE WITTEN CONJECTURE

,

, ,

Black Hills State University

March 22, 2012

Greg Naber

The differential topology of smooth 4-manifolds has witnessed a number of revolutionary ical

ll attempt

Abstract

upheavals in the recent past, no few of which have resulted from an encounter with theoretphysics. One of the most spectacular of these was the emergence of the Seiberg-Witten invariants from Edward Witten’s quantum field theory interpretation of the Donaldson invariants. The terrain one must traverse to follow this development is arduous and I wito provide only a brief aerial view, but references will be provided for those who wish to explore the territory in earnest.

Mathematical Framework for Classical Gauge Theories:

 

         • A smooth, oriented, (semi-) Riemannian manifold (M, g) (space or spacetime) • A finite-dimensional vector space V (internal space) equipped with an inner product ( , )

• A Lie group G (internal structure group) together with a representation ρ:G → GL(V) of G on V that is orthogonal with respect to ( , ) • A smooth, principal G-bundle G P → M over M and a connection (gauge field) ω on P with curvature

cross-section ψ of the vector bundle P × V associated to

• An action functional S(ω, ψ), invariant under the action of G,

More details and many examples are available in Chapter 2 of [8].

(field strength) Fω • A ρ G P → M by the representation ρ (wave function) whose stationary points (ω, ψ), i.e., solutions to the Euler-

Lagrange equations (field equations), are the physical field configurations

Also see [TGP 1], pages 28-32.

Donaldson’s 1983 Theorem (Fields Medal):

Proof: ( More details are available in Appendix B of [7] )

or any Rie at is -an lls equations.

:

(global) gauge transformation of is an automorphism of the bundle that covers the identity on and two connections are gauge equivalent if they differ by such an automorphism.

or a generic choice of the moduli space , of gauge equivalence -anti-self-dual connections on looks like

Contrast this with Friedman’s 1982 classification of compaconnected, oriented, topological 4‐manifolds. Every unimo

near form is the intersectio. A corollary is that there ar

ct, simply dular, 

symmetric,  ‐valued bili n form of at least one such manifold e over 10 million topologically distinct compact, simply connected, oriented, topological 4‐manifolds   rank ( ; =32 that admit no smooth structure. 

with

mannian metric on , any connection on ti-self-dual satisfies the Yang-Mi

Fth

A

Fclasses of

Donaldson Polynomial Invariants:

, , , , …

: ;

, , , …

is a compact, simply connected, oriented, smooth 4-manifold with

1

etric , the moduli space

of gauge equivalence classes of -anti-self-dual connections on is either empty or a smooth, oriented manifold of dimension

, .

, sum of the signed points in ,

If , 0, then

and odd and

is the -bundle over with Chern class k > 0. For generic Riemannian m

,

: ;

is, morally at least, given by

,

where

is a certain map defined by Donaldson which, intuitively at least, associates 2-forms on , to surfaces in .

e are not topological invariants, but are invariant under orientation serving diffeomorphism is reason, they can be used to distinguish

manifolds that are homeomorphic, but not diffeomorphic.

: ; , ;

This “definition” is naïve in the extreme, even after yothe Donaldson μ‐map is, but setting it all straight invo

u know what lves a 

tremendous amount of technical labor. Most of the difficulties arise from the fact that  , is generally not compact. Naïve as it is, it conveys something of the correct flavor and is ofte the way the invariants are thought of in physics. For a sketch of the “real” definition, see [TGP 2], page 69‐76. 

n 

s 

Thpre . For th

There are infinitely many Donaldson invariants , , , , …, they are extraordinarily difficult to compute, and the computational labor increases exponentially with . However, in the Spring of 1994, there was a breakthrough.

Combine all of the Donaldson invariants , , , , … into the Donaldson (formal power) series

!

Kronheimer-Mrowka Structure Theorem:

is of “K ; (KM-asic classes) that

If b

M-simple type”, then there exist , … , and rational numbers , … , (coefficients) such

,,

The point is that, despite all expectations to the contrary, is

he Structure Theorem was announced in the Spring of 1994. In the Fall of 994 a bombshell was dropped and the breakthrough was rendered moot. To ll the story we must return to 1988.

determined by a finite amount of data.

T1te

Donaldson Invariants as Intersection N

, ,

↓

umbers:

↓ ↑

/ / [ω]

oduli Space of Associated vector bundle with Section

/ /

Is there an analogous Gauss-Bonnet-Chern type

integral representation of the Donaldson invariants?

“This is such stuff as quantum field theory is made of.”

nna call”? -Ghostbusters

Mall connections fiber ,

Recall that the Euler number of an oriented real vector bundle E→X of fiber dimension 2k can be 

1. (Poincaré‐Hopf) The intersection number  , where   is a 

2. (Gauss‐Bonnet‐Chern)  , where   is the Euler class. 

dimension 2k over a compact, oriented manifold of computed in a number of ways, e.g.,  

generic section and   is the 0‐section. 

-Nigel Hitchin

so, “who ya go

Ed Witten [10] found such a (formal) integral representation of the Donaldson certa

Topological Quantum Field Theory (TQFT)

Gauge Field (Connection) +

“Matter Fields”

Bosonic Fermionic

, ,

, ,

+ n ”

invariants as expectation values of in observables in his 1988

Field Content:

,

, , , , ,

“Donaldson-Witten Actio

,

,

and the corr

/

where is a coupling co t) measure on the space of fields (this is a formal path integral). Witten’s derivation of in [10] is physically motivated. He “proves” that is independent of the metric and the coupling constant, computes it in the weak coupling limit , where the integral “localizes” to the anti-self dual moduli space giving, when ,

.

Atiyah and Jeffrey [1] show that can be obtained by a formal application to the infinite-dimensional vector bundle , of Donaldson

esponding Partition Function

nstant and is a (nonexisten

theory of an expression for the Euler class

athai-Quillen in the context of equivariant cohomology (see 59-94).

The localization to the moduli s dual connections then appears as an instance of an equivariant localization theorem that is well-known, at

itten also (formally) obtains the remaining Donaldson invariants as expectation values for a family of observables parametrized by ; .

developed by M[TGP 1], pages

pace of anti-self-

least in finite dimensions (see [TGP 2], pages 1-26). W

  Duality, Seiberg-Witten Invariants and the Witten Conjecture:

(For more details see [TGP 2], pages 26-79.)

“Duality” in Witten’s TQFT (1988-1994)

e → 0 ∞

Weak Coupling Strong Coupling

Perturbative Non-Perturbative

Computable Intractible

eiberg-Witten (1994): Exact Solutions in Strong Coupling

, " ‐ "

‐

A = connection on the associated -bundle

= section of the associated positive spinor bundle

e →

                       ?           Donaldson Invariants

S

Seiberg-Witten Invariants

Seiberg-Witten Equations:

tion of gauge equivsolutions and a corresponding moduli space of gauge equivalence classes of

Generically, the moduli space is either empty or a smooth, oriented manifold of dimension

Just as for Donaldson theory there is a no alence for

solutions.

and is always compact.

erg-Witten invariant

is empty.

When the moduli space is not 0-dimensional one can define Seiberg-Witten invariants by integrating over it, but it ero invariants arise only from 0-dimensional moduli spaces (manifolds for which this is true are said to be of SW-simple type).

To formulate Witten’s conjecture we let denote set of all (equivalence classes of) Spinc-structures for which and refer to elements of ; that are for some as SW-basic classes.

If this dimension is zero, then the 0-dimensional Seib

,

is the sum of the signed points in the moduli space if this is nonempty and 0 if it

is conjectured that nonz

The Witten Conjecture:

with 1 and odd. Then

d only if it is of SW

KM-basic classes coincide with SW-basic classes.

iberg-Witten invariants are related by

,

Let be a compact, simply connected, oriented, smooth 4-manifold

1. is of KM-simple type if an -simple type.

2.

3. The Donaldson and Se

,

, , , ,

where

.

Attitudes one might adopt to such a conjecture:

1. It should be rigorously proved.

• Pidstrigatch and Tyurin Leness

2. Rigorously true or not, the Seiberg-Witten invariants provide a much more tractable tool for the study of 4-manifolds so it makes good, practical sense to abandon the anti-self-dual equations in favor of the

• Essentially everyone else

vels, then mathematicians will want to take heed and turn their attention once again to their historical roots in

• Feehan and

Seiberg-Witten equations.

3. If physics is truly capable of casting such a penetrating light upon mathematics at its deepest le

physics.

• Atiyah

References

] Atiyah, M.F. and L. Jeffrey, Topological Lagrangians and Cohomology, J. Geo. Phys., 7(1990), 119-136.

] Donaldson, S.K., An Application of Gauge Theory to Four-Dimensional

[3] Donald ial Invariants for Smooth 4-Manifolds, Topology, 29(1990), 257-31.

lds

[1 [2 Topology, J. Diff. Geo., 18(1983), 279-315.

son, S.K., Polynom

[4] Feehan, P.M.N. and T.G. Leness, Witten’s Conjecture for Four-Manifo of Simple Type, arXiv:math.DG/0609530. [5] Kronheimer, P.B. and T.S. Mrowka, Embedded Surfaces and the Structure of

73-734.

riant

[7] Naber, G.L., Topology, Geometry, and Gauge Fields: Foundations, 2nd Edition, TAM 25, Springer, New York, 2010. [8] Naber, G.L., Topology, Geometry, and Gauge Fields: Interactions, 2nd Edition, AMS 141, Springer, New York, 2011.

] Seiberg, N. and E. Witten, Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD , Nucl. Phys. B 431(1994), 581-640. [10] Witten, E., Topological Quantum Field Theory, Comm. Math. Phys., 117(1988), 353-386. [11]     Witten, E., Monopoles and Four-Manifolds, Math. Res. Lett., 1(1994), 769-796.

lowing two papers are attached as Addenda to this file.

[TGP 1] Naber, G.L., Topology, Geometry and Physics: Background for the Witten Conjecture I, Journal of Geometry and Symmetry in Physics, 2(2004), 1-97.

GP 2] Naber, G.L., Topology, Geometry and Physics: Background for the Witten

, Journal of Geometry and Symmetry in Physics, 3(2005), 1-82.

Donaldson’s Polynomial Invariants, J. Diff. Geo., 3(1995), 5

[6] Mathai, V. and D. Quillen, Superconnections, Thom Classes and Equiva Differential Forms, Topology, 25(1986), 85-110.

[9

The fol

[T Conjecture II