Real and p-Adic Physics
154
Real and p-Adic Physics Brian Raymond Trundy A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by The Department of Physics Adviser: Steven S. Gubser November 2021 Copyright by Brian Trundy, 2021. All rights reserved.
Transcript of Real and p-Adic Physics
Real and p-Adic Physics
Brian Raymond Trundy
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance by
The Department of
Physics
Adviser: Steven S. Gubser
November 2021
© Copyright by Brian Trundy, 2021.
All rights reserved.
Abstract
This thesis examines the consequences of non-Archimedean geometry in hologra-
phy and many-body physics. In this framework real space is replaced by a non-
Archimedean field which is almost always a p-adic field or an algebraic extension of a
p-adic field. The corresponding bulk geometry is replaced by a discrete tree. The re-
sulting theories are not only useful as toy models of real theories - they help elucidate
which parts of physics do and do not depend on the choice of underlying field.
Chapter 1 sets the stage for the rest of the thesis and gives a brief review of
non-Archimedean mathematics and physics.
Chapter 2 is based on [1] coauthored with Steven S. Gubser, Christian Jepsen,
and Ziming Ji. We exhibit a sparsely coupled classical statistical mechanical lattice
model that interpolates between real and p-adic geometry when varying a spectral
exponent. Holder continuity conditions in both real and p-adic space allow us to
quantify how smooth or ragged the two-point Green’s function is as a function of the
spectral exponent. This model was motivated by proposed cold atom experiments [2]
and serves as our bridge into the non-Archimedean world.
Chapter 3 is based on [3] coauthored with Steven S. Gubser, Christian Jepsen,
and Ziming Ji. We study melonic tensor models over real and non-Archimedean
fields in tandem. Much attention is paid to the combinatorial structure of potential
interaction terms and the perturbative expansion. The Schwinger-Dyson equation is
solved exactly in the p-adic case for a subset of these field theories.
Chapter 4 is based on [4] coauthored with Steven S. Gubser and Christian Jepsen.
We examine the bulk dual p-adic field theories whose Green’s functions are non-
trivial sign characters. These theories constitute the simplest known notion of spin in
p-adic physics. The construction is achieved by the introduction of an non-dynamical
U(1) gauge field on the discrete bulk geometry and the two point functions for dual
operators on the boundary are computed explicitly.
i
Acknowledgments
I would like to express my gratitude to all my friends and mentors at Princeton. I
will always look back fondly at my time spent on this campus. Thank you, Herman
Verlinde for seeing this project through with me to its end. I would like to thank Silviu
Pufu for serving as a reader of this thesis and for his valuable counsel. Thank you,
Lyman Page and Simone Giombi for serving as examiners on my FPO committee.
Without the following collaborators, this thesis would not exist: Sarthak Parikh,
whose work effort is rivaled only by his kindness and intellect. Christian Jepsen, one
of the only true polymaths I’ve ever met. Ziming Ji, whose unrelenting mind plumbs
the depths of every problem. Amos Yarom, a master of molding the complex into
the straightforward. Matthew Heydeman, a mind of unmatched curiosity. Bogdon
Stoica, whose creativity and breadth knows no bounds. Ingmar Saberi, a towering
intellect. Matilde Marcolli, a seeker of the most profound truths. Gregory S. Bentsen
and Monika Schleier-Smith, whose invaluable experimental insight bridges the gap
between real life and the most abstract theory.
I would like to thank my wife Brittany Holom-Trundy and our two dogs Mason
and Elvis. Our home together makes life worth living. Thank you, Mom and Dad,
for your unending support and encouragement.
Lastly, thank you Steven Scott Gubser. Your loss hit me more deeply than I
thought it could. Yet my loss pales in comparison to that felt by the whole of Prince-
ton, your family, and the scientific community. You were remarkable in every aspect
of your life and I fear that I will never be able to repay you for your guidance and
generosity.
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Contents
1 Introduction 1
1.1 The p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Bruhat-Tits Tree and its symmetry group . . . . . . . . . . . . . 7
1.3 Functions of a p-adic variable . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Integration and additive characters . . . . . . . . . . . . . . . . . . . 10
1.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Multiplicative characters and Gel’fand-Graev gamma functions . . . . 14
1.7 p-Adic field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 p-Adic CFT and the p-adic AdS/CFT correspondence . . . . . . . . . 19
2 Continuum Limits of Sparsely Coupled Ising-like Models 24
2.1 The Statistical Mechanical Framework . . . . . . . . . . . . . . . . . 27
2.2 Nearest neighbor coupling . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 p-Adic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Power-law coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Sparse coupling: From the Archimedean to the non-Archimedean . . 35
2.6 Continuity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 2-adic field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8 Archimedean field theory . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9 Numerical evidence: 2-adic Approximation of sparse results . . . . . . 46
2.10 Numerical evidence: Holder bounds in momentum space . . . . . . . 47
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2.11 Numerical evidence: Holder bounds in position space . . . . . . . . . 50
2.12 The liminal region −1/2 < s < 1/2 . . . . . . . . . . . . . . . . . . . 52
2.13 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Higher Melonic Field Theories 60
3.1 Structure of higher melonic theories . . . . . . . . . . . . . . . . . . . 63
3.2 Symmetry groups of interaction vertices . . . . . . . . . . . . . . . . 68
3.3 Construction of interaction vertices . . . . . . . . . . . . . . . . . . . 73
3.4 One-factorizations and equivalent interaction terms . . . . . . . . . . 77
3.5 Finding the number of isomorphism classes of one-factorizations for
q = 12 using the orderly algorithm . . . . . . . . . . . . . . . . . . . 80
3.6 Isomorphism classes of ordered one-factorizations for q = 12 . . . . . 85
3.7 The two-point function and the Schwinger-Dyson equation . . . . . . 89
3.7.1 The IR solution . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7.2 The zoo of theories . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7.3 Full solution to the Schwinger-Dyson equation for direction-
dependent theories . . . . . . . . . . . . . . . . . . . . . . . . 97
3.8 The four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.8.1 Adelic product formula for the integral eigenvalues . . . . . . 104
3.9 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 Holographic Duals of Nontrivial Characters in p-adic AdS/CFT 107
4.1 Nearest neighbor actions . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1.1 Bosonic actions . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1.2 Fermionic actions . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2 The background geometries . . . . . . . . . . . . . . . . . . . . . . . 115
4.3 Bulk-to-boundary propagators . . . . . . . . . . . . . . . . . . . . . . 118
4.3.1 Scalars on Tp . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
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4.3.2 Scalars and fermions on the line graph . . . . . . . . . . . . . 119
4.4 Two-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4.1 Scalars on Tp . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4.2 Scalars on the line graph . . . . . . . . . . . . . . . . . . . . . 125
4.4.3 Fermions on the line graph . . . . . . . . . . . . . . . . . . . . 126
4.5 Gauge field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.6 Other sign characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
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1
Introduction
Among the remarkable properties of the crossing symmetric version of the Veneziano
amplitude [5]
A(s, t, u) = B(−α(s),−α(u)) +B(−α(s),−α(t)) +B(−α(t),−α(u)), (1.1)
is that it can be expressed [6] as an infinite product over all prime numbers p:
A(s, t, u) =∏p
Ap(s, t, u)−1, (1.2)
where Ap(s, t, u) is the four point amplitude for the open p-adic string [7, 8]. Calcula-
tions in p-adic string theory and p-adic quantum field theory are often more tractable
than their real counterparts, so at the time of its discovery (1.2) suggested an excit-
ing new way to dig up facts about strings. The above relation is an example of what
is known as an adelic identity. These are identities that treat the real line and the
p-adic numbers Qp for each prime impartially, and can have the flavor of expressing
real quantities in terms of their p-adic “building blocks”. The name is in reference to
the ring of adeles A, which is a (restricted) product of R and all of the p-adic number
fields. One can think of Qpprimes p and R as exhausting the collection of topologies
1
that can be given to a continuum, and the adeles treat all of these topologies in tan-
dem. We will cover the basics of the p-adic number fields in this chapter, and ease
the reader into the p-adic domain by considering a model that interpolates between
Archimedean and p-adic topologies in chapter 2.
Disappointingly, the 5-point amplitude for the open string was found not obey
such an identity [9], and excitement about p-adic string theory decreased as a result.
A sober view of the above adelic formula might be that it follows from the Veneziano
amplitude being expressible as a special function - a so called Gel’fand-Graev beta
function, that can be defined naturally over the ring of adeles. While the hope of adelic
miracles has been nearly retired, interest in p-adic physics has enjoyed a renaissance
due to the recent acknowledgement of a p-adic version [10, 11] of the AdS/CFT
correspondence [12, 13, 14]. One ends up finding formulas that are extremely similar
to those found in real AdS/CFT. These results would be unsurprising if not for the
fact that most calculations in p-adic AdS/CFT are accomplished via discrete sums
over a discrete version of AdS known as the Bruhat-Tits tree. That is, it is harder
to write off the similarities as being the result of a surface level resemblance between
the real and p-adic integrals that are used to calculate the observables. Moreover,
many of the tools that one employs to study Archimedean AdS/CFT can be ported
over to the p-adic version with varying levels of modification. This includes the use of
Mellin space to calculate amplitudes [15, 16, 17, 18, 19], holographic tensor networks
[20, 21, 22, 11], entanglement entropy and a Ryu-Takanagi-like formula [23, 24], and
a gravity-like action that describes the fluctuation of edge lengths within the p-adic
bulk [25].
Studying the p-adic counterpart of a real theory gives not only adelic formulas,
but it also reveals the aspects of theories which are invariant when one changes the
topology that is given to the continuum. This point is perhaps best appreciated in the
context of O(N) theory [26]. Moreover, sometimes the solution of the p-adic version
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of a theory is easier to come by than for the real theory. For instance, certain p-adic
tensor models can be solved exactly [27], and in chapter 3 we will encounter exact
solutions together with adelic formulas in the context of tensor models with more
complicated interaction terms. The p-adic analogue of spin and of higher dimensions is
possibly the most mysterious of all non-Archimedean subjects. While p-adic theories
with anti-commuting fields have been studied in the context of melonic tensor models,
p-adic superstrings [28, 29, 30, 31], and Gross-Neveu analogues [32], the relationship
between p-adic representations involving non-trivial sign characters and Archimedean
spin remains somewhat elusive. In chapter 4, we describe constructions on the line
graph of the Bruhat-Tits tree that are dual to p-adic CFTs that contain operators
that transform under these signed representations.
The rest of chapter serves as both an introduction to the thesis and as a repository
for foundational material that the rest of the chapters rely on - the organization is as
follows. In section 1.1 we give a brief review of the construction and properties of the
p-adic number line Qp. In section 1.2 we introduce the Bruhat-Tits tree construction,
identify P1(Qp) with its boundary, and discuss the relationship between its symmetry
groups and those of P1(Qp). In section 1.3 we discuss the general properties of func-
tions of a p-adic variable, including the analogues of smoothness and the lack of the
usual derivative. In section 1.5 we discuss multiplicative characters on Qp and there
relationship with the rich variety of quadratic extensions of Qp. There are many high
quality references that review the material in sections 1.1-1.4 of which [33, 34, 35, 36]
are a few. In section 1.7 we lay out the general features of the p-adic statistical field
theories that we will encounter in the rest of this thesis. Following this, we give a
brief review in section 1.6 of some special functions of a p-adic variable together with
their relationship with Archimedean special functions. The original references for this
material are [37, 38], with [34] being a handy manual for calculation. In section 1.8
we discuss the p-adic version of the AdS/CFT correspondence [10].
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1.1 The p-adic numbers
In elementary analysis we learn that the real numbers R are the completion of the
rational numbers Q. Implicit in this construction is the choice d(x, y) to define the
notion of a Cauchy sequence. What privileges the use of the usual notion of absolute
value |p/q| = |p|/|q| to define our distance d(x, y) = |x− y| on Q? In order for d(x, y)
to satisfy the usual axioms for a metric, we must have:
|x| ≥ 0 (non-negativity)
|x| = 0⇔ x = 0 (point seperating)
|x+ y| ≤ |x|+ |y| (triangle inequality)
Additionally, it is natural to require that | · | respect the multiplicative structure of
Q:
|xy| = |x||y|. (multiplicativity)
These are the axioms of a norm over the one dimensional vector space Q and it is
natural to ask whether | · | is the only such norm. The answer is given by Ostrowski’s
theorem [33]
All nontrivial norms on Q are topologically equivalent to either the real abso-
lute value | · | or to one of the p-adic norms | · |p, of which there is one for every
prime p.
The real absolute value is set apart by its respect for Q’s linear ordering and its
Archimedean property:
Given non-zero x ∈ Q there exists a natural number N such that |N · x| > 1.
4
By contrast, any norm that doesn’t satisfy the Archimedean property must instead
satisfy the ultrametric triangle inequality:
|x+ y|p ≤ max(|x|p, |y|p). (1.3)
One can go about proving this via the binomial theorem. Suppose we have a norm
for which |k| ≤ 1 for all k ∈ Z. Take x 6= 0, then:
|1 + x|n ≤n∑k=0
∣∣∣∣(nk)∣∣∣∣ |x|k (1.4)
≤ (n+ 1) max(1, |x|n). (1.5)
After taking then n’th root and then taking the limit n→∞, we obtain the inequality
|1 + x| ≤ max(1, |x|). The ultrametric inequality (1.3) quickly follows.
The p-adic norms are defined as follows. By fiat, |0|p = 0. Given any nonzero
rational number x, consider the exponents vp(x) in its prime decomposition 1:
x = ±∏
primes p
pvp(x). (1.6)
One can check that vp(xy) = vp(x) + vp(y) and vp(x + y) ≥ min(vp(x), vp(y)). vp(x)
is known as the p-adic valuation of x, and we can use it to define the p-adic norm for
non-zero2 x:
|x|p = p−vp(x). (1.7)
It is conventional to identify the reals with p = ∞, and we will sometimes write the
real absolute value as | · |∞ to distinguish it from its p-adic counterparts. Just as
1This infinite product is well defined because co-finitely many vp(x) vanish.2One usually uses the conventions p−∞ = 0 and vp(0) = +∞ so that this formula holds for all
x ∈ Q.
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we can complete Q using the real absolute value to form the real numbers, we can
complete Q using the p-adic norm to form what is known as the p-adic numbers Qp.
And just as | · | extends to a norm defined on all real numbers, | · |p extends to a
norm defined on all p-adic numbers. Note that |pk|p = p−k, so 1, p, p2, . . . , pk, . . .
represents a sequence of exponentially smaller numbers with respect to the p-adic
norm. For this reason, we can make sense of p-adic series:
x = a0 + a1p+ a2p2 + a3p
3 + · · · . (1.8)
In fact, all elements x ∈ Qp have a unique representation as such a series [33]:
x = pvp(x)
∞∑n=0
anpn, (1.9)
where a0 6= 0 and an ∈ 0, . . . , p − 1. Note again the characteristic property of a
non-Archimedean absolute value: all integers N ∈ Z have |N |p ≤ 1. In fact, the
p-adic unit ball is known as the ring of p-adic integers
Zp = x ∈ Qp||x| ≤ 1. (1.10)
Any p-adic number x =∑∞
n=νp(x) anpn has a unique decomposition into an integer
part [x] =∑∞
n=0 anpn ∈ Zp and a fractional part x = x− [x]. The unit group of Zp
is identical to the unit sphere in Qp and is denoted by Up = x ∈ Qp||x| = 1.
The group of nonzero p-adic numbers (Q×p ,×) is isomorphic to the group Z× Up
via the correspondence:
Z× Up → Q×p (1.11)
(n, u) 7→ p−nu. (1.12)
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The related decomposition
Q×p =⊔n∈Z
p−nUp (1.13)
is sometimes useful. The collection of balls B = B(x, n)|x ∈ Qp, n ∈ Z where
B(x, n) = |x− y| ≤ pn (1.14)
= x+ p−nZp (1.15)
form a basis for the topology on Qp. Note that the balls defined above are usually what
are known as closed balls (as opposed to open balls), but |x−u| < pn = B(x, n−1)
so this distinction is a matter of convention. By definition B(x, n) are all open sets.
Since Up = tp−1i=1 (i+ pZp) is open and
Qp \ Zp =∞⊔n=1
p−nUp, (1.16)
we find that any B(x, n) is also closed. An interesting consequence of the ultrametric
inequality is that if |x− y| = r then B(x, s) = B(y, s) whenever s ≥ r.
1.2 The Bruhat-Tits Tree and its symmetry group
The uniqueness of the p-adic series expansion (1.9) gives rise to the geometric inter-
pretation of the p-adic numbers (plus infinity) as the boundary of a regular p+1-valent
tree - the Bruhat-Tits tree Tp. To see precisely how this identification comes about,
pick some vertex c ∈ Tp to serve as the tree’s center and fix a path 3 ` that is dou-
bly infinite. These assignments are shown in figure 1.1. We can identify the tree’s
boundary ∂Tp with the collection of infinite paths (or rays) that start at c. Given a
3We require that all edges in a path are distinct, so that there is no backtracking.
7
∞· · · · · · 0······
···
· · ·
······
···
· · ·
······
···
· · ·
······
···
· · ·
······
···
· · ·
p−2Up
p−2
p−1Up
p−1
p0Up
p0
p1Up
p1
p2Up
p2c `
Figure 1.1: The Bruhat-Tits Tree.
ray r starting at c, we can construct the following p-adic expansion:
x(r) = pν(r)
∞∑n=0
xn(r)pn, (1.17)
where ν(r) is the (directed) number of steps the ray takes before leaving ` and
xn(r) ∈ 0, . . . , p− 1 denotes the n’th choice of edge (starting at 0) that the ray
takes as it makes its way up the tree. One can see by construction that we actu-
ally only have p − 1 choices at the n = 0’th step, consistent with the requirement
x0(r) = 0. There are two rays of special interest that never leave `, the one going in
the positive ν direction converges to 0 and the one in the negative −ν direction to
∞. Indeed, the correspondence x : ∂Tp → P1(Qp) is one to one.
In the context of the p-adic AdS/CFT correspondence, we will be interested in the
possible symmetries of theories defined on the tree Tp as this will give us insight into
the analogue of the conformal group on Qp. The isometry group of Tp is enormous, and
it will be helpful to have a way of expressing distances between points on the tree in
terms of boundary points. Remarkably, one can write [39] the distance d(a, b) between
two points a, b ∈ Tp in terms of the cross ratio of boundary points x, y, z, w ∈ P1(Qp),
where the intersection of the (undirected) paths x→ y and w → z is exactly a→ b:
p−d(a,b) = |x, y; z, w|p =|x− z|p|y − w|p|x− w|p|y − z|p
. (1.18)
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It’s a simple exercise to show that the cross ratio is invariant under the group
PGL(2,Qp) of linear fractional transformations
x→ ax+ b
cx+ dwhere ad− bc 6= 0. (1.19)
Armed with (1.18) one can identify the isometry group of Tp with the group of func-
tions f on the boundary that preserve the cross ratio [40]:
|f(x), f(y); f(z), f(w)|p = |x, y; z, w|p. (1.20)
We can characterize this group as follows: let f(x) satisfy (1.20). Set h(x) =
f(g−1(x)) where g(x) is the linear fractional transformation satisfying g(f−1(0)) = 0,
g(f−1(1)) = 1, and g(f−1(∞)) = ∞. Then h(x) fixes x = 0, 1, and ∞. Moreover
|h(x)|p = |h(x), h(1);h(0), h(∞)|p = |x, 1; 0,∞|p = |x|p, so h(x) is an isometry. We
follow [40] and identify the conformal group of Qp with cross-ratio preserving trans-
formations:
Conf(Qp) = g h| g ∈ PGL(2,Qp) and h is an isometry fixing 0 and 1. (1.21)
1.3 Functions of a p-adic variable
Since the p-adic numbers are, among other things, a topological space we can talk
about continuous functions with p-adic domain or co-domain. Indeed, if V is any
topological space, f : Qp → V is continuous at x if it takes open neighborhoods
U ⊆ V of f(x) to open pre-images f−1(U) ⊆ Qp. In the case V = R, this is
equivalent to the usual definition from analysis: given ε > 0 there exists δ > 0 such
that |f(x)− f(y)|∞ < ε whenever |x− y|p < δ.
However, the notion of a smooth function f : Qp → R does not generalize. This is
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mainly due to the difficulty of defining an appropriate derivative df(x)/dx. Indeed,
the derivative is meant to be a linear approximation of a function f(x) − f(0) =
f ′(0)x+O(x). But if f(x)−f(0) ∈ R and x ∈ Qp, what then is f ′(0)? An appropriate
replacement for smooth functions turns out to be locally constant functions. For
instance, locally constant functions with compact support are used to construct p-
adic Schwartz spaces in p-adic harmonic analysis [36].
We can define local constancy quite generally: a map g between a topological
space V and a set X is locally constant if every v ∈ V has an open neighborhood O
such that g(o) = g(v) whenever o ∈ O. Local constancy is rather uninteresting in the
Archimedean regime - a locally constant function f : R→ R is simply constant. This
is not so for functions of a p-adic variable. Consider the composition
f(x) = h(|x|p), (1.22)
where h : R→ R is any function. Then f is locally constant - it is constant over each
open sphere SN = x ∈ Qp||x|p = p−N.
1.4 Integration and additive characters
Since the p-adic numbers constitute a locally compact Hausdorff topological group
(under addition) we can uniquely define a Harr measure on Qp by fixing its normal-
ization. The standard choice is to take
µ(Zp) =
∫Zpdx = 1. (1.23)
Using the identity Zp = tp−1i=0 (i + pZp) together with the translation invariance and
additivity of µ, we obtain the equation µ(pZp) = p−1. More generally, multiplying by
10
pn scales the volume of a set by p−n. In particular, since Up = Zp \ pZp, we have:
∫Updx = 1− 1
p. (1.24)
Using (1.11) and countable additivity of the measure, we arrive at:
∫Qpf(x)dx =
∑n∈Z
p−n∫Upf(pnu)du. (1.25)
The p-adic integral breaks up into an infinite sum of integrals over the p-adic unit
group.
The additive group Qp is Pontryagin self-dual - all additive characters χ : Qp → S1
are of the form:
χ(x) = exp(2πikx), (1.26)
where k ∈ Qp. Heuristically, the fractional part of kx is used since the remaining
terms in the p-adic expansion are positive integers and don’t change the value of
the exponential. Therefore, sometimes the braces are dropped and one uses the
convention exp(2πiz) = 1 for z ∈ Zp. We define the Fourier transform over Qp with
the following conventions:
f(k) =
∫Qpdx χ(kx)f(x) and f(x) =
∫Qpdk χ(−kx)f(k) (1.27)
Just as the Gaussian is a fixed point of the Fourier transform on R, we can define
a p-adic Gaussian that is a fixed point of (1.27). It turns out that the appropriate
11
quantity is the indicator function of Zp:
γp(x) =
1 x ∈ Zp
0 otherwise
. (1.28)
We also have a p-adic delta function:
δ(x) =
∫Qpdk χ(−kx), (1.29)
satisfying the usual delta function identity, this time over Qp:
f(x) =
∫dyf(y)δ(x− y). (1.30)
1.5 Field extensions
By the fundamental theorem of algebra, the only nontrivial algebraic extension of
the real numbers is C. The story is more complicated for Qp, as it turns out to
have nontrivial degree n extensions for all n > 1. Suppose F/Qp is a degree n field
extension. Given a ∈ Q, the mapping x 7→ ax is Qp linear and we define the field
norm (not to be confused with | · |p) of a to be the absolute value of its determinant:
NF/Qp(a) = det(x 7→ ax). (1.31)
Note that NF/Qp is Qp valued. In fact, we can use this to define an absolute value on
F
|a|F = |NF/Qp(a)|p. (1.32)
12
Note that for x ∈ Qp we have |x|F = |x|np . We can use this absolute value to define
analogues of Zp, Up and pZp for F
ZF = x ∈ F ||x|F ≤ 1 (1.33)
UF = x ∈ F ||x| = 1 (1.34)
PF = x ∈ F ||x|F < 1. (1.35)
Considered as a subset of the ring Zp, the ideal x ∈ Qp||x|p < 1 is principal -
generated by p. Similarly, one can show that PF is principal and generated by some
ω known as the uniformizing element of PF . Moreover ZF/ωZF = Fq is a finite field
with characteristic p (indeed p ∈ ωZF ) and |ω|F = 1/q. Since Fq has characteristic p,
we have q = pf for some f . The uniformizing element essentially plays the same role
in F that p did for Qp, for instance any a ∈ F has a unique representation
a = ωνF (a)∑n≥0
anωn, (1.36)
where a0 6= 0 and an ∈ ZF/ωZF .
Note that |p|F < 1, so there must be some e ≥ 1 such that p = ωeu where u ∈ UF .
This lead us to the following identity: p−n = |p|F = |ωe|F = p−ef , so n = ef . The
integer e is known as the ramification degree of the extension. The case e = 1 is that
of an unramified extension, whose uniformizing element can be chosen to be p. The
case e = n is that of a totally ramified extension, whose uniformizing element can be
chose to be p1/n. For a general ramification index, the uniformizer can be chosen to
be p1/e. Many of the contents of this thesis have natural generalizations to extensions
of Qp, but we have chosen to stick to Qp in these situations for simplicity.
The case n = 2 will be of special interest to us, so we develop it further here.
The quadratic extensions of Qp together with the trivial extension Qp(1) = Qp are
13
in one to one correspondence with the factor group Q×p /(Q×p )2. We can refine the
decomposition (1.13) further by noting that Up ' F×p × U1 where U1 = 1 + pZp is a
subgroup of Up. One can show [36] that for odd p, we have U21 = U1 so that
[Q×p : (Q×p )2] = 4. (p > 2). (1.37)
There are three corresponding nontrivial quadratic extensions of Qp(√τ) for p > 2.
They are generated by√τ =√p,√ε, and
√pε where ε is a (p − 1)’th root of unity
in Qp.
The case of p = 2 is trickier. To see why, suppose we are given 1 + 2z where
|z|2 < 1. Then (1 + 2z)2 = 1 + 4z + 4z2 = 1 + O(23). In fact, for p = 2 we have
U21 = 1 + 23Zp. We then have
[Q×2 : (Q×2 )2] = 8. (p = 2). (1.38)
There are seven corresponding nontrivial quadratic extensions Q2(√τ) of Q2. The
generators can be chosen to be√τ where τ = −1,±2,±3, and ± 6.
1.6 Multiplicative characters and Gel’fand-Graev
gamma functions
The goal of this section is to introduce special functions due to Tate, Gel’fand, and
Graev [37, 38]. The presentation is clearest if let K = R,C, and Qp and treat all
three cases in tandem. We gave a classification of the additive characters over Qp in
(1.26). By a multiplicative character, we mean a group homomorphism π : K× → C×.
For a given additive character χ : K → C, the Gel’fand-Graev gamma function
14
associated with a multiplicative character π : K → C is defined by
Γ(π) =
∫K
dx
|x|Kχ(x) π(x) . (1.39)
The multiplicative characters that are most relevant to this thesis are
πs(t) ≡ |t|sK , πs,sgn(t) ≡ |t|sK sgn(t) . (1.40)
where a sign function sgn(t) is any multiplicative character taking only the values ±1.
It is straightforward to show using the definition (1.39) that the Fourier transforms
of the multiplicative characters (1.40) are given by
F [πs](ω) = Γ(πs+1) π−s−1(ω) , F [πs,sgn](ω) = Γ(πs+1,sgn) π−s−1,sgn(ω) . (1.41)
In order to write down explicit expressions for the Gel’fand-Graev gamma func-
tions associated with R and Qp and the multiplicative characters (1.40), it is expedient
to introduce the local zeta functions ζ∞, ζp : C→ C:
ζ∞(s) = π−s2 ΓE
(s2
), ζp(s) =
1
1− p−s, (1.42)
where ΓE(s) is the familiar Euler gamma function. ζp(s) = (1 − p−s)−1 is so named
because of the adelic relation ζ(s) = Πpζp(s).
For K = R, the Gel’fand-Graev gamma functions are given by
Γ(πs) =ζ∞(s)
ζ∞(1− s), Γ(πs,sgn) = i
ζ∞(1 + s)
ζ∞(2− s). (1.43)
For K = C there are no sign functions, and the Gel’fand-Graev gamma function
15
is given by
Γ(πs) = (2π)−2s (ΓE(s))2 sin(πs) . (1.44)
As we saw in section 1.5, for K = Qp, there are multiple distinct quadratic extensions
Qp(√τ). Associated with each of these extensions, there is a sign function, which
takes the value 1 on the image of the field norm NQp(√τ)/Qp
sgnτ (x) =
1 x = a2 − τb2 for some a, b ∈ Qp
−1 otherwise
. (1.45)
We can rephrase this definition as follows: given z = a +√τb ∈ Qp(
√τ), define the
conjugate z∗ = a−√τb. Then sgn(x) = 1 if and only if x = z∗z for some z ∈ Qp.
For p = 2, there are seven distinct non-trivial sign functions corresponding to
τ = −1, ±2, ±3, and ±6, and the gamma functions evaluate to
Γ2(πs) =ζ2(s)
ζ2(1− s),
Γ2(π(−3)s ) =
ζ2(1− s)ζ2(2s)
ζ2(2− 2s)ζ2(s),
Γ2(π(−1)s ) = Γ2(π(3)
s ) = i4s
2,
Γ2(π(−2)s ) = −Γ2(π(6)
s ) = i8s√
8,
Γ2(π(2)s ) = −Γ2(π(−6)
s ) =8s√
8.
(1.46)
For p > 2 there are three distinct non-trivial sign functions, which can be labeled
by τ equal to p, ε, and εp, where ε is an integer that is not a square modulo p. The
16
Gel’fand-Graev gamma functions are given by
Γp(πs) =ζp(s)
ζp(1− s), Γp(π
(ε)s ) =
ζp(1− s)ζp(2s)ζp(2− 2s)ζp(s)
,
Γp(π(p)s ) =
ps√p,
−i ps
√p,
Γp(π(εp)s ) =
− ps√p
for p ≡ 1 mod 4 ,
ips√p, for p ≡ 3 mod 4 .
(1.47)
1.7 p-Adic field theory
Statistical field theory over the p-adic numbers can be defined in the same way as
field theory over R. We consider an action functional S(φ) on a space of fields φ(x)
of a p-adic variable x ∈ Qp and form the partition function
Z(J) = Z(0)
∫Dφ exp(−S(φ) + φ · J), (1.48)
where φ · J =∫Qp φ(x)J(x)dx is shorthand for the inner product between functions
formed by the p-adic integral. There is no clear p-adic equivalent of the derivative on
R, so it is easier to start by writing p-adic actions in momentum space. The quadratic
part of the action for a scalar field takes the form:
Squad(φ) =1
2
∫Qpφ(−ω)(|ω|s + r)φ(ω). (1.49)
Introducing the Vladimirov derivative of weight s
Dsφ(x) =
∫Qpdyφ(x)− φ(y)
|x− y|s+1, (1.50)
17
one can write the action in position space
Squad(φ) =1
2
∫Qpdx(φ(x)Dsφ(x) + rφ(x)2). (1.51)
The free two-point function associated with the action (1.49) is simply
〈φ(ω1)φ(ω2)〉0 = G(ω1)δ(ω1 + ω2), (1.52)
where
G(ω) =1
|ω|s + r. (1.53)
We will also have occasion to consider actions involving N ≥ 1 real commuting or
anti-commuting fields ψi. To that end, we take (1.49), generalize to K = R,C, or Qp,
drop the mass term, replace |k|s with an arbitrary multiplicative character π, and add
index structure:
Sπ(ψ) =1
2
∫dωψi(−ω)Ωijπ(ω)ψj(ω). (1.54)
where π : K× → C× is a multiplicative character.
We will choose Ωij to be either symmetric or anti-symmetric. For the symmetric
case we define σΩ = 1 and set Ω = δij. For the anti-symmetric case we let σΩ = −1
and set Ω = σ2 ⊗ 1N/2 to be a standard symplectic form. Note that N must even in
order for the index structure to be non-singular since det(ΩT ) = (−1)N det(Ω). We
encode the statistics of ψ in the following identity ψiψj = σψψjψi. So σψ = 1 for
commuting fields, and σψ = −1 for anti-commuting fields. Then making the change
18
of variable ω → −ω, we write
Sπ(ψ) =1
2
∫dωψi(−ω)Ωijπ(ω)ψj(ω) (1.55)
=1
2
∫dωψi(ω)Ωijπ(−ω)ψj(ω) (1.56)
= π(−1)σΩσψSπ(ψ). (1.57)
Specializing to characters of the form πs,sgn(ω) = |ω|s sgn(ω), we obtain the constraint
σΩσψ sgn(−1) = 1. (1.58)
The free two-point function associated with (1.54) is:
G(ω) =Ωij
πs,sgn(ω)= Ωij sgn(ω)|ω|−s, (1.59)
where ΩijΩjk = δij.
1.8 p-Adic CFT and the p-adic AdS/CFT corre-
spondence
Not long after Freund and Olson’s discovery of the p-adic string [7], an axiomatic
framework for p-adic conformal field theory was suggested in [41]. In Melzer’s formu-
lation, PGL(2,Qp) plays the role of the global conformal group. Anticipating p-adic
AdS/CFT, we note here that the symmetry group of the Bruhat-Tits tree is larger
(1.21) than just PGL(2,Qp). However, we will only find need to consider representa-
tions that are trivial on the subgroup of isometries h(x) with h(0) = 0 and h(1) = 1.
Melzer’s framework consists of locally constant fields φ : Qp → C. Because of a
lack of a derivative on these fields, one cannot use the standard techniques of quantum
theory that make use of the Lie algebra of ones symmetry group. In other words,
19
we cannot make use of infinitesimal symmetry transformations. Thus Melzer calls
primary operators those φ(x) that transform under PGL(2,Qp) as
φ′(x′) =
∣∣∣∣ ad− bc(cx+ d)2
∣∣∣∣−∆φ
p
φ(x) for x′ =ax+ b
cx+ d. (1.60)
That is, in p-adic CFT we have no descendants and only have pseudo-primary oper-
ators. We simply call these operators primary in the p-adic context. As standard in
conformal field theory, using the representation (1.60) one can easily determine the
form of the (normalized) two-point function:
〈φi(x)φj(0)〉 = δ∆i,∆j
1
|x|2∆ip
. (1.61)
In p-adic CFT without descendants we have the following OPE:
φi(x)φj(0) =∑k
Cijk|x|−∆i−∆j+∆kp φk(0), (1.62)
where the sum runs over p-adic primaries. The three point function therefore takes
the form, for |x2 − x3|p < |x1 − x2|p:
〈φi(x1)φj(x2)φk(x3)〉 =∑`
Cij`|x23|−∆j−∆k+∆`p 〈φi(x1)φ`(x3)〉 (1.63)
= Cijk|x12|−∆ij,kp |x13|
−∆ik,jp |x23|
−∆jk,ip , (1.64)
where the last equality follows from the identity |x12| = |x13| and we use the shorthand
∆i1...in,j1...jm = ∆i1 + · · · + ∆in − ∆j1 − · · · − ∆jm together with xij = xi − xj. This
result can also be obtained by performing an appropriate PGL(2,Qp) transformation
on 〈φi(∞)φj(0)φk(1)〉. In p-adic CFT, the conformal blocks take a very simple form.
20
Consider the following four-point function for |x| ≤ |1− x| ≤ 1. We have:
〈φi(∞)φj(1)φk(0)φ`(x)〉 =∑m
CijmCk`m|x|−∆k`,mp . (1.65)
Similar formulas result from taking |1 − x|p ≤ |x|p = 1 and 1 ≤ |x|p = |1 − x|p, and
taking x = 1 we can derive the consistency condition [41]
∑m
CijmCklm =∑m
CilmCkjm =∑m
CikmCjlm. (1.66)
The OPE associativity condition boils down to an associativity condition on the
structure constants of a commutative algebra.
We will will have occasion to consider more general projective representations of
PGL(2,Qp), that satisfy
φ′(x′) =
√sgn
(ad− bc
(cx+ d)2
) ∣∣∣∣ ad− bc(cx+ d)2
∣∣∣∣−∆φ
p
φ(x) for x′ =ax+ b
cx+ d, (1.67)
and lead to the two point function
〈φ(x)φ(0)〉 = sgn(x)1
|x|−2∆φ. (1.68)
Whether φ must be commuting or anti-commuting in this context depends on sgn(−1)
and its behavior under Hermitian conjugation.
Looking back, the roots of p-adic AdS/CFT lie in a paper by Zabrodin [42]. He
showed that the non-local p-adic string action
S(O) ∼∫dxdy
(O(x)−O(y))2
|x− y|2p(1.69)
21
could be obtained by considering a lattice field theory
S(ϕ) =1
2
∑A∼B
(ϕA − ϕB)2, (1.70)
where A ∼ B indicates that the sum ranges over all neighboring vertices on Tp.
One integrates out the interior of Tp and obtains (1.69). In hindsight, this can be
interpreted as a holographic computation of the two-point function 〈O(x)O(0)〉 ∼
|x|−2.
In [10] a full framework for holographic calculations on Tp was developed. Many
of the results are well expressed by adopting the following coordinate system on the
tree. We note that any vertex in the tree can be uniquely identified by a pair (x, z)
where x ∈ Qp is a boundary point and z = pn indicated the “depth” of the vertex
along the tree. Formally, we have the vertex equality A = B when zA = zB and
|xA − xB| ≤ |zA|.
In the simplest variation of p-adic AdS/CFT we consider the action (1.70) and
add a mass term since we anticipate it to be related to the scaling dimension of the
dual boundary operator:
S(ϕ) =1
2
∑A∼B
(ϕA − ϕB)2 +∑A∈Tp
1
2m2φ2
A. (1.71)
The Green’s function for A,B ∈ Tp is
GAB = ζp(2∆)p−∆d(A,B), (1.72)
where m2∆ = (−ζp(−∆)ζp(∆− 1))−1. Compare this to the real case m2
∆ = ∆(∆− n).
The bulk to boundary propagator K(A, x) satisfying the normalization condition
∫QpK(A, x)dx = |zA|1−∆
p , (1.73)
22
can be written
K(A, x) =ζp(2∆)
ζp(2∆− 1)
|zA|∆pmax(|zA|p, |xA − x|p)2∆
. (1.74)
In Fourier space:
K(A, k) =
(|zA|1−∆ + |zA|∆|k|2∆−1
p
ζp(1− 2∆)
ζp(−1 + 2∆)
)γp(kzA). (1.75)
The correlators of the boundary theory can then be computed using the standard
AdS/CFT dictionary:
⟨exp
(∫Qpdx φ0(x)O(x)
)⟩= e−Son-shell(φ), (1.76)
where the on-shell action is evaluated on φ satisfying
limzA→0
|zA|∆−1φ(zA, x) = φ0(x). (1.77)
In practice, the correlation functions are a sum of diagrams. To compute the dia-
grams, one sums the relevant bulk-to-boundary propagators over all of Tp [10]. More
advanced methods for computing these diagrams can be found in [43] together with
the Mellin space approach taken in [18, 19].
23
2
Continuum Limits of Sparsely Coupled Ising-like
Models
This chapter is based on [1] coauthored with Steven S. Gubser, Christian Jepsen, and
Ziming Ji. We thank S. Hartnoll for getting us started on this project by putting us
in touch with M. Schleier-Smith’s group, and we particularly thank G. Bentsen and
M. Schleier-Smith for extensive discussions.
The study of statistical field theory over the p-adic numbers arguably began with
Dyson’s hierarchical model [44]. A rigorous study of Dyson’s phase transition was
carried out in [45] and the identification of p-adic field theory with the continuum
limit of Dyson’s model was made in [46]. The ideas found in this chapter can be
understood, for p = 2, in terms of rephrasing of Dyson’s original work.
Consider the “furthest neighbor” Ising model. By this we mean start with 2N
Ising spins labeled by sites 0, . . . , 2N − 1 and strongly couple each spin σi to the spin
that is sequentially furthest from it σi+2N−1 , where the addition takes place in Z/2NZ.
This produces 2N−1 pairs of strongly coupled spins where each pair is decoupled from
all other pairs. In the interest of seeking a more interesting thermodynamic limit, we
proceed by coupling each pair of spins with the pair of spins that is furthest away
from it. We then couple pairs of pairs, and so on. At each stage of this process we
24
0
1
2
3
4
5
6
7
0 4 2 6 1 5 3 7
000 100 010 110 001 101 011 111
000 001 010 011 100 101 110 111
Monna map M
h
M(h)
Figure 2.1: Left: A furthest neighbor coupling pattern among eight spins. The
thickness of lines indicates the strength of the coupling between spin 0
and the other spins. The coupling pattern is invariant under shifting
by a lattice spacing, so for example spins 1 and 5 are as strongly
coupled as spins 0 and 4. The blue circle is to guide the eye and does
not indicate additional couplings.
Right: A hierarchical representation of the couplings between spins.
Above each spin’s label we have given the base 2 presentation of the
spin number, and we have shown how the Monna map acts on these
numbers by reversing digits in the base 2 presentation.
reduce the coupling strength by a fixed factor 21+s, where s ∈ R is what we will call
the spectral parameter. The overall picture is illustrated in figure 2.1.
It is natural to view the furthest neighbor model as a hierarchy of spin clusters,
also shown in figure 2.1. This tree of clusters gives a particularly clear understanding
of the 2-adic distance. Indeed, if we define d(i, j) to be the number of steps required
to go from i to j via a path through the tree, then we have the following identity:
|i− j|2 = 2−N+d(i,j)/2 (2.1)
for distinct i and j. A quick comment on how to interpret this formula is now
appropriate. For i ∈ Z/2NZ we let |i|2 = |i∗|2 where i∗ is the representative of i in
0, . . . , 2N − 1. Note that this convention sets |0 + 2NZ|2 = 0 consistent with the
usual |0|2 = 0 for 0 ∈ Q but not with (2.1) for coincident points. It’s also important
to note that while | · |2 is not an absolute value on Z/2NZ (e.g. multiplicativity
25
fails), the statement |u|2 = 1 is actually independent of the representative chosen for
u ∈ Z/2NZ. Green’s functions in the furthest neighbor Ising model depend only on
the 2 -adic distance |i− j|2, and this is a consequence of the partition function being
invariant under a relabeling of the lattice sites i→ ui+ b where |u|2 = 1. Intuitively,
|u|2 is a rotation and rotational invariance implies that physical quantities depend
only on the norm |i− j|2.
The sparse coupling pattern that we study in this chapter couples spins i, j ∈
Z/2NZ only if i−j is a power of 2 (modulo 2N). With such a coupling, the number of
spins coupling to spin 0 grows as the logarithm of the system size N = log2(2N). We
will see that when the spectral parameter of this model is made large and negative
that, to a good approximation, only nearest neighbor couplings survive.
Less obvious is that when the spectral parameter is large and positive, the model
behaves to a good approximation like the furthest neighbor model. Here’s an expla-
nation. When s 0, the coupling between spins 0 and 2N−1 produces very tightly
coupled pairs. When we proceed to the next layer of the tree pairs of pairs are also
tightly coupled. Proceeding to the next layer, we couple 0 weakly to ±2N−3 but not
±3× 2N−3. However, what matters is that we are coupling the already tightly bound
quartets 0, 2N−1,±2N−2 and ±2N−3,±3× 2N−3. This logic works at general lay-
ers of the hierarchy - while we aren’t coupling all constituents of one spin cluster to
all of the constituents of another spin cluster, what matters is that the clusters are
already tightly bound relative to the the strength of the inter-cluster coupling.
The Green’s function of nearest neighbor model with 2N spins is well-approximated
at large N by a continuum Green’s function that can be extracted from statistical
field theory over R. This two-point function is smooth away from the origin, and
this smoothness gives a good way of understanding how the continuous topology of
R emerges from a discrete lattice. As reviewed in section 1.3, the accepted analogue
of a smooth function f : Qp → R is a locally constant function. Indeed, the Green’s
26
functions arising from the p-adic field theories in section 1.7 are locally constant.
When we study sparse coupling patterns, we will find that the Green’s functions
are well approximated by locally constant functions in the limit of large positive s.
However, we will have to use a more subtle measure of strong continuity - Holder
continuity - to characterize the transition from large positive s to large negative
s. The long and short of it is that as s is varied from positive to negative, the
Green’s function transitions from being 2-adically continuous to being continuous in
the Archimedean sense.
The organization of the rest of this chapter is as follows. In section 2.1 we describe
the class of one-dimensional statistical mechanical models that we will study and give
a general account of the form of the two-point function in these models. In sections
2.2-2.5 we treat four models within this class. In order: Nearest neighbor interactions,
p-adic interactions, power-law interactions, and finally the sparse coupling model
which interpolates between the nearest neighbor and 2-adic models. After a brief
discussion of Holder continuity in section 2.6 we derive Holder continuity bounds for
the continuum limit of the sparsely coupled Green’s function in sections 2.7-2.8. In
sections 2.9-2.11 we show through numerical studies that to a good approximation, the
continuity of the discrete momentum space Green’s function is characterized by these
Holder bounds. The continuity of the position space Green’s function is characterized
partly by the continuum Holder bounds. Indeed, a stronger form of continuity than
one might expect from field theoretical results emerges in particular regimes.
2.1 The Statistical Mechanical Framework
We wish to study the general properties of non-local couplings. Hence, we consider
an elementary class of statistical mechanical models consisting of L real degrees of
27
freedom φi together with a Hamiltonian
H = −1
2
∑ij
Jijφiφj −∑j
hjφj, (2.2)
which is a sum of a quadratic form in φi together with an external source. We view
the degrees of freedom as sitting on L lattice sites and impose translation invariance
of the coupling, i.e. Jij = Ji−j where i − j is understood modulo L. As with any
quadratic form, we can rearrange the coefficients to be symmetric: Jij = Jji. In the
single subscript notation Jh = J−h.
The translation invariance and symmetry of J guarantee that we can unitarily
diagonalize J via a discrete Fourier transform. Specifically, we define an orthonormal
basis for CL:
(uk)j =1√L
exp
(2πikj
L
)for k = 0, 1, 2, . . . , L− 1. (2.3)
We then use the Fourier transform convention Xk = u†kX for X ∈ CL. The uk are
eigenvectors of J :
Juk =√LJkuk, (2.4)
where Jk is the Fourier transform of the coupling strength vector (J0, . . . , JL−1). The
eigenvalue equation (2.4) allows us to write the Hamiltonian in momentum space
H = −√L
2
L−1∑k=0
Jkφ−kφk −L−1∑k=0
h−kφk. (2.5)
In order for the integral defining the partition function
Z(h) =
∫dLφ exp(−βH(φ, h)) (2.6)
=
∫dLφ exp
(β
[1
2φTJφ+ hTφ
])(2.7)
28
to converge, J must be negative definite. However, we want to study the massless case
in which the Hamiltonian depends only on the squared differences of the parameter
between lattice sites (φi−φj)2. This is equivalent to invariance under the target space
translation φ → φ + cu0 where u0 = (1, . . . , 1) and c is any constant. Therefore, we
require a vanishing eigenvalue J0 = 0 of J . To overcome this infrared divergence, we
will regulate by setting J = −µ where µ > 0 and take the limit µ→ 0 after discarding
uninteresting constants in our formulas.
Having established it’s convergence, the partition function takes the standard
form:
Z(h) = Z(0) exp
(−β
2hTJ−1h
)(2.8)
= Z(0) exp
(− β
2√L
L−1∑k=0
1
Jkh−khk
). (2.9)
The two-point function Gij = 〈φiφj〉 is also translation invariant Gij = Gi− j and
can be calculated directly:
Gh = − 1
βL3/2
L−1∑k=1
1
Jkexp
(2πikh
L
)− 1
µβL3/2. (2.10)
Dropping the constant divergent as µ → 0, we arrive at the formula for the Green’s
function that we will use for the rest of this chapter:
Gh = − 1
βL3/2
L−1∑k=1
1
Jk
(2πikh
L
). (2.11)
It will also be convenient to have a condition on the position space couplings Jh that
ensures Jk < 0 for k 6= 0 and J0 = 0. One can check it is enough to have∑L−1
h=0 Jh = 0
and Jh ≥ 0 for h 6= 0 with the strict inequality holding for at least one h.
As mentioned at the begging of the chapter, we will be studying a sparsely cou-
29
pled model that interpolates between Archimedean and p-adic coupling. Before in-
troducing it, we will introduce the models it interpolates between, starting with the
Archimedean model.
2.2 Nearest neighbor coupling
Dyson showed in [44] that phase transitions could occur in a one dimensional Ising
model given the right coupling strength. While there is still a lack of a phase transition
in the one-dimensional Ising model whose sites are coupled along a line, Dyson’s model
describes sites that are coupled hierarchically. If we take the view that the coupling
determines the topology of model instead of the indexing of the fields, then the nearest
neighbor coupling
JNNH = J∗(δh+1 + δh−1 − 2δh), (2.12)
describes the (Archimedean) topology of degrees of freedom distributed on a circle.
The corresponding two point function is:
GNNh =
1
4βJ∗L
L−1∑k=1
e2πikh/L
sin2 πkL
. (2.13)
Taylor expanding the 1/ sin2 term, we arrive at:
1
LGNNh =
1
4βJ∗L2
L−1∑k=1
e2πikh/L
(L
πk
)2∑n≥0
ζ(2n)(2n− 1)
(k
L
)n(2.14)
=1
βJ∗
L−1∑k=1
e2πikh/L
8π2k2+
1
β4J∗
∑n≥0
ζ(2n)(2n− 1)1
Ln
L−1∑k=1
kn−2e2πikh/L. (2.15)
30
The last term is O(1/L), so we can take the L→∞ limit and write:
GNNh ∼ L
βJ∗G(h/L), (2.16)
where the continuum Green’s function on the circle R/Z for a representative x ∈ [0, 1]
is:
G(x) =1
2
(x− 1
2
)2
− 1
24. (2.17)
We have the Green’s function identity G′′(x)− 1 = −δ(x). We also have∫R/ZG = 0.
2.3 p-Adic coupling
We now introduce the p-adic coupling. Fix a prime number p and consider a model
with L = pN lattice sites. We can then define the following all-to-all coupling for
h 6= 0:
Jp−adich = J∗|h|−s−1
p . (2.18)
This coupling endows the model with the hierarchical topology that we saw in section
2. We will write this as Jph for the rest of the section so as to not encumber the reader
with verbose notation. Using the constraint Jp0 = 0 one can derive the following
recursion relation on N : Jp0 |N+1 = p(J0|N − J∗(1− p−1)ps(N+1)). Solving this with a
geometric sum, we find for L = pN
Jp0 = −J∗Lζp(−s)
ζp(1)ζp(−sN). (2.19)
Let us turn our attention to calculating the Fourier transform of the coupling. It
will be helpful to first compute the transform of f(N)h =
√L|h|−s−1
p (1− δh) over CpN
since it will reappear in our calculations for the Fourier transform of the Green’s
function. It’s easy to show that f(N)k depends only on |k|p, so it is enough to find
31
it’s value for k = 0, 1, p, . . . , pN−1. We first derive a recursion relation for f(N)h . Let
0 ≤ −logp|k| < N , then kp < pN and:
f(N)kp = f
(N−1)k +
pN−1∑h=pN−1
|h|−s−1p e2πihk/pN−1
(2.20)
= pf(N−1)k + pN+(N−1)s. (2.21)
This simplifies if we express it using the following normalized quantities g(N)k =
p−N(s+1)f(N)k . We then have:
gNkp = p−s(gN−1k +
1
ζp(1)
). (2.22)
We now make the ansatz g(N)k = AN |k|sp + BN . Note that this conforms with our
intuition from Archimedean physics, where the Fourier transform of |x|−s−1 scales
under k → λk like |λ|s+1−d where d is the dimension of the space. The recursion
relation (2.22) implies that AN = A is independent of N and gives the relation
BN = p−s(BN−1 + 1/ζp(1)). This has constant (B = BN) solution
B = −ζp(−s)ζp(1)
. (2.23)
Lastly, we use A = B − g(1)1 to find
A =ζp(−s)ζp(1 + s)
. (2.24)
Putting this all together, we have for k 6= 0:
f(N)k = Ls+1ζp(−s)
( |k|spζp(1 + s)
− 1
ζp(1)
). (2.25)
32
For k = 0 one has immediately f(N)0 = −Jp0/J∗. We can use these results to easily
compute the Fourier transform of Jph = J∗(f(N)h /√L− f (N)
0 δh):
Jpk =J∗√L
(f(N)k − f (N)
0 ). (2.26)
From this equation we manifestly have J0 = 0. For k 6= 0 our result simplifies to
Jpk = J∗√Lζp(−s)ζp(1 + s)
(∣∣∣∣ kL∣∣∣∣s − ζp(1 + s)
ζp(1)
). (2.27)
We will be interested in the regime s > 0 where the first term in (2.27) is larger than
the second. We can therefore write for k 6= 0:
Gpk = − 1
βLJpk= − 1
βL3/2J∗
ζp(1)
ζp(−s)∑n≥1
(ζp(1 + s)
ζp(1)L−s
)n|k|−nsp . (2.28)
We can now make use of the results derived in this section to apply the inverse Fourier
transform and obtain
Gph = − 1
βL2J∗
ζp(1)
ζp(−s)∑n≥1
ζp(1 + s)n
ζp(1)n
[ζp(−ns+ 1)
ζp(ns)
(|h|ns−1
p − ζp(ns)
ζp(1)
)(1− δh)
(2.29)
− ζp(−ns+ 1)
ζp(1)ζp(N(−ns+ 1))δh
]. (2.30)
Note that L2Gph is independent of L for h 6= 0. Indeed, only the allowed range of h
changes as we increase L = pN . Taking L→∞ corresponds to the range of h becom-
ing dense in the of p-adic integers Zp = x ∈ Qp||x|p ≤ 1. It is worth noting that
since L2Gph depends on h only through its norm |h|p, the continuum Green’s function
is locally constant. This local constancy guarantees a locally constant extension of
L2Gph to all of Zp. More explicitly, given x ∈ Zp we can simply evaluate the Green’s
function at |x|p.
33
2.4 Power-law coupling
The p-adic coupling introduced in the last section and the sparse coupling to be intro-
duced in the section following this one both include an adjustable spectral exponent
s. Since the nearest neighbors coupling has no such exponent, we modify it to include
one:
Jpowerk = − J∗
2s√L
sin−s(πk
L
). (2.31)
This reduces to the nearest neighbor model when s = −2. Let us justify this coupling’s
name. For s < 1, and L 1 we can approximate
Jh = −J∗2s
1
L
L−1∑k=0
sin−s(πk
L
)e−2πikh/L (2.32)
' −J∗2s
∫ 1
0
sin−s(πx) exp(−2πihx)dx (2.33)
= −J∗π
sin(πs/2)Γ(1− s)Γ(h+ s/2)
Γ(1 + h− s/2). (2.34)
Using Stirling’s approximation for 1 h L, we verify that the coupling obeys a
power law:
Jpowerh ∼ −J∗
πsin(πs/2)Γ(1− s)hs−1. (2.35)
Given s < −1 and in the limit L 1, the position space Green’s function can be
expressed in terms of the polylogarithm Lin(x) =∑∞
k=1 xk/kn:
Gpowerh =
21+sπs
βJ∗L1+sRe Li−s(e
2πih/L). (2.36)
34
2.5 Sparse coupling: From the Archimedean to
the non-Archimedean
We now exhibit the primary model that we are interested in studying. Let L = 2N
for some integer N ≥ 1 and consider the following coupling:
J sparseh = J∗
N−1∑n=0
2ns(δh−2n + δh+2n − 2δh). (2.37)
We refer to this coupling as sparse because any particular degree of freedom will be
coupled to at most O(logL) other degrees of freedom. It is immediately apparent that
for s sufficiently negative this sparsely coupled model will become well approximated
by a nearest neighbor model. In particular, with suitable normalization, we expect
Gsparseh to closely approximate GNN
h . What is not so immediately apparent is the
following:
1. For s sufficiently positive, G2-adich closely approximates Gsparse
h .
2. As s is varied from negative to positive values, Gsparseh transitions from exhibiting
Archimedean continuity to exhibiting 2-adic continuity. A precise statement
about the form that this continuity takes will be made in the next few sections
when we discuss Holder bounds and study the continuum Green’s function of
the sparsely coupled model.
We are interested in the notion of closeness not only in the Archimedean regime, but
in the p-adic regime as well. We therefore find it useful to employ a discrete variation
of the Monna map. We can express any h ∈ Z/pNZ in a p-adic expansion:
h =N−1∑n=0
hnpn where hn ∈ 0, 1, . . . , p− 1. (2.38)
35
The hn in the above expression are unique, so we can define a single valued function:
M(h) =N−1∑n=0
hN−1−npn. (2.39)
Our discrete Monna map is an involutionM−1 =M. Suppose h and g are p-adically
close, i.e. hn = gn for n < M where 1 M < N . Then M(h) and M(g) are
sequentially close:
|M(h)−M(g)|∞ =
∣∣∣∣∣N−1∑n=M
(hn − gn)pN−1−n
∣∣∣∣∣ ∼ pN−M−1 (2.40)
The converse is not true: sequentially close numbers can be mapped to numbers that
aren’t p-adically close.
2.6 Continuity conditions
In the following sections we will investigate continuum field theories with sparse cou-
pling patterns. In section 2.7 we study 2-adic field theories and derive Holder con-
tinuity bounds for the two-point function in both position and momentum space.
We follow this up in section 2.8 with similar bounds in a sparsely coupled bilocal
Archimedean field theory.
Since the bounds we will encounter in the following sections are all Holder bounds,
it will be useful to review the appropriate definitions. Let X and Y be metric spaces.
In this chapter, we will generally have Y = R and X ⊆ Qp or X ⊆ R. The metric
on X is simply dp(n,m) = |n−m|p for subspaces of Qp and d∞(x, y) = |x− y|∞ for
subspaces of R. Let α > 0 be given. A function f : X → Y is said to be α-Holder
continuous on X if there exists some K > 0 such that given any x1, x2 ∈ X, the
36
s = -1Gh
sparseGh
power
10 20 30 40 50 60h
-0.2
0.2
0.4
0.6
Gh/G0
s = -1Gh
sparseGh
2-adic
1 2 3 4 5 6log2(h)
-0.2
0.2
0.4
0.6
Gh/G0
s = -0.3Gh
sparseGh
power
10 20 30 40 50 60h
-0.1
0.1
0.2
0.3
Gh/G0
s = -0.3Gh
sparseGh
2-adic
1 2 3 4 5 6log2(h)
-0.1
0.1
0.2
0.3
Gh/G0
s = 0.3Gh
sparseGh
power
10 20 30 40 50 60h
-0.1
0.1
0.2
0.3
0.4
Gh/G0
s = 0.3Gh
sparseGh
2-adic
1 2 3 4 5 6log2(h)
-0.1
0.1
0.2
0.3
0.4
Gh/G0
s = 1Gh
sparseGh
power
10 20 30 40 50 60h
-0.2
0.2
0.4
0.6
0.8
1.0
Gh/G0
s = 1Gh
sparseGh
2-adic
1 2 3 4 5 6log2(h)
-0.2
0.2
0.4
0.6
0.8
1.0
Gh/G0
Figure 2.2: Left: Gsparseh and Gpower
h versus h. This column shows how close
Gsparseh is to a smooth function in the usual Archimedean sense, and
confirms that Gsparseh ≈ Gpower
h when s is sufficiently negative.
Right: Gsparseh and G2−adic
h versus log2 M(h). This column shows how
close Gsparseh is to a smooth function in the 2-adic sense, and confirms
that Gsparseh ≈ G2−adic
h when s is sufficiently positive.
37
following inequality is satisfied:
dY (f(x1), f(x2)) ≤ KdX(x1, x2)α. (2.41)
f is said to be α-Holder continuous over some subset O ⊂ X if f |O is α-Holder
continuous. f is said to be locally α-Holder continuous at x ∈ X if f is α-Holder
continuous over some open neighborhood of x. Lastly, the function f as a whole is
said to be locally α-Holder continuous if it is locally α-Holder continuous at every
x ∈ X.
The strength of Holder continuity depends on the exponent α: the larger the ex-
ponent the more restrictive the condition. Holder continuity is stronger than uniform
continuity for any positive α. The case α = 1 is equivalent to Lipschitz continuity.
It’s unusual to have α > 1 in the real case. Indeed, if α > 1 and X ⊆ R then any
α-Holder continuous function f must be constant on each connected component of X.
However, for X ⊆ R there is a rich variety of Holder continuous functions that have
α > 1. One such example is any linear combination of functions∑
i ai|x − xi|α for
constants xi ∈ Qp. We will make use of local Holder continuity. Indeed, we will find
instances in which the two point function is globally Holder with a smaller exponent
than it is locally Holder away from the origin.
2.7 2-adic field theory
We begin with the 2-adic field theory. The basics of p-adic field theory were given in
section 1.7. We will specialize to p = 2 and consider the bilocal action
S(φ) = −∫Q2
dxdy1
2φ(x)J(x− y)φ(y), (2.42)
38
where
J(x) = J∗∑n∈Z
2ns(δ(x− 2n) + δ(x+ 2n)− 2δ(x)). (2.43)
As in section 2.1, the action can be diagonalized in Fourier space:
S(φ) = −1
2
∫Q2
dkφ(−k)J(k)φ(k). (2.44)
We can immediately write down the Fourier space two point function G(k) = −1/J(k).
To avoid having to carry J∗ around everywhere, we will use the convenient value
J∗ = 1/4. The Fourier space coupling may be written
J(k) = J∗∑n∈Z
2ns(χ(2nk) + χ(−2nk)− 2) (2.45)
= −∑n∈Z
2ns sin2(π2nk). (2.46)
We can restrict the infinite sum to n < ν(k)− log2 |k|2, since k2n = 0 otherwise.
An amusing connection to population dynamics can be observed at this point.
Recall the logistical map, x→ rx(1−x). If xnn∈Z is a solution to this iterated map,
then we can think of xn as (proportional to) the population of a species at generation
number n. For r = 4, a solution is xn = sin2(π2nk) where k is a real number. However,
this is not the most general solution, because it has the property xn → 0 as n→ −∞.
Consider instead xn = sin2(π2nk) where k is a 2-adic number. Then xn = 0 for all
n ≥ −v(k), but we need not have xn → 0 as n → −∞. Thus we see that the 2-adic
number k parametrizes the routes to extinction under the r = 4 logistical map, and
the 2-adic norm of k predicts the moment of extinction: n∗ = log2 |k|2. To make
the discussion simple, suppose now that k is a 2-adic integer, so that extinction has
occurred by the time n = 0. Further suppose that each generation leaves an imprint
on its environment proportional to xn, and that this imprint dissipates over time with
a half life of 1/s generations. So the environmental imprint at time 0 of generation n
39
(with n < 0 since extinction occurs no later than time 0) is In = α2nsxn, where α is
the constant of proportionality. Then I = −αJ(k) as computed in (2.45) is the total
environmental imprint of the species, summed across all generations and measured
at time 0.
Since n reaches arbitrarily negative numbers, the coupling weights 2ns will give a
well defined J(k) if and only if s > 0. This is encouraging since s > 0 is the regime
in which we expect 2-adic continuity to emerge. Given s > 0 we can thus write
J(k) = −|k|s2Ψ(k) where k = |k|2k is a 2-adic unit and
Ψ(k) =∑m≥1
2−ms sin2(π2−mk). (2.47)
For clarity, we have used the following change of summation index: m = −(n+v(k)).
We will now show that Ψ is bounded and s-Holder continuous over the 2 adic units
U2. First boundedness:
|Ψ(k)| ≤∑m≥1
2−ms (2.48)
= −ζ2(−s). (2.49)
In fact, we can obtain a stronger bound from below. Note that since it is a 2-adic
unit, k = 1 + k12 + · · · where k1 = 0 or 1. Therefore the first two terms of the
sum (2.47) are non-vanishing and give the bound 2−s + 2−2s−1 ≤ Ψ(k). To show
s-Holder continuity note that fractional parts of 2−mk1 and 2−mk2 are exactly when
m ≤ ν(k1 − k2). We therefore have:
|Ψ(k1)−Ψ(k2)| ≤ |k1 − k2|s∞∑j=0
2−js| sin2(π2−j k1)− sin2(π2−j k2)| (2.50)
≤ Ks|k1 − k2|s, (2.51)
40
where Ks = −2ζ2(−s). Since
|1/Ψ(k1)− 1/Ψ(k2)| = |Ψ(k1)|−1|Ψ(k2)|−1|Ψ(k1)−Ψ(k2)|, (2.52)
one can show using the fact that Ψ is bounded below by 2−s+2−2s−1 > 0 that 1/Ψ(k)
is s-Holder continuous. Therefore given k 6= 0, the momentum space Green’s function
G(k) =1
|k|s2Ψ(k)(2.53)
is the product of a locally constant function and an s-Holder continuous function.
Hence G(k) is locally s-Holder continuous away from the origin.
Let’s turn our attention to the position space two-point function G(x). As in
analysis over R, we expect that if G(k) is integrable at large |k| that G(x) will be
continuous. We therefore focus on the regime s > 1 for which G ∈ L1(Qp). We run
into an IR divergence if we try to calculate G(x) outright, therefore we introduce an
IR regulator by integrating over the region |k| ≥ |kIR| = 2−νIR . We find
G(x) = ζ2(s− 1)|kIR|1−s2
∫U2
dk
Ψ(k)+ |x|s−1
2 g(x), (2.54)
where
g(x) = ζ2(s− 1)
∫U2
dk
Ψ(k)+∞∑n=1
2(1−s)n∫U2
dkχ(2−nkx)
Ψ(k). (2.55)
Dropping the divergent term in (2.54) is equivalent to working with
G(x) =
∫Q2
χ(kx)− 1
|k|s2Ψ(k)= |x|s−1
2 g(x). (2.56)
We now derive a local (2s−1)-Holder bound for g(x). Note that by the same logic we
used to derive the Holder bound for Ψ(k), when calculating the difference g(x1)−g(x2)
41
we can restrict the sum to n > ν(x1 − x2). Therefore:
g(x1)− g(x2) =∑
n>ν(x1−x2)
2(1−s)n∫U2
dk
(χ(2−nkx1)
Ψ(k)− χ(2−nkx2)
Ψ(k)
)(2.57)
=∑
n>ν(x1−x2)
2(1−s)n∫U2
dq χ(2−nq)
(1
Ψ(q/x1)− 1
Ψ(q/x2)
)(2.58)
In the second line we’ve used changes of variable q = kxi. Note that we don’t pick
up any Jacobian factor since |xi|2 = 1. Also note that |1/x1 − 1/x2|2 = |x1 − x2|2.
Now we use the fact that Ψ is s-Holder together with |q|2 = 1:
|g(x1)− g(x2)| ≤ 2(1−s)ν(x1−x2)
∞∑n=1
2(1−s)n∫U2
dq
∣∣∣∣ 1
x1
− 1
x2
∣∣∣∣s2
(2.59)
= Ms|x1 − x2|2s−12 , (2.60)
where Ms = −ζ2(1−s)/ζ2(1). Now G(x) = |x|s−12 g(x) where |x|s−1
2 is locally constant
and globally s− 1-Holder continuous and g(x) is (2s− 1)-Holder continuous. We can
therefore conclude that G(x) is globally Holder continuous with exponent s− 1 and
locally Holder continuous with exponent 2s− 1.
2.8 Archimedean field theory
We now carry out the analogous analysis in real field theory. We have the action
S = −1
2
∫Rφ(x)J(x− y)φ(x) = −1
2
∫Rφ(−k)J(k)φ(k), (2.61)
where we define J(x) using the same formula as in (2.43). Our Fourier transform
convention is f(x) =∫R exp(2πikx)f(k). The two point function satisfies G(k) =
42
−1/J(k). Again using J∗ = 1/4 we have
J(k) = −∑n∈Z
2ns sin2(π2nk). (2.62)
For s ≤ −2, the sum diverges as n → −∞. But looking back at the definition
(2.43), this means that the coupling is concentrated at x = 0. Indeed, if we regulate
the sum by imposing a cutoff and re-scale J∗ while removing it, then we can attain
J(x) → −δ′′(x). The theory becomes local. If s ≥ 0 - the region of s that Holder
bounds were proved on the 2-adic side - then the sum diverges as n→∞. Thus s ≥ 0
is a region where the coupling is dominated by arbitrarily long-ranged interactions.
We therefore restrict our attention to the regime −2 < s < 0 where the sum
(2.62) converges and the theory is not perfectly local. We can ask what properties
the function J(k) satisfies analogous to the ones found for Ψ(k) in the 2-adic case. In
fact, we claim
1. −J(k) ≈ |k|−s∞ , meaning that there exist positive constants K1 and K2, inde-
pendent of k, such that K1|k|−s∞ < −J < K2|k|−s∞ for all k ∈ R\0.
2. For −1 < s < 0, J(k) is globally Holder continuous with exponent −s.
3. For −2 < s < −1, the derivative J ′(k) = dJ(k)/dk is globally Holder continuous
with exponent −s − 1. (Note that J(k) itself cannot have Holder continuity
exponent greater than 1 without being constant. So the derivative condition
we claim here is the best that can be expected.) It follows that J(k) is globally
1-Holder continuous on any bounded domain.
To arrive at the estimate −J(k) ≈ |k|−s∞ , we define
nk ≡ − log2(π|k|∞) . (2.63)
43
Then we have
−J(k) =∑n<nk
2ns sin2(π2nk) +∑n≥nk
2ns sin2(π2nk) ≈∑n<nk
2ns(π2nk)2 +∑n≥nk
2ns
≈ (π|k|∞)−s [−ζ2(−s− 2) + ζ2(−s)] .(2.64)
To derive the Holder condition on J(k) for −1 < s < 0, set δ = |k1 − k2|∞ and note
that
| sin2(π2nk1)− sin2(π2nk2)|∞ ≤ min1, π2nδ . (2.65)
Defining
nδ = − log2(πδ) , (2.66)
we see that
|J(k1)− J(k2)|∞ ≤∑n∈Z
2ns min1, π2nδ =∑n<nδ
2n(s+1)πδ +∑n≥nδ
2ns
≈ (πδ)−s [−ζ2(−s− 1) + ζ2(−s)] ,(2.67)
where again ≈ means equality to within fixed multiplicative factors, independent in
this case of δ. The last expression in (2.67) is the desired Holder bound, valid when
−1 < s < 0. If instead −2 < s < −1, then we may calculate
J ′(k) = −π∑n∈Z
2n(s+1) sin(π2n+1k) . (2.68)
By the same method as in (2.67) we arrive at the Holder continuity condition for
J ′(k) with exponent −s− 1.
By combining the property −J(k) ≈ |k|−s∞ with the Holder bounds, we see that
G(k) is locally Holder with exponent −s for −1 < s < 0. Also, G′(k) is locally Holder
away from k = 0 with exponent −s − 1 for −2 < s < −1, implying that G(k) is
locally 1-Holder away from k = 0.
44
Now let’s investigate smoothness of the Green’s function in position space. We
naively define
G(x) =
∫Rdk
e2πikx
−J(k). (2.69)
As in the 2-adic case, our intuitive understanding is that G(x) will be continuous
everywhere iff G(k) is integrable at large k, which is the case iff s < −1. Because
−J(k) ≈ |k|s∞, the UV-integrable regime is −2 < s < −1, and in this regime the
integral (2.69) has an IR divergence. Again as in the 2-adic case, the infrared diver-
gence results in an overall additive constant in G. It does not matter much how this
constant is removed; one option is to alter (2.69) to
G(x) ≡∫Rdke2πikx − 1
(−J(k)). (2.70)
For the purposes of a Holder continuity condition we must estimate
G(x1)−G(x2) =
∫R
dk
(−J(k))(e2πikx1 − e2πikx2) . (2.71)
Setting δ = |x1 − x2|∞, we have
|G(x1)−G(x2)|∞ ≤∫R
dk
(−J(k))min2, 2π|k|∞δ
= 2π
∫|k|<1/δ
dk
(−J(k))|k|∞δ + 2
∫|k|>1/δ
dk
(−J(k))
≈∫ 1/δ
0
dk ks+1δ +
∫ ∞1/δ
ks
≈ δ−s−1
[1
s+ 2− 1
s+ 1
].
(2.72)
In short, for −2 < s < −1, we have obtained a global Holder bound with exponent
−s− 1.
45
log maxG˜κsparse
/G˜κ2-adic
log minG˜κsparse
/G˜κ2-adic
0.5 1.0 1.5 2.0 s
-0.5
0.5
1.0
N
6
7
8
9
10
11
12
13
14
15
16
log maxG˜κsparse
/G˜κ2-adic
log minG˜κsparse
/G˜κ2-adic
6 8 10 12 14 16 N
-0.5
0.5
1.0
1.5
s
0.1
0.2
0.3
0.4
0.5
Figure 2.3: Left: Optimal values of the constants K1 and K2 appearing in (2.73)
as functions of s for fixed N .
Right: Optimal values of the constants K1 and K2 as function of
N for fixed s. The expectation is that provided s > 0, K1 and K2
asymptote to constants at sufficiently large N .
2.9 Numerical evidence: 2-adic Approximation of
sparse results
We begin our numerical discussion by considering the question of how well the two-
point function of the 2-adic model approximates the two-point function of the sparse
model. From the results in section (2.7), we expect to find s dependent constants
K1(s), K2(s) > 0 such that
K1(s) <Gsparsek
G2−adick
< K2(s) (2.73)
in the region s > 0. Numerical evidence for these bounds is show in figure 2.3, where
the optimal values of Ki(s) have been computed for various values of s and N . As
s → 0, it is not apparent whether the K2(s) remains bounded as N → ∞. The
algorithm used to find Ki(s) is O(N2N), so barring the discovery of a better way
to calculate K2(s), we are limited by our ability to calculate using sufficiently high
values of N . Based on empirical analysis on the curves on the left side of figure 2.3,
we find Ki(s) ≈ 1 + 2−2sκi(s) where κi(s) must vary slowly with s. For reference, the
46
steps used to compute Ki(s) for given values of s and N are listed below:
1. Compute Gsparsek and G2−adic using the methods of sections 2.5 and 2.3.
2. Normalize each Green’s function in position space by adjusting J∗ to satisfy
G0 = L−1/2
L−1∑k=1
Gk = 1. (2.74)
3. Compute K1(s) and K2(s) via
K1(s) = mink 6=0
Gsparsek
G2−adick
and K1(s) = maxk 6=0
Gsparsek
G2−adick
. (2.75)
2.10 Numerical evidence: Holder bounds in mo-
mentum space
Next we would like to numerically investigate the local Holder continuity bounds that
we found in sections 2.7 and 2.8. We start with the momentum space bounds, which
we summarize here:
For 0 < s, G : Q2 → R is locally s-Holder continuous away from the origin.
For −1 < s < 0, G : R → R is locally (−s)-Holder continuous away from the
origin.
For s < −1, G : R → R is locally 1-Holder continuous (i.e. locally Lipschitz)
away from the origin.
After normalizing Gsparse0 = 1 and G2−adic
0 = 1, we expect that Gsparsek /G2−adic
k to obey
a local s-Holder bound. It’s helpful to write the condition for a local Holder bound
47
explicitly for each pair of 2-adic nearest neighbors (k, k + L/2):
∣∣∣∣∣ Gsparsek
G2−adick
−Gsparsek+L/2
G2−adick+L/2
∣∣∣∣∣∞
≤ K2−adic(s)2−α2−adic(N,s)(N−1). (2.76)
For the moment we are being agnostic with regard to the value of α2−adic. Our goal
given the data is to approximate α2−adic(N, s) and compare it with the theoretical
result α2−adic = s. We expect the quantity
A2−adic(N, s) = log2 maxk odd
∣∣∣∣∣ Gsparsek
G2−adick
−Gsparsek+L/2
G2−adick+L/2
∣∣∣∣∣∞
(2.77)
to be approximately linear in N :
A2−adic(N, s) ≈ log2 K2−adic(s)− α2−adic(N, s)(N − 1). (2.78)
We now assume that α2−adic(N, s) is convergent as N → ∞, so that for sufficiently
large N we have estimate
α2−adic(N, s) ≈ −A2−adic(N, s) + A2−adic(N − 1, s). (2.79)
The same numerical analysis can be carried out on the Archimedean side, making only
a few adjustments. First, our local Holder bound will now be sensitive to sequential
nearest neighbors (k, k + 1). Second, we now expect Gsparsek /Gpower
k to obey a local
Holder bound. This is because Gpowerk is smooth away from the origin so forming
the ratio Gsparsek /Gpower
k doesn’t affect the local Holder exponent of Gsparsek . Moreover,
dividing by Gpowerk cancels out part of the trend of Gsparse
k to grow larger near k = 0 =
L, which makes it easier to accurately distill the local Holder exponent with a finite
48
4 6 8 10 12 N
-10
-5
5
A˜2-adic s
-0.75
-0.50
-0.25
0
0.25
0.50
0.75
4 6 8 10 12 14 16 N
-10
-5
5
10
A˜power s
-0.75
-0.50
-0.25
0
0.25
0.50
0.75
Figure 2.4: Left: 2-adic smoothness in momentum space. The dots are evalua-
tions of A2−adic(N, s) in (2.77), and the lines are plots of the linear
trajectories indicated in (2.78), with K(s) chosen so that the line goes
through the last data point.
Right: Archimedean smoothness in momentum space. The dots are
evaluations of Apower(N, s) in (2.80), and the lines are plots of the
linear trajectories indicated in (2.81), with K(s) chosen so that the
line goes through the last data point.
sample of points. We therefore use
Apower(N, s) = log2 maxL4<k< 3L
4
∣∣∣∣∣Gsparsek
Gpowerk
−Gsparsek+1
Gpowerk+1
∣∣∣∣∣∞
(2.80)
instead of (2.77). We expect this to follow linear trajectories:
Apower(N, s) ≈ log2 Kpower(s)− αpower(N, s)(N − 1). (2.81)
Our estimate for the exponent is
αpower(N, s) ≈ −Apower(N, s) + Apower(N − 1, s). (2.82)
In figure 2.4, the linear trajectories (2.78) and (2.81) are shown for a variety of values
of s. Interestingly, the trajectories persist in regions for which the Holder exponent,
which can be roughly identified with the slope of the lines, becomes negative.
49
2.11 Numerical evidence: Holder bounds in posi-
tion space
We can carry out the exact same analysis in position space. Specifically, given the
quantities
A2−adic(N, s) = log2 maxh odd
∣∣∣∣∣ Gsparseh
G2−adich
−Gsparseh+L/2
G2−adich+L/2
∣∣∣∣∣∞
(2.83)
Apower(N, s) = log2 maxL4<h< 3L
4
∣∣∣∣Gsparseh
Gpowerh
−Gsparseh+1
Gpowerh+1
∣∣∣∣∞, (2.84)
we have the large N estimates
α2−adic(N, s) = −A2−adic(N, s) + A2−adic(N − 1, s) (2.85)
αpower(N, s) = −Apower(N, s) + Apower(N − 1, s). (2.86)
We find good evidence that the estimates α2−adic(N, s), αpower(N, s) for the local
momentum space exponents and the estimates α2−adic(N, s), αpower(N, s) for the local
position space exponents all have finite large N limits. Our numerical results are
shown in figure 2.5 and are well described by the following piecewise linear functions:
For s < −1, αpower = αpower = 1.
For −1 < s < 0, αpower = −s and αpower = −2s− 1.
For 0 < s, α2−adic = s and α2−adic = 2s− 1.
The following caveats apply to these estimates:
For |s| > 1 it becomes difficult to obtain high quality numerical results, partic-
ularly for αpower and αpower. In this regime, the Green’s functions become very
smooth and one has to compute very small differences with high accuracy. Even
50
2-adicfield theory
real field theory
• α2-adic
• αpower
-2 -1 1 2s
-1
1
2
3
α
2-adicfieldtheory
real field theory
• α2-adic
• αpower
-2 -1 1 2s
-0.5
0.5
1.0
1.5
2.0
α
Figure 2.5: α versus s and α versus s in the 2-adic and Archimedean settings.
Field theory bounds derived in section 2.6 are shown in dashed black
and dashed blue. Dotted black and dotted blue show the naive ex-
trapolations of these bounds to negative α and α. Red and green
dots are numerical evaluations of α and α as defined in sections 2.10
and 2.11, respectively, with N = 20. Solid red and green lines show
the obvious piecewise linear trends which approximately match the
numerical evaluations. Open circles denote evaluations in which we
restricted 7L16 ≤ h <
9L16 ; otherwise we use half the available points as
explained in the main text. For s ≤ −2, convergence of the sparse
model to the nearest neighbor model implies that α = α = 1, but our
numerical scheme for picking out α and α becomes less reliable in this
region due to difficulty normalizing Gsparse and Gpower in a mutually
consistent way.
51
without the numerical challenges, it becomes a difficult to distinguish between
rapid smooth variation and the non-smooth behavior that characterizes Holder
exponents.
Numerical calculations diverge a bit from our piecewise linear description at
s = 0 and s = −1. This isn’t too surprising since the pre-factors K(s) in
the Holder inequalities diverge at these values. See [references to divergences].
More extensive numerical investigations are needed to establish reliable results
at these values of s. It’s possible that we need logarithmic corrections to the
Holder conditions at these values, or that the exponents are only approximately
described by piecewise linear functions.
2.12 The liminal region −1/2 < s < 1/2
Of special interest is the regime −1/2 < s < 1/2. In momentum space, our numerics
are consistent with there being a single exponent on the 2-adic side, α2−adic = s, which
describes both the global Holder continuity condition over all k and the local continu-
ity at each possible value of k. In other words, as far as we can tell, the function G(k)
is equally ragged everywhere. A similarly uniform story applies on the Archimedean
side, with αpower = −s. Numerical results are fully in accord with expectations from
field theory, where we were able to compute α2−adic and αpower analytically. The up-
shot is that the transition from Archimedean to ultrametric continuity happens rather
simply, with ordinary continuity failing just as 2-adic continuity emerges: i.e. αpower
becomes negative just as α2−adic becomes positive, at s = 0.
The numerically estimated position space exponents behave differently. Both the
2-adic and Archimedean exponents become negative in the region s ∈ (−1/2, 1/2),
indicated a loss of local Holder continuity. Indeed, the piecewise linear description
found in section 2.11 suggests that Gsparseh loses Archimedean continuity at s = −1/2
52
s = 0.3
0.6 0.7 0.8 0.9 1.0(h)/L
-1.0
-0.5
0.5
1.0
1.5
Ghsparse
/Gh2-adic
s = 0.3
0.6 0.7 0.8 0.9 1.0(κ)/L
0.5
1.0
1.5
G˜κsparse
/G˜κ2-adic
s = -0.3
0.6 0.7 0.8 0.9 1.0(h)/L
-60
-40
-20
20
Ghsparse
/Gh2-adic
s = -0.3
0.6 0.7 0.8 0.9 1.0(κ)/L
1
2
3
4
5
6
7
G˜κsparse
/G˜κ2-adic
Figure 2.6: Plots of Gsparseh /G2−adic
h and Gsparseκ /G2−adic
κ over the Monna map of
the odd integers. As s becomes more positive, the numerical data is
closer to a 2-adically continuous curve when N is large. Blue points
are for N = 6, while the red curves are for N = 10.
53
s = 0.3
0.3 0.4 0.5 0.6 0.7h/L
-1500
-1000
-500
Ghsparse
/Ghpower
s = 0.3
0.3 0.4 0.5 0.6 0.7κ/L
2
4
6
8
10
12
G˜κsparse
/G˜κpower
s = -0.3
0.3 0.4 0.5 0.6 0.7h/L
-8
-6
-4
-2
2
4
Ghsparse
/Ghpower
s = -0.3
0.3 0.4 0.5 0.6 0.7κ/L
0.5
1.0
1.5
2.0
2.5
3.0
G˜κsparse
/G˜κpower
Figure 2.7: Plots of Gsparseh /Gpower
h and Gsparseκ /Gpower
κ over the middle half of
points. As s becomes more negative, the numerical data is closer to
a continuous curve when N is large. Blue points are for N = 6, while
the red curves are for N = 10.
54
and does not gain 2-adic continuity until s = +1/2.
Based on these figures and related studies, the scenario we regard as most likely is
that for −1/2 < s < 0, the continuum limit of Gpowerh defines an absolutely continuous
measure, G(x)dx, with respect to ordinary Lebesgue measure dx, but for s > 0 any
such continuum limit would necessarily have a singular term in its Radon-Nikodym
decomposition. Similarly, we suggest that for 0 < s < 1/2, the continuum limit of
G2−adich defines an absolutely continuous measure with respect to the standard Haar
measure on Q2 while for s < 0 any such continuum limit would have a singular term
(with respect to the Haar measure on Q2) in its Radon-Nikodym decomposition. We
find support for the claim of absolutely continuous measures in the above-mentioned
regimes when we study the scaling of the height of the spikes in figures 2.6 and 2.7
as a function of N : the weight of each spike (meaning the integral over a small region
including the spike) distinctly appears to tend to zero with increasing N . When
singular terms in Radon-Nikodym decompositions do exist, we conjecture that they
have as their support sets which are dense in position space.
One way in which singular terms in Radon-Nikodym decompositions could arise
is for the continuum limit G(x) to include delta functions. Inspection of figure 2.6 is
consistent with there being a dense set of delta function spikes in G(x) as a function
of 2-adic x when s = −0.3, but none when s = 0.3. Similarly, figure 2.7 is consistent
with there being a dense set of delta function spikes in G(x) as a function of real x
when s = 0.3, but none with s = −0.3. The discerning reader may note, however,
that the spikes on the Archimedean side are stronger at s = 0.3 than the ones on the
2-adic side at s = −0.3. This asymmetry manifests itself in the scaling of the height of
these spikes with N , for the weight of each spike grows with N on the Archimedean
side for s = 0.3, but may be trending very slowly toward 0 on the 2-adic side at
s = −0.3. A related effect appears in figure 2.5: α2−adic ≈ −1 for s < 0, while
αpower ≈ −1− s for s > 0.
55
Inspection of figures 2.6 and 2.7 reveals some self-similarity in the Green’s func-
tions both before and after the Monna map is applied. We have not investigated
this fractal behavior in detail; however, we note that similar behavior has been found
independently in band structure calculations in connection with proposed cold atom
experiments [47].
2.13 Outlook
For decades, p-adic numbers have been considered as an alternative to real numbers
as a notion of continuum which could underlie fundamental physics at a microscopic
scale; see for example [35]. This chapter has shown how the large system size limit
of an underlying discrete system naturally interpolates between a one-dimensional
Archimedean continuum and a 2-adic continuum as we vary a spectral exponent. By
focusing on a free field example, we are able to solve the model through essentially
trivial Fourier space manipulations. The correlators of the theories we study are all
determined in terms of the two-point function through application of Wick’s theorem.
The two-point function is continuous in an Archimedean sense when s is sufficiently
negative, and in a 2-adic sense when s is sufficiently positive. The transition from
these two incompatible notions of continuity can be precisely characterized in terms of
Holder exponents characterizing the smoothness of the two-point function. We have
found the dependence of these exponents on s through a combination of analytical
field theory arguments and numerics on finite but large systems.
Quite a wide range of generalizations of our basic construction can be contem-
plated:
1. We can generalize to primes p > 2. One significant subtlety arises when doing so,
namely the structure within Z/pZ of sparse couplings. The simplest alternative
is for spin 0 to couple to spins ±θpn with a strength pns, where θ runs over all
56
elements of 1, 2, 3, . . . , p−1. This coupling pattern is featureless within Z/pZ
because it treats all values of θ the same. One can however contemplate other
possibilities. For example, if p = 5, an interesting alternative is to introduce
couplings only for θ = 1 and θ = 4 (the quadratic residues). More generally,
one could expand the dependence of couplings on θ in a sum of multiplicative
characters over Z/pZ.
2. This chapter has focused entirely on bosonic spins φi, but there is no reason
not to consider fermions ci instead. Then the coupling matrix Jij would have to
be anti-symmetric, and likewise the two-point Green’s function would be odd.
Within this framework one could consider a variety of sparse coupling patterns.
3. Higher-dimensional examples are not hard to come by. Consider real bosonic
spins φ~ı labeled by a two-dimensional vector ~ı = i11 + i22, where i1 and i2 are
in Z/3NZ. Suppose we establish a coupling matrix J~ı~ = J~ı−~ where
J~h =
3minn1,n2s if h1 = ±3n1 and h2 = ±3n2
3n1s if h1 = 0 and h2 = ±3n2
3n2s if h2 = 0 and h1 = ±3n1 ,
(2.87)
with all other entries vanishing except J0, whose value we choose in order to
have the Fourier coefficient J~κ vanish when ~κ = 0. Then for sufficiently negative
s we have effectively a nearest neighbor model which approximates the massless
field theory S =∫d2x 1
2(∇φ)2. For s sufficiently positive, one obtains instead a
continuum theory over Z3×Z3, which can be understood as the ring of integers
in the unramified quadratic extension of Q3, see section 1.5.
All the examples above remain within the paradigm of free field theory. Still easy to
formulate, but obviously much harder to solve, are interacting theories with sparse
57
s = 0 112
23
−1 −12
−23
φ4re
leva
ntφ
6re
leva
nt
φ 4relevant
φ 6relevant No relevant deformations
ULTRAMETRICARCHIMEDEAN
Gaussian Gaussians
Figure 2.8: Conjectured pattern of fixed points of the renormalization group for
interacting field theories of a single bosonic scalar field with φ→ −φsymmetry.
couplings. For example, one could start with any of the models introduced in sec-
tions 2.2-2.5 and add a term∑
i V (φi) to the Hamiltonian describing arbitrary on-site
interactions. To get some first hints of what to expect these interactions to do, recall
in 2-adic field theory that G(x) ≈ |x|s−12 at small x. Comparing this to the standard
expectation G(x) ≈ |x|2∆φ
2 , we arrive at ∆φ = (1− s)/2 as the ultraviolet dimension
of φ. When describing perturbations of the Gaussian theory, we can use normal UV
power counting: [φn] = n∆φ. Thus φn is relevant when s > 1− 2/n. If we impose Z2
symmetry, φ→ −φ, then in the region s < 1/2, the Gaussian theory has no relevant
local perturbations, but as s increases from 1/2 to 1, first φ4 and then higher powers
of φ2 become relevant. It is reasonable to expect some analog of Wilson-Fisher fixed
points to appear. Possibly as s → 1 these fixed points extrapolate to analogs of
minimal models. An analogous story presumably applies on the Archimedean side to
power-law field theories controlled by s in the range (−1, 0), with G(x) ≈ |x|−s−1∞ and
therefore ∆φ = (1 + s)/2. See figure 2.8.
The sparse coupling theories are sufficiently similar to 2-adic field theories for
s > 0 and to power-law field theories for s < 0 that it is reasonable to conjecture that
the same pattern of renormalization group fixed points arises. However, this line of
reasoning is incomplete. In particular, one would require an understanding of how
58
the improved local Holder smoothness arises and how it might affect renormalization
group flows. A Monte Carlo study of the phases of the sparsely coupling Ising model
might help shed light on the renormalization group flows available to interacting
models, particularly in the range −2/3 < s < 2/3 where no powers of φ higher than
φ4 are relevant—according at least to naive power counting.
59
3
Higher Melonic Field Theories
This chapter is based on [3] coauthored with Steven S. Gubser, Christian Jepsen, and
Ziming Ji. We thank Igor Klebanov for useful discussions and suggestions.
In the last chapter, we developed a sparsely coupled statistical model that interpo-
lated between Archimedean and p-adic continuity. In this chapter we will explore the
diversity that arises when we consider field theory over R and Qp in tandem. In this
context, we will classify a large collection of melonic theories with q-fold interactions,
demonstrating that the interaction vertices exhibit a range of symmetries of the form
Zn2 for some n ∈ 0, 1, . . .. The number of different theories (over both Archimedean
and non-Archimedean spaces) proliferates quickly as q increases above 8 and is related
to the problem of counting one-factorizations of complete graphs. The symmetries
of the interaction vertex lead to an effective interaction strength that enters into the
Schwinger-Dyson equation for the two-point function as well as the kernel used for
constructing higher-point functions.
Melonic theories [48, 49, 50, 51, 52] are an interesting class of quantum field the-
ories whose essential property is that in an appropriate large N limit, the dominant
Feynman diagrams can be generated by iterating on the replacement of a propagator
by a melonic insertion, as shown in figure 3.1 for a melonic version of scalar φ4 theory.
Melonic theories are interesting for two related reasons: 1) The melonic large N limit
60
is relatively tractable because its Green’s functions can be determined through func-
tional techniques including Schwinger-Dyson equations; and 2) The simplest Green’s
functions are the same as for the Sachdev-Ye-Kitaev (SYK) model [53, 54], widely
studied because of its proposed relationship to AdS2.
It is well recognized that melonic theories exist not just with quartic interactions,
but also higher order interactions [55, 56, 57, 58]. The aim of this chapter is to take
some steps toward an understanding of what sorts of higher order interactions are
possible. We focus on the Klebanov-Tarnopolsky model [52] where the interaction is of
order q—meaning that q propagators meet at each interaction vertex. Inquiries in this
direction were initiated in [59]. Starting at q = 8, the number of different interaction
vertices proliferates quickly. Some of them are symmetrical under a subgroup of
permutations of the propagators leading into them; others have no such symmetry.
Interaction vertices do not mix with one another in the leading melonic limit: The
diagrams that would permit this are subleading. We are therefore content to restrict
to theories with only one type of interaction vertex—and each different interaction
vertex gives a different theory. An interaction vertex of order q can be constructed
starting from a coloring of the complete graph with q vertices such that each of
q − 1 colors is incident once on each vertex. The questions addressed in this chapter
regarding these interaction vertices are:
1. What are the possible symmetry groups of these interaction vertices?
2. How does the number of distinct interaction vertices grow with q?
We claim in sections 3.2 and 3.3 to completely settle question #1: The possible
3 1
5 7
2
6
4 0
3 1
5 7
2
6
4 0
S4
φ4
tr Φ4
Z4
Z2 Z2
ψa0a1a2ψa0b1b2ψb0a1b2ψb0b1a2
1
Figure 3.1: A melonic insertion.
61
symmetry groups are Zn2 , where 0 ≤ n ≤ v if q = 2v and 0 ≤ n < v if q = u2v
where u is odd and larger than 1. (It is easy to show that q must be even.) This
demonstration will be constructive, in that we produce vertices with each possible
symmetry. Question #2 turns out to be difficult, and it is essentially the problem
of counting so-called one-factorizations of complete graphs, where results are gen-
erally available up to q = 14 [60, 61, 62, 63, 64, 65, 66]: results spanning over a
century! In section 3.4, we summarize how these results can be combined and mod-
estly extended to give complete results on the number of distinct interaction vertices
up to q = 14. The symmetry analysis undertaken in this chapter suggests a new
twist on the counting problem: In addition to counting all one-factorizations, one
can count one-factorizations with a given symmetry group. An explicit example of
this symmetry-constrained counting is presented in section 3.5, and some additional
conceptual points are discussed in section 3.6.
The symmetry group of the interaction vertex leads to an effective coupling that
enters into the Schwinger-Dyson equation for the two-point function and the ladder
operator used for computing the four-point function, which will be exhibited explicitly
in all cases in sections 3.7 and 3.8. We note here that we assume every interaction
vertex we construct leads to a theory with a melonic limit, and all indications are
consistent with this. This being said, one will not find a fully rigorous proof of this
claim here. Over the p-adic numbers Qp it has already been shown in [27] that there
is quite a variety of melonic theories, depending on what sign function one chooses
over Qp. For approximately half of these melonic theories over Qp, the Schwinger-
Dyson equation can be solved exactly, not just in the infrared, but at all scales, in
terms of the solution to a q-th order polynomial equation. Remarkably, as shown in
section 3.8.1, there is an adelic product formula relating the eigenvalues of the ladder
operator integral equation across real and p-adic theories.
62
3.1 Structure of higher melonic theories
The action of the simplest Klebanov-Tarnopolsky model [52] is
S =
∫dt
(i
2ψa0a1a2∂tψ
a0a1a2 +g
4ψa0a1a2ψa0b1b2ψc0a1b2ψc0b1a2
), (3.1)
where each index, which we think of as a color index, runs independently from 1 to
N and each ψa0a1a2 is a Majorana fermion. Each index is separately O(N) invariant
and ψa0a2a3 transforms under O(N)3 as a triple tensor product of the fundamental
representation of O(N). We seek to understand the color structure of higher rank
models with higher degree interactions. We will simultaneously generalize to models
defined over p-adic numbers and to models with either O(N) or Sp(N) indices. The
models we consider will take the form
S =σψ2
∫K
dωψA(−ω)ΩAB|ω|s sgn(ω)ψB(ω) (3.2)
+ (σψ)q4g
|G|
∫K
dtΩA(0)A(1)...A(q−1) , (3.3)
where K is either R or Qp, G is the automorphism group of the interaction vertex, to
be discussed further in section 3.2. The spectral parameter s would usually be chosen
to be 1 for fermionic theories over R or 2 for bosonic theories over R, but for theories
over Qp it is more natural to let it vary continuously over positive real values. We
set σψ = −1 for fermionic theories, and σψ = +1 for bosonic theories. By sgnω we
mean a sign character, which is to say a multiplicative homomorphism of non-zero
elements of K to 1,−1. Capital indices are really groups of q− 1 lowercase indices,
each N -valued. For example, to recover (3.1) as a special case, we would set q = 4,
63
so that A = a0a1a2; we would set ΩAB = δa0b0δa1b1δa2b2 ; we would set
ΩABCD = δa0b0δa1c1δa2d2δb1d1δb2c2δc0d0 ; (3.4)
and of course we would set K = R and s = 1. It is well recognized (see e.g. [49,
51, 52]) that the structure ΩABCD in (3.4) corresponds to a coloring of the edges of
the tetrahedron so that only three colors are used, and opposite edges have the same
color.1
When considering theories over Qp, as explained in section 1.7 and in [27], we
must allow ΩAB to be symmetric (σΩ = 1) or anti-symmetric (σΩ = −1); and we
must choose sgnω to be one of the several multiplicative sign characters over Qp,
which are in one-to-one correspondence with the quadratic extensions of Qp. To get
a real, non-vanishing kinetic term, ΩAB must be Hermitian, and we must have
σψσΩ = sgn(−1) . (3.5)
(Surprisingly, non-trivial sign characters over Qp can have either sgn(−1) = −1 or
+1.)
For q = 4, still following [27], the obvious adaptation of the Klebanov-Tarnopolsky
model to a theory over Qp is to set ΩAB = Ωa0b0Ωa1b1Ωa2b2 where
Ωab =
1N×N for σΩ = 1
σ2 ⊗ 1N2×N
2for σΩ = −1 ,
(3.6)
1It is useful to clarify here one point of terminology: We use the terms “vertex” and “edge” todescribe the inner structure of an interaction vertex like ψa0a1a2ψa0b1b2ψc0a1b2ψc0b1a2 . From thispoint of view, an interaction vertex is a graph unto itself, with q vertices when the interaction termhas q powers of ψ. A full Feynman diagram consists of propagators connecting interaction vertices,and as is familiar from earlier work including [49], the inner structure of a propagator is q−1 threadswhich flow into the edges inside an interaction vertex.
64
and to set
ΩABCD = Ωa0b0Ωa1c1Ωa2d2Ωb1d1Ωb2c2Ωc0d0 . (3.7)
(Note that if σΩ = −1, then because of (3.6), N must be even.) If indeed Ωab is
antisymmetric, then one needs a direction on all the edges of the tetrahedral graph in
order to decide the order of the indices in each factor on the right hand side of (3.7).
However, flipping the direction on any one edge flips the sign of ΩABCD, and so can
be compensated for by changing the sign of g.
Generalizing the kinetic term to q > 4 is easy: We need only set
ΩAB =
q−2∏i=0
Ωaibi . (3.8)
Generalizing the interaction tensor turns out to be more subtle, and laying the ground-
work for finding suitable generalizations is the focus of the rest of this section.
Up to the minor issue of directedness, constructing a rank q interaction tensor
ΩA(0)A(1)...A(q−1) as a product of q(q − 1)/2 factors Ωa
(i)r a
(j)r
corresponds to a coloring
problem on the complete graph of q points (and therefore q(q−1)/2 edges), where we
use q − 1 colors (each one labeled by a value of r) and require that each of the q − 1
links incident on a given vertex (each one labeled by a value of i) must be a different
color. A special case of Baranyai’s theorem guarantees that this can always be done
provided q is even. It is impossible when q is odd. One can map the problem onto the
scheduling of a round-robin tournament, where each link is one game, each vertex is a
contestant, and each color is a round, during which each contestant plays exactly one
game. This phrasing makes it obvious that no coloring is possible for q odd, because
in a given round one must pair up all q contestants in two-person games. For q even,
65
there is a canonical solution, which is
rij ≡ i+ jmod q − 1 if 0 ≤ i < q − 1, 0 ≤ j < q − 1, and i 6= j
ri,q−1 ≡ 2imod q − 1 for 0 ≤ i < q − 1 .
(3.9)
Here, rij is the color of the edge from vertex i to vertex j. Vertex labels i and j take
values from 0 to q − 1, while color labels r run from 0 to q − 2. The corresponding
interaction tensor is
ΩA(0)A(1)...A(q−1) =∏
0≤i<j≤q−1
Ωa
(i)rija
(j)rij
. (3.10)
In (3.10), we took care of the directedness issue by requiring i < j, which is the same
as alphabetizing the lowercase indices in (3.7).
For q = 2, 4, and 6, the canonical solution (3.9) is the only solution. Starting at
q = 8, there are multiple solutions: That is, rij can be chosen differently from (3.9)
but still consistent with the requirement that we use only q − 1 colors and have one
edge of each color coming together at each vertex. For any such rij, we can still use
(3.10) to construct the interaction tensor. We exhibit the six different solutions for
q = 8 in figure 3.2. By “different,” we mean that there is no way to relabel the colors
and/or the vertices to map any of the six rij into one another. A striking point is
that the solutions have different symmetry groups, composed of up to three factors
of Z2.2
For q = 10, there are (we claim) 396 different interaction vertices, and none of
them have any symmetry. To justify this claim, and to proceed to larger q, we need to
give a more conceptually organized presentation. We do so in the next three sections,
starting with constraints on the symmetry groups in section 3.2, continuing with an
explicit construction in section 3.3, and concluding with a summary of the problem
2By Z2 we mean the integers modulo 2, or equivalently the multiplicative group 1,−1—notthe 2-adic integers.
66
canonical asymmteric, 960
3 1
5 7
2
6
4 0
fully symmteric, 30
3 1
5 7
2
6
4 0
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
1
(a) G = Z 32
30 one-factorizations
Order of Aut(F) is 1344
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
1
(b) G = Z 22
630 one-factorizations
Order of Aut(F) is 64
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
1
(c) G = Z 22
420 one-factorizations
Order of Aut(F) is 96
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
Z22, 420
3 1
5 7
2
6
4 0
1
(d) G = Z2
2520 one-factorizations
Order of Aut(F) is 16
canonical asymmteric, 960
3 1
5 7
2
6
4 0
fully symmteric, 30
3 1
5 7
2
6
4 0
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
1
(e) G = 1
1680 one-factorizations
Order of Aut(F) is 24
canonical asymmteric, 960
3 1
5 7
2
6
4 0
fully symmteric, 30
3 1
5 7
2
6
4 0
asymmteric, 1680
3 1
5 7
2
6
4 0
Z2, 2520
3 1
5 7
2
6
4 0
1
(f) G = 1
960 one-factorizations
Order of Aut(F) is 42
Figure 3.2: The six inequivalent melonic interactions and their symmetry fac-
tors for q = 8. Each interaction can be identified with an isomor-
phism class of one-factorizations, and we also list the number of one-
factorizations in each class as well as |Aut(F)|, the number of permu-
tations in S8 that preserve the one-factorizations in a given isomor-
phism class (see section 3.4 for an explanation of this notation). The
canonical coloring is the bottom right.
67
3 1
5 7
2
6
4 0
3 1
5 7
2
6
4 0
S4
φ4
tr Φ4
Z4
Z2 Z2
ψa0a1a2ψa0b1b2ψb0a1b2ψb0b1a2
1
3 1
5 7
2
6
4 0
3 1
5 7
2
6
4 0
S4
φ4
tr Φ4
Z4
Z2 Z2
ψa0a1a2ψa0b1b2ψb0a1b2ψb0b1a2
1
3 1
5 7
2
6
4 0
3 1
5 7
2
6
4 0
S4
φ4
tr Φ4
Z4
Z2 Z2
ψa0a1a2ψa0b1b2ψb0a1b2ψb0b1a2
1
Figure 3.3: Left: The symmetry group of φ4 theory is the symmetric group S4.
Middle: The symmetry group of tr Φ4 theory is the cyclic group Z4.
Right: The symmetry group of the quartic tensor interaction is the
Klein group Z22.
of counting distinct interaction vertices in section 3.4.
3.2 Symmetry groups of interaction vertices
In ordinary scalar field theory where the scalar φ is real-valued, the symmetry group
of a q-fold interaction vertex φq is the permutation group Sq, with order q!, because
all propagators leading into the interaction vertex are equivalent and can be per-
muted arbitrarily without changing the structure of the interaction. In a matrix field
theory based on a Hermitian N × N matrix Φ, the symmetry group of a tr Φq in-
teraction vertex is the group Zq of cyclic permutations of the propagators. In the
q = 4 Klebanov-Tarnopolsky model, the symmetry group is Z2 × Z2 (not Z4), gen-
erated by the permutations (12)(34), and (13)(24) and sometimes referred to as the
Klein group. As should be clear from figure 3.3, a permutation in the Klein group
reorders propagators leading into the interaction vertex in such a way that we get
back to exactly the same diagram that we started with. This sort of permutation is
what we will call a coloring automorphism. If we look at the inner structure of the
interaction vertex, we see that a coloring automorphism permutes the vertices (each
one corresponding to an incoming propagator) in such a way as to preserve the colors
of each edge.
68
To be precise: When we say that rij is a coloring of the complete graph Kq with
q vertices, what we mean is that ij labels an edge (so i 6= j), and each rij is chosen
from the set of “colors” 0, 1, . . . , q − 2, with the constraints that rij = rji and that
for fixed i, rij is a bijection from the q − 1 vertices that remain after i is omitted to
the set of colors. In other words, each edge leading into a given vertex is a different
color. A coloring automorphism is defined as a map i→ π(i) such that rπ(i)π(j) = rij
for all i and j. Let the group of coloring automorphisms be G. The main purpose of
the rest of this section is to limit the possibilities for G. Then in section 3.3 we will
show that all groups G not ruled out by the arguments of this section actually can
be realized.
Our first claim is that any coloring automorphism is an involution. Denote the
coloring automorphism by π. Assume that π is not the identity, since otherwise the
claim is trivial. For some vertex i, we have j ≡ π(i) 6= i. Then rij = rπ(i)π(j) = rjπ(j),
where in the first equality we remembered that π is a coloring automorphism. From
rji = rjπ(j) we can conclude that i = π(j) because, as noted previously, the coloring
r must be a bijection, for fixed j, from vertices i 6= j to colors.
Next we remember an elementary result of group theory: Any finite group G
consisting only of involutions is isomorphic to Zn2 for some n. First let’s show that
the group is abelian. Therefore let g and h be group elements. Because g and h are
involutions, we have (gh)−1 = h−1g−1 = hg. But because gh is also an involution,
(gh)−1 = gh. So gh = hg as required. Now the fundamental theorem of finite abelian
groups tells us that G must be a direct product of cyclic subgroups of prime-power
order. Because all elements of G are involutions, any cyclic subgroup must be a copy
of Z2, and the result is proven.
Another elementary point to note is that if a coloring automorphism preserves
any vertex, then it is necessarily the trivial automorphism that maps all vertices to
themselves. To see this, suppose π(i) = i for some vertex i, and consider any other
69
G = Z2
1 0
G = Z 22
1 0
2 3
G = Z 32
5
6
4
7
1 0
2 3
1
G = Z2
1 0
G = Z 22
1 0
2 3
G = Z 32
5
6
4
7
1 0
2 3
1
G = Z2
1 0
G = Z 22
1 0
2 3
G = Z 32
5
6
4
7
1 0
2 3
1
Figure 3.4: The first three instances of maximally symmetrical interaction ver-
tices, with q = 2v and G = Zv2. We omitted one color from the q = 8
case, namely a seventh color with edges running from each corner to
the diametrically opposite corner, for example from 0 to 6.
vertex j. We have rij = rπ(i)π(j) = riπ(j), and because r is a bijection, for fixed i, from
vertices j 6= i to colors, we can conclude π(j) = j. It follows that any permutation
that gives a coloring automorphism consists of q/2 two-cycles.
We also note that color automorphisms that have a two-cycle in common must
be identical. For suppose there are color automorphisms π and π and a vertex i such
that π(i) = π(i). Then for any j we have that rij = rπ(i)π(j) = rπ(i)π(j) but also
rij = rπ(i)π(j). Since r is a bijection, it follows that π(j) = π(j) for all j.
It helps our imagination to think of the group G = Zn2 of coloring automorphisms
as reflections through n orthogonal planes which do not pass through any vertices. If
q = 2v and n = v, this line of thinking suggests that we can produce a coloring of Kq
whose automorphism group is Zv2: See figure 3.4 for the first few instances.
With these preliminaries in hand, we now come to the main result of this section:
If q = u2v where u is odd, then the largest that G can be is Zv2 if u = 1, or Zv−12
if u > 1. The arguments in the remainder of this section do not demonstrate the
existence of interaction vertices with any particular symmetry group G; rather, they
rule out larger symmetry groups.
The proof of our main result relies on the orbit-stabilizer theorem, which we
summarize here for the purposes of a self-contained presentation. If a group G acts
70
on a set X, then the stabilizer stab i of an element i ∈ X is the subgroup of G of
elements which preserve i. Meanwhile, the orbit orb i is the subset of X consisting of
all images of i under the action of elements of G. The theorem says
| orb i|| stab i| = |G| . (3.11)
As a first application, let X be the set of q vertices, assume that G = Zn2 . The
stabilizer of a vertex i is the trivial group: This is because any π ∈ G that maps i
to itself must also map all other vertices to themselves. Invoking (3.11) we see that
each orbit contains 2n points. The union of all vertex orbits is all of X, and so there
must be
q ≡ u2v−n (3.12)
distinct vertex orbits. Already, (3.12) shows that G cannot be larger than Zv2, because
if n > v, q is not an integer.
To finish proving the main result, all we need to do is to exclude the possibility
that G = Zv2 when u > 1. This turns out to require somewhat more subtle reasoning
than we have used so far, but the essential idea is to consider the quotient of the
interaction vertex by G and show that, as a graph with q = u vertices, it leads to
an impossible coloring problem. Clearly, we could just set v = n throughout the
following paragraphs, but we refrain from doing so because keeping n ≤ v general
allows us to see some first hints on how to actually construct graphs with any allowed
symmetry.
Let’s start with a second application of the orbit-stabilizer theorem. Consider the
set of all edges of a fixed color. For any fixed color, there are q/2 such edges. Edges
of a fixed color are permuted among themselves by coloring automorphisms, and for
each edge we have
| orb e|| stab e| = |G| . (3.13)
71
Following a standard trick, we divide both sides of (3.13) by | stab e| and then sum
over all distinct orbits to get
q
2=∑orbits
|G|| stab e|
. (3.14)
This can be written more usefully as
u = 2n−v+1∑orbits
1
| stab e|. (3.15)
To get proper mileage out of (3.15), we need some knowledge of the values of | stab e|—
which need not be unity! Consider however the following division of edges (still of
any fixed color) into two classes. There are edges which join two vertices which are
in the same vertex orbit; we will call these “internal” edges. All other edges we
will refer to as “external” edges. We observe that stab e is the trivial group for any
external edge, because if it weren’t, then there would be some coloring automorphism
that exchanges the edge’s two ends, and that would make the edge internal. So
| stab e| = 1 for external edges.
Next we want a count of external edges of a fixed color. It’s easier to start by
considering external edges of any color, i.e. the disjoint union of external edges of
each fixed color. There are(q2
)·22n such edges, because to specify one we must choose
a pair of vertex orbits, and then from each of those two vertex orbits we must choose
one vertex. Because the stabilizer of an external edge is trivial, its orbit must have
2n elements. So the count of external edge orbits of any color is
Ne =
(q
2
)· 2n = q
(u2v−n−1 − 1
2
). (3.16)
Suppose now 0 ≤ n < v. Then 2v−n−1 is a whole number, and because we have only
q− 1 colors to work with, there must be at least one color—call it red—with at least
72
u2v−n−1 external edge orbits. Restricting to red edges only, we recall that | stab e| = 1
for each external edge, and so at least u2v−n−1 terms in the sum on the right hand side
of (3.15) must be unity. Comparing to (3.15), we see that—for red edges—the sum
over orbits works out perfectly with only external edges, implying that there can’t be
any red internal edges. This is informative and useful for constructing examples.
Now suppose n = v. If also u = 1, then from (3.16) we see that Ne = 0: There are
no external edges at all! This makes sense because there is only one vertex orbit, and
indeed the cubical vertex illustrated in figure 3.4 shows that it is entirely consistent
to have n = v and u = 1. Where things get dangerous is if n = v and u > 1. Then,
from (3.16), Ne = q(u − 1)/2. Since we have only q − 1 colors to work with, there
must be at least one color—again call it red—with at least (u + 1)/2 external edge
orbits. Considering only red edges the sum on the right hand side of (3.15) restricted
to external edges gives u + 1. This is disastrous, because adding in the contribution
of internal edges (if any) results in the absurd inequality u ≥ u+ 1. Another way to
put it is that if n = v and u is odd and greater than 1, then we can’t color even the
external edges consistently with only q − 1 colors—let alone the internal edges. See
figure 3.5.
3.3 Construction of interaction vertices
Since for any q we can construct an interaction vertex with the canonical ordering,
which has no coloring automorphisms, what remains to be shown is that for q equal
to twice times an even number, it is always possible to construct an interaction vertex
with any allowed non-trivial symmetry group. Figure 3.2 explicitly shows that this
is true for q = 8. We now prove inductively that it is also true for q > 8.
Let q′ = q/2 and assume that rij is a coloring of Kq′ with symmetry group G′
(which may be the trivial group). Split the q vertices into two sets of q′ vertices;
73
3
4
1
0
2
5
q = 6
Z2
25 34
01
q = 3
1
Figure 3.5: A partial coloring for q = 6 which illustrates what goes wrong when
one demands too much symmetry. Here we propose Z2 symmetry,
which according to our general result is too much. On the left we
show a partial coloring with Z2 symmetry. On the right, we show
the same coloring modded out by the Z2 symmetry. The dots on the
right are vertex orbits, and the double lines of each color are edge
orbits. The problem is that we need two edge orbits between each
pair of vertex orbits, and all of them (in this simple case) must have
different colors. That means we need six colors just for the external
edges, and we only have five to work with—leaving us without a
second color to use in connecting the 01 and 34 orbits. Internal edges
would run within a vertex orbit, for example from 0 to 1.
74
342
051
9
108
117
6
342
051
9
108
117
6
342
051
9
108
117
6
1
(a)
342
051
9
108
117
6
342
051
9
108
117
6
342
051
9
108
117
6
1
(b)
342
051
9
108
117
6
342
051
9
108
117
6
342
051
9
108
117
6
1
(c)
Figure 3.6: Constructing an interaction vertex with q = 12 and G = Z2 in three
steps. (a) Separate the vertices into two groups of 6, shown here as
inner and outer rings. We’ve connected each group only cyclically in
order to avoid clutter, but in the full construction, we start with two
copies of the complete graph K6, one colored according to rij and the
other according to rij = rij + q′ − 1. (b) Reconnect the 12 vertices
according to the prescription shown in the first two lines of (3.17).
(c) Add the last color according to the third line of (3.17). In the
parlance of section 3.2, these last edges are the internal edges, while
the ones colored in the previous step are the external edges.
notationally this can be done by identifying a vertex first by indicating which set it’s
in, say with a Greek index α = 0 or 1, and then which vertex within the set it is, say
with a Roman index i ∈ 0, 1, . . . , q′ − 1. Let rij ≡ rij + q′ − 1. We now claim that
the coloring rαi,βj of Kq defined by
r0i,0j = r1i,1j = rij ,
r0i,1j = r1i,0j = rij ,
r0i,1i = r1i,0i = q − 1 ,
(3.17)
has symmetry group G = G′ × Z2. See figure 3.6 for a diagrammatic illustration of
how this coloring is generated. That the coloring (3.17) inherits the G′ symmetry is
clear from the fact that, for any π ∈ G′, we have that
rαπ(i),βπ(j) = rαi,βj . (3.18)
75
But by construction the coloring (3.17) is also invariant under the permutation τ that
acts on the Greek indices as
α→ α + 1 mod 2 , β → β + 1 mod 2 , (3.19)
while leaving the Roman indices unchanged. Furthermore, the coloring (3.17) has no
other symmetries. For coloring automorphisms that only swap around Roman indices
are in one-to-one correspondence with the coloring automorphisms of rij. And if a
coloring automorphism σ changes the Greek index of any index pair αi such that
σ(0i) = 1i′, then for any j
rσ(0i),σ(0j) = r1i′,σ(0j) . (3.20)
But since r0j,0k 6= r1j′,0k′ for all j, j′, k, k′, it follows that σ must change the Greek
index of all index pairs and so must be of the form σ = π τ for some π ∈ G′. This
completes the inductive proof.
In general, for a given symmetry, there are multiple interaction vertices different
from the one generated by the prescription (3.17). An exception, however, occurs
for the maximally symmetric vertex when q = 2v. In this case there is only one
vertex with G = Z v2 . For G consists of the identity element and q − 1 permutations
that commute amongst each other and each consists of q/2 two-cycles. And amongst
permutations in Sq consisting of q/2 two-cycles, one can at most form a set of q − 1
elements that commute amongst each other but don’t share a two-cycle; and any two
such sets are equivalent by conjugation. But if we consider the edges of a given color,
say red, in a maximally symmetric colored graph, then this sub-graph is invariant
under the permutation π that swaps vertices connected by a red edge. But π must
commute with all permutations in G, and so it follows that π ∈ G. Moreover, if we
explicitly write π = (a0a1)(a2a3) . . . (aq−1aq), then we recognize that each two-cycle in
π corresponds to a red edge; in other words, each non-trivial element of G is precisely
76
associated with a one-factor. Since there are q− 1 non-trivial elements of G, the one-
factors are all specified once G is specified. And we have argued that G is essentially
unique.
3.4 One-factorizations and equivalent interaction
terms
In this section we determine the conditions under which two theories with actions
of the form (3.2) are equivalent. As discussed briefly in section 3.1, the choice of an
interaction tensor ΩA(0)A(1)···A(q−1) corresponds to a coloring problem on the complete
graph Kq on q vertices.
More precisely, these interaction tensors are in one-to-one correspondence with
ordered one-factorizations of Kq. A one-factor of Kq is a set F of edges such that each
vertex of Kq belongs to a unique edge in F . A one-factorization of Kq is a partition
of the edge-set of Kq into q − 1 one-factors Fi. We denote a one-factorization as
F = F0, . . . , Fq−2, and when speaking of one-factorizations we do not distinguish
between different orderings of the Fi. An ordered one-factorization is a (q − 1)-tuple
F = (F0, . . . , Fq−2), i.e. we have imposed a particular order on a one-factorization.
To see that ordered one-factorizations correspond to interaction vertices, recall that
the fields ψa0...aq−2 don’t have any built-in symmetry under interchange of indices ai.
Each such index is associated with a definite color: For example, we could say that
red is associated with the first index a0, then green with a1, blue with a2, and so
forth. And each color is associated with a one-factor: red for F0, green for F1, blue
for F2, etc.
One can check that any graph automorphism π : Kq → Kq carries (ordered) one-
factorizations into other (ordered) one-factorizations via the actions πF = (π(F0), . . . , π(Fq−2))
and πF ≡ πF , respectively. Two ordered one-factorizations F and G are said to be
77
isomorphic if there exists a graph automorphism π such that F = πG. Similarly,
F ' G means there exists π so that F = πG. We can also consider the natural action
of any τ ∈ Sq−1 on ordered one-factorizations τF = (Fτ(0), . . . , Fτ(q−2)). This action
simply permutes the colors associated to each one-factor. In this language, F ' G if
and only if there exists a graph automorphism π and a permutation of colors τ such
that πF = τG.
Given an ordered one-factorization F , we can form the following interaction tensor:
ΩFA(0)...A(q−1) ≡q−2∏r=0
∏〈ij〉∈Fr
Ωa
(i)r a
(j)r, (3.21)
where 〈ij〉 is the edge in Kq running between the vertices i and j and ar is the color
index associated to the one-factor Fr. In writing 〈ij〉, we implicitly assume i < j.
Conversely, any degree q interaction term in which each constituent ψA has a single
color index contracted with each other ψA arises from an interaction tensor of the
form (3.21).
Note that for distinct ordered one-factorizations F and G we end up with distinct
interaction tensors ΩF 6= ΩG. We will however show that if their underlying one-
factorizations F and G are isomorphic then ΩF and ΩG give rise to equivalent theories.
Our argument proceeds in two parts: First we discuss equivalence under the action of
τ ∈ Sq−1 permuting colors, and then we consider the action of a permutation π ∈ Sq
of vertices.
Suppose we make the following field redefinition: φA = ψτ−1A, where τ−1 ∈ Sq−1
and τA ≡ aτ−1(0) . . . aτ−1(q−2). Then:
ΩFA(0)...A(q−1)φA(0) · · ·φA(q−1)
= ΩτFA(0)...A(q−1)ψ
A(0) · · ·ψA(q−1)
. (3.22)
So, modulo a linear field redefinition implemented by a permutation matrix, the
interaction term only depends on the underlying one-factorization F .
78
Gq 2 4 6 8 10 12 14
1 0 0 1 2 396 526,910,769 ∼ 1.13× 1018
Z2 1 0 0 1 0 4851 0Z2
2 0 1 0 2 0 0 0Z3
2 0 0 0 1 0 0 0
Table 3.1: The number of isomorphism classes of one-factorizations of Kq with
given symmetry group G = Aut(F). The values for q = 12 and 14
were determined using [65] and [66], respectively; the exact count for
q = 14 is quoted in the main text.
Now view a graph automorphism as a permutation on q vertices, π ∈ Sq. With
the change of indices B(i) = Aπ(i) one can show:
(σψ)πΩFA(0)...A(q−1)ψA(0) · · ·ψA(q−1)
= ΩFA(0)...A(q−1)ψAπ−1(0) · · ·ψAπ
−1(q−1)
= ΩπFB(0)...B(q−1)ψ
B(0) · · ·ψB(q−1)
.
(3.23)
So any two isomorphic ordered one-factorizations give the same interaction term, up
to a sign.3 We can get rid of this sign by sending g → −g.4
We should point out here that isomorphic one-factorizations give equivalent inter-
actions to all orders in N . As we will show in sections 3.7 and 3.8, even non-isomorphic
one-factorizations lead to the same two-point and four-point functions in the melonic
limit, up to a rescaling of g by a power of the order of the coloring automorphism
group G.
For future reference, we note that by Aut(F) we mean all graph automorphisms π
such that πF = F . Aut(F) is the exactly the vertex automorphism group described
in section 3.2. In contrast, Aut(F) consists of all graph automorphisms π such that
πF = F , that is all π that can be undone by a permutation τ ∈ Sq−1 of the colors
3In the presence of multiple interaction terms in the lagrangian, this argument that a permutationof vertices does not lead to a new theory continues to hold true. But it will no longer be true thatan interaction term only depends on the underlying one-factorization F , see section 3.6.
4See [67] and [68] for a careful discussion, in the context of the q = 4 fermionic tensor model,of how the Hamiltonian even at the quantum level transforms in the degree 1 sign representationunder permutations of indices and of how this affects the spectrum of the theory.
79
πF = τF . Clearly Aut(F) ≤ Aut(F). In general Aut(F) has a much richer group
structure than Aut(F). See for instance table 3.2 in section 3.5 where, for q = 12, we
list the ten possible values for |Aut(F)| when |Aut(F)| = 2.
In the case q = 12, the vast majority of isomorphism classes of one-factorizations
have trivial Aut(F). It is known in the literature [65] that there are a total of
526,915,620 non-isomorphic one-factorizations (the sum of the two entries in the q =
12 column in figure 3.1) and exactly 252,282,619,805,368,320 distinct one-factorizations
of K12. Implementing the orderly algorithm used in [64] and [65] on the set of one-
factors invariant under a given involution π ∈ S12, we find that there are 1,008,649,635,840
one-factorizations of K12 with Aut(F) = Z2 and that they fall into 4851 isomorphism
classes under permutations of vertices, see section 3.5.
For q = 14, Aut(F) can only be trivial and there are 1,132,835,421,602,062,347
nonisomorphic one-factorizations of K14 [66].
For q ≤ 14, a summary of the number of isomorphism classes of one-factorizations
and therefore the number of inequivalent interaction terms is given in table 3.1. It is
known that for sufficiently large q, the number of non-isomorphic one-factorizations
N(q) satisfies logN(q) ∼ 12q2 log q [69].
3.5 Finding the number of isomorphism classes of
one-factorizations for q = 12 using the orderly
algorithm
In this section we discuss in detail how to find the numbers of melonic interactions
with trivial and non-trivial automorphism groups for q = 12.
We have seen that melonic interaction vertices can be identified with isomorphism
classes of one-factorizations of complete graphs. For q = 12 the total number of
80
isomorphism classes was found in [65]. To classify melonic theories it is desirable to
also know how many of these isomorphism classes correspond to melonic interactions
with a non-trivial automorphism group. A one-factorization corresponds to such an
interaction if there exists a vertex permutation π ∈ S verticesq that leaves all one-factors
invariant. We will now show, for q = 12, how to employ the orderly algorithm of [64]
and [65] to find the total number of isomorphism classes of one-factorizations as well
as the number of isomorphism classes corresponding to melonic interactions with
non-trivial automorphisms.
A one-factor Fi can be represented by a set of six pairs
Fi =a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11
(3.24)
where all the ai are distinct and belong to the set 0, 1, ..., 11.
By forming a set of I pair-wise disjoint one-factors, we get a partial one-factorization
HI ,
HI = F0, F1, ..., FI−1 (3.25)
where 1 ≤ I < 12. A (full) one-factorization is a partial one-factorization with I = 11.
In the notation of section 3.4, H11 = F .
A key point in implementing the orderly algorithm consists in imposing a lexico-
graphic order amongst partial one-factorizations. One first imposes an order amongst
the one-factors as follows:
Associate to each one-factor Fi a unique ordered 12-tuple Fi by writing it in the
form (3.24) with a2i < a2i+1 and a2i < a2i+2 for all i. Then take F to be given
by
(a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11) . (3.26)
Given two distinct one-factors Fi and F ′i , we impose an order such that Fi < F ′i
81
if the left-most digit ai by which Fi and F ′i differ is smaller for Fi.
Having ordered the one-factors, we can now order the partial one-factorizations:
Associate to each partial one-factorization HI a unique ordered I-tuple HI by
writing it in the form (3.25) with Fi < Fi+1 for all i. Then take HI to be given
by
(F0, F1, ..., FI−1) . (3.27)
Given two distinct partial one-factorizations HI and H ′I , we impose an order
such that HI < H ′I if the left-most digit ai by which HI and H ′I differ is smaller
for HI .
Now, given a partial one-factorization, we can act on it with a permutation π ∈ S12
by permuting the digits of all the one-factors:
ai → π(ai) . (3.28)
With these preliminaries in place, we are ready to describe the orderly algorithm.
The algorithm works by constructing non-isomorphic partial one-factorizations HI
and then using these to construct larger non-isomorphic partial one-factorizations
HI+1. The way one ensures that all constructed partial one-factorizations HI are
non-isomorphic is by always selecting from any isomorphism class only the one with
the lowest lexicographic ordering. The steps of the algorithm can be described as
follows:
1. Consider all partial one-factorizations H1 containing exactly one-factor and dis-
card the ones for which there exists a permutation π ∈ S12 such that πH1 < H1.
Denote the set of remaining partial one-factorizations H1 by H1.
2. Consider all partial one-factorizations H2 obtained by adjoining to each member
H1 of H1 a one-factor disjoint from the one-factor in H1. Discard the ones for
82
which there exists a permutation π ∈ S12 such that πH2 < H2. Denote the set
of remaining partial one-factorizations H2 by H2.
3. Continue on as in step 2 to obtain sets Hn of ordered partial one-factorizations
for n = 3, 4, . . . , 10.
4. Consider all one-factorizations H11 obtained by adjoining to each member H10
of H10 a one-factor disjoint from all the one-factors in H10. Discard the ones for
which there exists a permutation π ∈ S12 such that πH11 < H11. Denote the
set of remaining one-factorizations H11 by H11. This set will contain exactly
one one-factorizations for each isomorphism class of one-factorizations, namely
the one with the lowest lexicographic ordering.
The efficiency of the above algorithm can be improved in several ways. In consid-
ering in step 1 the partial one-factorizations H1 containing one one-factor, one can
restrict consideration to one-factors containing the pair 0, 1. For the one-factors
adjoined to the members of H1 in step 2, one can restrict consideration to one-factors
containing the pair 0, 2. Similarly one can consider restricted sets of one-factors
in implementing the rest of step 3 as well as step 4, where one needs only adjoin
one-factors containing 0, 11 to the members of H10.
The algorithm as described above will find all isomorphism classes of one-factorizations
as was done in [65]. To find only the isomorphism classes corresponding to melonic
interactions with non-trivial automorphism groups, the algorithm must be modified.
As we have seen in section 3.2, for q = 12 the only possible non-trivial automorphism
group is Z2. So for any one-factorization there can be at most a single non-trivial
permutation π ∈ S12 that leaves all one-factors invariant, and we have also seen in
section 3.2 that such a permutation must consist of six two-cycles. Let S12 denote
the subset of permutations in S12 that consist of six two-cycles. In total there are 11!!
such permutations.
83
For any one-factorization H11 whose one-factors are invariant under a permu-
tation π ∈ S12, one can act on it with some permutation σ ∈ S12 to generate a
one-factorization σH11 whose one-factors are invariant wrt. any other permutation
π′ ∈ S12. The isomorphism classes of one-factorizations corresponding to a melonic
vertex with G = Z2 each contains a number of elements divisible by 11!! as each of
them contains the same number of one-factorizations for each member of S12. For
the purpose of finding the number of such isomorphism classes by constructing ex-
actly one example from each class, we can therefore restrict attention to one-factors
invariant under one specific permutation π ∈ S12, say
π = (0, 1)(2, 3)(4, 5)(6, 7)(8, 9)(10, 11) . (3.29)
The modifications by which we need to adjust steps 1 to 4 above, then, consist of:
Throughout the implementation of the algorithm, one should only consider one-
factors that are invariant under the permutation (3.29). In total there are 331
such one-factors.
Throughout steps 1 to 4, when checking for each partial one-factorization HI
whether there exists a permutation π such that πHI < HI , one should only
consider permutations π that respect the symmetry under (3.29), ie. permuta-
tions that only mix the 331 one-factors invariant under π amongst themselves,
i.e. permutations that commute with π. This subgroup of S12, the centralizer
CS12(π), contains 46,080 = 12!11!!
= 6! · 26 permutations.
These two restrictions make the orderly algorithm run much faster. One finds that
the total number of isomorphism classes corresponding to G = Z2 melonic interaction
vertices is 4851. Having generated one representative of each of these classes, one can
act on the representatives with permutations in S12 and delete duplicates to find that
there are 97,032,192 one-factorizations consisting of one-factors that are each invariant
84
|Aut(F)| 2 4 6 8 10 12 16 24 48 240n 3697 944 13 104 1 38 36 10 6 2
Table 3.2: The number of non-isomorphic one-factorizations n with |Aut(F)| = 2
for each automorphism group order.
under (3.29). Multiplying this number with 11!! one finds the total number of one-
factorizations all of whose one-factors are invariant with respect to some non-trivial
permutation, namely 1,008,649,635,840.
An alternative way to obtain this number of one-factorizations is to use the
orbit-stabilizer theorem in the form (3.14). Acting on the 4851 non-isomorphic one-
factorizations with the centralizer CS12(π), one can find the order of the automorphism
group Aut(F) for each of these one-factorizations. One can then carry out the sum
4851∑i=1
|S12||Aut(F i)|
, (3.30)
and recover the number 1,008,649,635,840. The terms in the sum (3.30) assume but
ten different values, and the sum can be evaluated straightforwardly using the data in
table 3.2, which lists the number of non-isomorphic one-factorizations corresponding
to interaction vertices with Z2 symmetry for any order of Aut(F).
3.6 Isomorphism classes of ordered one-factorizations
for q = 12
For a complete graph of order q, we can consider the set of all ordered one-factorizations.
The orbits of this set under permutations τ ∈ Sq−1 of the one-factors are equal to the
one-factorizations of the graph. The orbits of the one-factorizations under permuta-
tions π ∈ Sq of the vertices are equal to the isomorphism classes that we identified
with melonic interaction vertices and tabulated in figure 3.1. But another set one can
consider is the orbits of ordered one-factorizations under permutations π ∈ Sq of the
85
vertices. The relations between these sets and orbits are depicted diagrammatically
in figure 3.7.
Figure 1: The number of equivalence classes of ordered one-factorizations under permutationsof edges and vertices for q = 12.
automorphism group ofthe interaction vertex 1 Z2
ordered one-factorizations 11! · 252,281,611,155,732,480 11! · 1,008,649,635,840one-factorizations 252,281,611,155,732,480 1,008,649,635,840
isomorphism classes ofordered one-factorizations 21,023,467,596,311,040 168,108,272,640
isomorphism classes ofone-factorizations 526,910,769 4851
ordered one-factorizationsS one-factorsq−1
S one-factorsq−1
one-factorizations
S verticesq S vertices
q
ordered one-factorizations
isomorphism classes of isomorphism classes of
one-factorizations
1
Figure 3.7: The relationship between ordered one-factorizations, one-
factorizations, and their isomorphism classes under vertex per-
mutations. An arrow labeled by a group indicates that the set at the
head end can be identified with the orbits of the set at the tail end
under the action of the group.
We argued in section 3.2 that while an ordered one-factorization of a graph can
immediately be translated into a melonic interaction vertex, permutations of the ver-
tices of the graph correspond to commuting fields past each other, and permutations
of the one-factors correspond to field redefinitions, neither of which operations yield
physically distinct theories, so that in counting the number of theories via ordered
one-factorizations one should mod out by S verticesq and S one-factors
q−1 . But this reasoning
only applies to lagrangians with a single interaction term. For theories with multiple
interaction terms, one can still commute the fields around in each interaction term,
and so one should still mod out by S verticesq . But field-redefinitions can only be ap-
plied globally, and so one should no longer count the number of interaction terms by
modding out by S one-factorsq−1 . A crude upper bound on the number of distinct theories
with n interaction vertices of order q is the n-th power of the number of isomorphism
classes of ordered one-factorizations. The number of isomorphism classes of ordered
one-factorizations, call it C, is therefore not without interest in the context of melonic
86
models. In the remainder of this section we will derive this number in the case of
q = 12 and for interaction vertices with and without trivial automorphism groups.
Letting X denote a set acted on by a finite group G, Xg denote the set of elements
in X that are left invariant by g ∈ G , and X/G denote the set of orbits of X under
G, Burnside’s lemma tells us that
|X/G| = 1
|G|∑g∈G
|Xg| . (3.31)
By taking X to be the set of ordered one-factorizations and G to be S12 acting on
vertices, we can use Burnside’s lemma to find C. If in the sum over g ∈ G in (3.31)
we take g to be the identity element, then |Xg| is equal to |X| itself, that is the total
number of ordered one-factorizations, which in turn is equal to 11!M , where M is the
total number of one-factorizations. From [65] we know that
M = 252,282,619,805,368,320 . (3.32)
If in the sum over g ∈ G in (3.31) we consider the sum over the permutations g ∈ S12,
with S12 defined in section 3.5, then this partial sum will count each ordered one-
factorization whose one-factors are all invariant under some permutation g ∈ S12
exactly once. This number is equal to 11!M2, where M2 is the total number of one-
factorizations whose one-factors are all invariant under some permutation g ∈ S12.
We found in section 3.5 that this number is equal to
M2 = 1,008,649,635,840 . (3.33)
If in the sum over g ∈ G in (3.31), g is not the identity element nor belongs to S12,
then |Xg| = 0.
87
In conclusion, Burnside’s lemma tells us that
C =1
12!(11!M + 11!M2) = 21,023,635,704,583,680 . (3.34)
This, the number of isomorphism classes of ordered one-factorizations, can be split
into C1, the number of classes corresponding to interaction vertices with trivial auto-
morphism group, and C2, the number of classes corresponding to interaction vertices
with automorphism group Z2,
C = C1 + C2 . (3.35)
To find out how C splits into C1 and C2 we can use the orbit-stabilizer theorem in
the form of (3.14) and (3.30),
|X| =∑
orbits o
|G||stab(o)|
. (3.36)
The orbits of ordered one-factorizations under S12 acting on vertices fall into two sets.
There are C1 orbits o for which |stab(o)| = 1 and C2 orbits o for which |stab(o)| = 2.
Hence,
11!M = 12!
(C − C2
1+C2
2
). (3.37)
Solving for C2, one finds that
C2 = 168,108,272,640 . (3.38)
The numbers of ordered and un-ordered one-factorizations and the numbers of iso-
morphism classes of these under vertex permutations for q = 12 are summarized in
the table in table 3.3.
88
automorphism group ofthe interaction vertex trivial Z2
ordered one-factorizations 11! · 252,281,611,155,732,480 11! · 1,008,649,635,840
one-factorizations 252,281,611,155,732,480 1,008,649,635,840
isomorphism classes ofordered one-factorizations 21,023,467,596,311,040 168,108,272,640
isomorphism classes ofone-factorizations 526,910,769 4851
Table 3.3: The number of equivalence classes of ordered one-factorizations under
permutations of one-factors and vertices for q = 12.
3.7 The two-point function and the Schwinger-Dyson
equation
The kinetic term in the action (3.2) gives rise to a free propagator that in momentum
space is given by
Ga0a1...aq−2b0b1...bq−2
0 (ω) =√
sgn(−1)sgn(ω)
|ω|sq−2∏i=0
Ωaibi ≡ G0(ω)
q−2∏i=0
Ωaibi , (3.39)
where the matrix Ωab is the inverse of Ωab, that is, ΩabΩbc = δca. With the normaliza-
tion of the kinetic term as given in (3.2), we must choose
√sgn(−1) =
1 if sgn(−1) = 1
i if sgn(−1) = −1 .
(3.40)
Once we include an interaction term, the propagator picks up loop corrections,
but in the limit where g → 0 and N → ∞ such that g2N(q−1)(q−2)
2 is kept fixed, only
melonic diagrams survive. The melonic contributions to the free propagator can all
be obtained by iteratively applying the melonic insertion to the free propagator as
shown on figure 3.8.
Adding together the free propagator and all the melonic corrections yields the
89
dressed propagator
Ga0a1...aq−2b0b1...bq−2(ω) ≡ G(ω)
q−2∏i=0
Ωaibi . (3.41)
Just as in the cases of the SYK model [70] and the Klebanov-Tarnopolsky tensor
model with q = 4 [52], the dressed propagator satisfies the Schwinger-Dyson equa-
tion depicted schematically in figure 3.9. It first seems surprising that the right-most
propagator in figure 3.9 has no melonic insertion. This is correct because any Feyn-
man diagram contributing to the dressed propagator must have a right-most melonic
insertion, and a free propagator attaches to it from the right.
To work out the exact mathematical expression for the Schwinger-Dyson equation,
it is necessary to consider the automorphism group of the interaction as well as the
number of sign flips involved in index contraction.
The first melonic correction to the free propagator, depicted on the right-hand
side of figure 3.8, contains two interactions each of which is described by a colored
graph with vertices labeled from 0 to q − 1. For a given Feynman diagram corre-
sponding to the first melonic correction, define a permutation σ ∈ Sq by requiring
that if an external propagator enters the left interaction vertex at vertex i, then an
external propagator enters the right interaction vertex at vertex σ(i); and likewise
requiring that an internal propagator connected to vertex j in the left interaction
i σ(i)
j
k
σ(j)
σ(k)
i σ(i)
j
k
σ(j)
σ(k)
...
...︸ ︷︷ ︸q− 1 undressed propagators
= +
...
...
︸ ︷︷ ︸q− 1 dressed propagators
1
Figure 3.8: A melonic insertion.
90
i σ(i)
j
k
σ(j)
σ(k)
...
...︸ ︷︷ ︸q− 1 undressed propagators
= +
...
...
︸ ︷︷ ︸q− 1 dressed propagators
1
Figure 3.9: The Schwinger-Dyson equation for the dressed propagator.
vertex connects to vertex σ(j) in the right interaction vertex. See figure 3.10.
We claim now that such a Feynman diagram is melonic if and only if σ belongs
to the automorphism group G of the interaction. The claim can be proved by the
following reasoning:
The two interactions are connected by q−1 internal propagators, each of which
carry q − 1 threads, giving a total of (q − 1)2 internal threads.
The q − 1 threads of the left external propagator must connect to the q − 1
threads of the right external propagator, so there are at most (q − 1)(q − 2)
internal threads that can partake in index loops.
Since at least two internal threads are needed for an index loop, there can be at
i σ(i)
j
k
σ(j)
σ(k)
i σ(i)
j
k
σ(j)
σ(k)
...
...︸ ︷︷ ︸q− 1 undressed propagators
= +
...
...
︸ ︷︷ ︸q− 1 dressed propagators
1
Figure 3.10: To get a melonic diagram, the external legs must be connected to
vertices in the two interactions that are related by an automorphism.
91
i σ(i)
j
k
σ(j)
σ(k)
i σ(i)
j
k
σ(j)
σ(k)
...
...︸ ︷︷ ︸q− 1 undressed propagators
= +
...
...
︸ ︷︷ ︸q− 1 dressed propagators
1
Figure 3.11: All index loops can be brought to have a uniform orientation by
an even number of arrow flips. To get a uniform orientation of any
external thread an odd number of arrows flips are requisite.
most (q− 1)(q− 2)/2 index loops—the number required for a melonic diagram.
This maximum number of loops is achieved when and only when the external
threads only pass once through the internal propagators and the each index
loop contains exactly two internal threads.
Labeling the vertex that the left external propagator is incident to by i, the fact
that the external threads must pass only once through the internal propagators
is equivalent to the statement that rij = rσ(i)σ(j) for all j 6= i.
The fact that each index loop must contain exactly two internal threads is
equivalent to the statement that for any two indices j and k different from i,
rjk = rσ(j)σ(k).
We see then that the melonic diagrams are exactly those for which rlm = rσ(l)σ(m)
for all l,m, ie. those for which σ ∈ G. Since in writing down the Feynman diagram
corresponding to the first melonic correction we may take the left external propagator
to be incident to any vertex i, while the right external propagator can only be incident
to vertices i′ for which i′ = σ(i) for some σ ∈ G, the total number of Feynman
diagrams contributing is q|G|. And for each additional melonic insertion we pick up
an extra copy of this factor.
The other subtlety to consider before writing down the Schwinger-Dyson equation
is whether the contraction of index loops causes a sign difference between O(N) and
92
Sp(N) symmetric models. To account diagrammatically for the possibility of Sp(N)
symmetry, we replace each thread with an arrow. The orientation of any arrow can
be flipped at the cost of a factor of σω, and index loops of arrows with uniform
orientation can be contracted to give a factor of N without picking up a sign.
For the first melonic correction to the free propagator we may take the threads
of all propagators to point from left to right as in figure 3.11. For a given Feynman
diagram described by a permutation σ ∈ G as explained above, we may for any l
and m take the the arrow between vertices l and m to have the same orientation as
the arrow between vertices σ(l) and σ(m) since any automorphism involves an even
number of arrow flips. In this case, all index loops consist of two arrows with one
orientation and two arrows with the opposite orientation, and so all the loops may
be contracted without picking up a sign. But to obtain a uniform orientation of the
external threads running from left to right and recover the index structure of the free
propagator, it is necessary for each of the q− 1 colors to flip one thread in one of the
two interactions. Hence, we must in total flip an odd number off arrows, and so we
conclude that we pick up a factor of σΩ with each melonic insertion. This fact was
shown in [27] to apply for q = 4, but we see now that it holds true for all q.
Taking into account the symmetry of the interaction and the index structure of
the corrections, we arrive finally at the Schwinger-Dyson equation
G(t1 − t2) = G0(t1 − t2)
+ q|G|(g
|G|
)2
σΩ N(q−1)(q−2)
2
∫dt dt′G(t1 − t)G(t− t′)q−1G0(t′ − t2) .
(3.42)
To slightly shorten expressions and remove the explicit dependence on G, we find it
useful to introduce the definition
g ≡√
q
|G|N
(q−1)(q−2)2 g . (3.43)
93
3.7.1 The IR solution
The Schwinger-Dyson equation can be expressed in term of convolutions as
G(t) = G0(t) + σΩg2(G ∗Gq−1 ∗G0)(t) . (3.44)
In the infra-red limit, g is large, and to leading order in 1/g we can set G(t) = 0. To
solve (3.44) in the IR, we will adopt the methodology of [27], and so we introduce
multiplicative characters defined as
πs(t) ≡ |t|s, πs,sgn(t) ≡ |t|s sgn(t) . (3.45)
The Fourier transform of these multiplicative characters are given by
F [πs](ω) = Γ(πs+1) π−s−1(ω) , F [πs,sgn](ω) = Γ(πs+1,sgn) π−s−1,sgn(ω) , (3.46)
where Γ(πs) and Γ(πs,sgn) are instances of Gel’fand-Graev gamma functions, see sec-
tion 1.6. Using the above, we may express the bare propagator in momentum and
position space as
G0(ω) =√
sgn(−1) π−s,sgn(ω) , G0(t) =√
sgn(−1) Γ(π1−s,sgn) πs−1,sgn(t) .
(3.47)
As an ansatz for solving (3.44) with the left-hand side set to zero, we choose
G(t) = b π− 2q,sgn(t) , (3.48)
and so the IR Schwinger-Dyson equation in position space reads
−π1−s,sgn = σΩ g2 bq π− 2
q,sgn ∗ π 2(1−q)
q,sgn∗ π1−s,sgn . (3.49)
94
Using (3.46) to Fourier transform the multiplicative characters, we get the momentum
space Schwinger-Dyson equation
−πs,sgn = σΩ g2 bq Γ(π q−2
q,sgn)Γ(π 2−q
q,sgn) π 2−q
q,sgn ∗ π (q−2)
q,sgn∗ πs,sgn . (3.50)
The multiplicative characters all cancel, and we read off directly that
bq =−1
σΩ g2 Γ(π q−2q,sgn)Γ(π 2−q
q,sgn)
. (3.51)
The s dependence is seen to cancel out entirely. Insofar as the ansatz (3.48) describes
the true IR behavior, the dressed propagators of the various non-local theories with
different values of s all flow to the same function in the IR. In the case of bosonic
theories, however, where the potential can be unbounded from below, the physical
significance of the IR solution (3.48) is uncertain. Nonetheless, we explicitly include
bosonic theories under the scope of theories we are subjecting to formal perturbation
theory since bosonic theories are required in order to write down an adelic relation
connecting Archimedean and p-adic theories, as we will see in section (3.8.1).
In section 3.7.3 we will see that for a large subset of the p-adic theories, the full
Schwinger-Dyson equation (3.42) can be solved exactly, and in these cases one can
verify by inspection of the full solution that the two-point function flows to the IR
solution described by equations (3.48) and (3.51).
3.7.2 The zoo of theories
As we are working in Euclidean space, the reality of the action (3.2) dictates that
G(t) should be real too. The ansatz (3.48) can therefore only provide an accurate
description of the IR behavior if b is real, that is, if the right-hand side of (3.51) is
positive. In other words, the sign of Γ(π q−2q,sgn)Γ(π 2−q
q,sgn) must be opposite to that
of σΩ. For any choice of number field R or Qp and any choice of sign function in
95
K condition τ Γ(π q−2q,sgn)Γ(π 2−q
q,sgn) σΩ σψ explanation
R 1 − 2πqq−2
tan(πq
)1 1 O(N) bosonic
R −1 − 2πqq−2
cot(πq
)1 −1 O(N) fermionic
C 1 − π2q2
(q−2)2 1 1 O(N) bosonic
Qp p odd 1 − (p2/q−1)(p2−p2/q)
p(p−p2/q)2 1 1 O(N) bosonic
Qp p odd ε (p2/q+1)(p2+p2/q)
p(p−p2/q)2 −1 −1 Sp(N) fermionic
Qp p ≡ 1 mod 4 p 1/p −1 −1 Sp(N) fermionic
Qp p ≡ 1 mod 4 εp 1/p −1 −1 Sp(N) fermionic
Qp p ≡ 3 mod 4 p −1/p 1 −1 O(N) fermionic
Qp p ≡ 3 mod 4 εp −1/p 1 −1 O(N) fermionic
Q2 1 −5·41/q−4−161/q
2(41/q−2)2 1 1 O(N) bosonic
Q2 −1 −1/4 1 −1 O(N) fermionic
Q2 2 1/8 −1 −1 Sp(N) fermionic
Q2 −2 −1/8 1 −1 O(N) fermionic
Q2 3 −1/4 1 −1 O(N) fermionic
Q2 −3 (1+41/q)(4+41/q)
2(2+41/q)2 −1 −1 Sp(N) fermionic
Q2 6 −1/8 1 −1 O(N) fermionic
Q2 −6 1/8 −1 −1 Sp(N) fermionic
Table 3.4: Table of melonic theories, Archimedean or ultrametric. Shaded rows
indicate theories for which an exact solution to the Schwinger-Dyson
equation is given in subsection 3.7.3.
the action, this requirement together with the sign constraint (3.5) uniquely specifies
whether the theory must be bosonic or fermionic and O(N) and Sp(N) symmetric.
Generalizing the table of theories in [27] to all values of q, table 3.4 lists the melonic
theories with renormalization group flow from a free theory in the UV to a strongly
interacting fixed point in the IR. The choice of τ in table 3.4 amounts to a choice of
the sign function, as explained in section 1.6. Often, the value of τ involves ε, which
stands for any integer that is not a square modulo p.
96
3.7.3 Full solution to the Schwinger-Dyson equation for direction-
dependent theories
Recall that any p-adic number x can be represented by a series expansion
x = pv(x)
∞∑m=0
cm pm (3.52)
where cm ∈ 0, 1, ..., p− 1 and c0 6= 0.
It was shown in [27] for q = 4 that the Schwinger-Dyson equation (3.44) can be
solved exactly for p-adic theories when the sign function is direction-dependent, that
is, when the sign function sgn(x) depends not only on the norm |x| but also on the first
p-adic digit c0 in the expansion (3.52). These theories are indicated by shaded rows
in table 3.4, and the solutions of [27] generalize straightforwardly to higher values of
q.
For each of the theories, the solution can be written as
G(t) = G(|t|) sgn(t) , (3.53)
where G(|t|) is the unique real root of a q-th order polynomial.
For the direction-dependent theories with odd p, ie. those with a sign function
defined by τ = p or τ = εp, G(|t|) is given by the real solution to the equation
G(|t|) =√
sgn(−1)Γ(π1−s,sgn)|t|s−1
(1− g2
p|t|2 G(|t|)q
). (3.54)
For p = 2 and τ = −1 or τ = 3, G(|t|) is given by the real solution to
G(|t|) =√
sgn(−1)Γ(π1−s,sgn)|t|s−1
(1− g2
22|t|2 G(|t|)q
). (3.55)
97
And p = 2 and τ = ±2 or τ = ±6, G(|t|) is given by the real solution to
G(|t|) =√
sgn(−1)Γ(π1−s,sgn)|t|s−1
(1− g2
23|t|2 G(|t|)q
). (3.56)
In all cases, we choose√
sgn(−1) as in (3.40).
3.8 The four-point function
As shown in [52], the four-point function of the Klebanov-Tarnopolsky tensor model
with rank three fermions has the same structure as the SYK model [70], and this
result generalizes to values of q > 4 [55]. Working to sub-leading order in the melonic
limit, the four-point correlator can be decomposed as
ΩA1A2ΩA3A4
⟨T (ψA1(t1)ψA2(t2)ψA3(t3)ψA4(t4))
⟩=
N2(q−1)G(t12)G(t34) +N q−1
(Γ(t1, t2, t3, t4) + σψΓ(t1, t2, t4, t3)
)+O(N q−2) ,
(3.57)
where Γ stands for a sum of so-called ladder Feynman diagrams that can be expanded
as Γ =∑
n Γn according to the number n of sets of q−2 rungs in the ladder diagrams.
Schematically, for q = 8:
Γ(t1, t2, t3, t4) =
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
+
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
+
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
+ ...
(3.58)
Here black lines represent propagators while gray lines stand for contraction through
the matrix ΩAB. But for a more exact understanding of the four-point function, we
98
must also consider the threads running within each interaction vertex and endow each
line with an orientation. For the index contraction in (3.57), the arrows of the first
ladder diagram are oriented as
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
. (3.59)
By flipping two arrows, each corresponding to q−1 interchanges of indices in a matrix
Ωab or Ωab, we get an oriented diagram,
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
. (3.60)
Since an even number of arrow flips were performed, there is no σΩ dependence so
that
Γ0(t1, t2, t3, t4) = σψG(t1 − t3)G(t2 − t4) . (3.61)
The absence of a factor of σΩ in (3.61) owes directly to the choice of contraction with
matrices ΩA1A2ΩA3A4 rather than say ΩA1A2ΩA4A3 in (3.57), and so is ultimately the
result of a convention. The important question to ask is whether the insertion of
each new set of rungs in a ladder diagram leads to a factor of σΩ. Consider therefore,
as in figure 3.12, the diagram obtained by appending an extra set of rungs to some
oriented ladder diagram.
The propagator that forms part of the top rail of the ladder immediately to the
left of the newly appended set of rungs may be incident to any vertex i of the top
interaction vertex. The propagator that forms part of the top rail immediately to the
right of the appended sets of rungs may be incident to any other vertex j 6= i of the
99
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A1
A2
A3
A4
A3
A4
...
...
A1
A2
i j
σ(i) σ(j)
k
l
σ(k)
σ(l)
1
Figure 3.12: Insertion of an extra set of rungs in an oriented ladder. Not all
internal threads in the interactions nor all rungs have been drawn
so as to not clutter up the figure.
topmost interaction vertex. But for a given choice of i and j, in order to get a ladder
diagram that is not suppressed in the melonic limit, the two propagators of the bottom
rail immediately to the left and right of the bottom interaction vertex must be given,
respectively, by σ(i) and σ(j) for some σ ∈ G. And similarly the appended rungs
must connect vertex l of the top interaction vertex to vertex σ(l) of the bottom one for
all l 6= i, j. These facts follow immediately from the above discussion of the two-point
function once we note that 1) the threads of the propagators incident to vertices i
and σ(i) all partake of the same index loops; and 2) from the point of view of index
contraction we may look upon the propagators incident to vertices i and σ(i), the
arrows running to and between A3 and A4, and the two appended interaction vertices
and the rungs running between them as all forming part of a melonic insertion into
one single propagator. Since there are q choices of vertex i, (q − 1) choices of vertex
j, and |G| choices of permutation σ, the total number of melonic Feynman diagrams
that contribute to the ladder diagram in question is q(q − 1)|G|.
As in our consideration of the two-point function, we can assume that the orien-
100
tation of an arrow between any two vertices i′ and j′ of the top interaction vertex is
the same as the orientation of the arrow between σ(i′) and σ(j′) since automorphisms
induce an even number of arrow flips. As to the q−2 arrows that make up the rungs,
these all have the same orientation prior to flipping any arrows, as one can find by
computing the four-point function via functional differentiation. As in figure 3.12 we
will take these arrows to point downwards rather than upwards, but this is an arbi-
trary choice that does not affect the parity of arrow flips needed to make all index
loops have a uniform orientation of arrows.
To determine whether appending an extra set of rungs gives rise to an overall
factor of σΩ, we need to consider all the index loops involved. These fall into four
types:
1. The index loops running from i through one of the rungs of the ladder and
through σ(i) without passing through A3 and A4, as illustrated in blue on
figure 3.12. Because there are q − 2 such loops and they all require the same
number of arrow flips to obtain uniform orientation, no net factor of σΩ is
introduced on account of these index loops.
2. The index loops running from j through A3 and A4 and back to j through one
of the rungs in the ladder, illustrated in red on figure 3.12. Again these q − 2
loops are even in number and require the same number of arrow flips for uniform
orientation, so again no net factor of σΩ is introduced.
3. The index loops that run between the two interaction vertices and consist of
four threads each, two in two rungs and two within the two interaction vertices.
There are (q − 2)(q − 3)/2 such loops, an example of which is illustrated in
yellow on figure 3.12. Each of these index loops consists of two threads with
one orientation and two index loops with the opposite orientation, so once again
no net factor of σΩ is introduced in bringing about uniform orientation.
101
4. Lastly, there is the index loop that runs between i and j and σ(i) and σ(j),
illustrated in green on figure 3.12. Since we are assuming that the index loops
of the original ladder diagram, prior to insertion of an extra set of rungs, each
had a uniform orientation of arrows, two arrows must be flipped in order to give
this index loop a uniform orientation (the arrow between σ(i) and σ(j) and the
arrow between σ(j) and A4 on figure 3.12) and again we do not pick up any net
factor of σΩ.
In summary, there are no relative sign differences in ladder diagrams between SO(N)
and Sp(N) symmetric tensor models.
Having worked out the subtleties relating to Sp(N) symmetry and the automor-
phism group of the interaction vertex, we are ready to write down the recursive
relation describing ladder diagrams:
Γn+1(t1, t2, t3, t4) =
∫dt dt′K(t1, t2, t, t
′)Γn(t, t′, t3, t4) (3.62)
where the integration kernel is given as
K(t1, t2, t, t′) = σψq(q − 1)|G|
(g
|G|
)2
N(q−1)(q−2)
2 G(t1 − t)G(t2 − t′)G(t− t′)q−2
= σψ(q − 1)g2G(t1 − t)G(t2 − t′)G(t− t′)q−2 .
(3.63)
In the IR, we can plug in the expression for the dressed propagator derived in
section 3.7.1 to obtain
K(t1, t2, t3, t4) =− σψσΩ(q − 1)
Γ(π q−2q,sgn)Γ(π 2−q
q,sgn)
π− 2q,sgn(t13) π− 2
q,sgn(t24) π 2(2−q)
q,sgn
(t34) .
(3.64)
Following [71] and [52] and defining
v(t1, t2) ≡ πh− 2q,sgn(t12), (3.65)
102
the integral eigenvalue-equation to solve in order to find the scaling dimensions of
two-particle operators is given by
v(t1, t2) =1
g(h, q)
∫dt3 dt4K(t1, t2, t3, t4) v(t3, t4) . (3.66)
Changing variables to
u ≡ t13 , v ≡ t42 , (3.67)
makes it manifest that the integral in the eigenvalue equation is a convolution since
∫dt3 dt4 π− 2
q,sgn(t13)π− 2
q,sgn(t24) π
h+2(1−q)q−h,sgn
(t34) =
sgn(−1)
∫du dv π− 2
q,sgn(u) π− 2
q,sgn(v) π
h+2(1−q)q
,sgn(t12 − u− v) .
(3.68)
Invoking the sign constraint (3.5), the eigenvalue equation can therefore be written
as
πh− 2q,sgn = − (q − 1)
g(h, q)Γ(π q−2q,sgn)Γ(π 2−q
q,sgn)
π− 2q,sgn ∗ π− 2
q,sgn ∗ πh+
2(1−q)q
,sgn. (3.69)
Fourier-transforming this equation using (3.46), the multiplicative characters cancel,
as do two of the gamma functions, and we find that
g(h, q) = −(q − 1)Γ(π q−2
q,sgn)Γ(πh+ 2−q
q,sgn)
Γ(π 2−qq,sgn)Γ(πh+ q−2
q,sgn)
. (3.70)
This formula is valid for real as well as p-adic numbers, and for an action (3.2) with
a kinetic term with any sign function.
On the real numbers, selecting the usual sign function in the action (3.2), the
above equation reproduces the fermionic result of the SYK and melonic tensor models,
equations (3.73) in [71] and (6.8) in [55]. Selecting the trivial sign on the reals, the
above equation reproduces the bosonic result, equation (4.14) in [52] with d = 1.
103
On the p-adic numbers, there are multiple inequivalent sign functions, each of
which can be labeled by a p-adic number τ as explained in section 1.6. For the sign
functions characterized by τ = p or τ = εp, g(h, q) reduces to 1− q. For τ = ε, g(h, q)
is a non-constant function in h, but the equation g(h, q) = 1 never has a solution for
the p-adic theories.
3.8.1 Adelic product formula for the integral eigenvalues
In [30] it is demonstrated how, by invoking the functional equations of suitably cho-
sen Dirichlet L-functions, one can derive an adelic product formula for the signed
Gel’fand-Graev gamma functions. By selecting a fixed rational number τ , one picks
out a sign function for each of the number fields R and Qp. For each of these fields
with associated sign function sgnτ (x) there is a signed character πs,sgn(x) and an asso-
ciated gamma function Γ(πs,sgn). Taking the product over all these gamma functions
for any fixed complex number s, one gets the usual sign function of τ :
∏K=R,Qp∀p
Γ(πs,sgn) =
1 for τ > 0
−1 for τ < 0 .
(3.71)
Note that for any choice of τ there will be many fields K for which sgnτ (x) is
the trivial sign character, ie. unity, so that πs,sgn reduces to πs. For example, this
happens for K = R when τ is positive.
As an immediate consequence of (3.71), we find that for fixed rational τ , complex
h, and even number q, the integral eigenvalues g(h, q) satisfy the adelic product
formula
∏K=R,Qp∀p
g(h, q)
1− q= 1 . (3.72)
For most choices of τ , this product mixes bosonic and fermionic, and O(N) and Sp(N)
104
symmetric theories.
3.9 Outlook
Given the relative uniqueness and simplicity of melonic theories with quartic and sex-
tic vertices, it is surprising that melonic theories proliferate and diversify for larger
orders q of the interaction vertex. Already at q = 8, there are six different types of
vertices, with up to Z32 symmetry. The situation can be compared with matrix mod-
els, where if we restrict to quartic vertices, the most commonly studied interactions
are tr Φ4 and (tr Φ2)2. The first of these has Z4 symmetry, and the second has Z32
symmetry. If we restrict to only one of these two vertices, then we get remarkably
different behavior in the large N limit: tr Φ4 leads to a planar limit, while (tr Φ2)2
leads to bubble diagrams. Mixing the two gives some interesting modifications of
scaling exponents of the pure tr Φ4 theory [72, 73, 74, 75]. In contrast, for melonic
theories the treatment of the two-point and four-point functions proceeds almost iden-
tically for the myriad theories we can construct. The one salient difference among
the theories considered here is that the effective coupling constant that enters into
the self-consistent treatment of two- and four-point functions includes the inverse half
power of the order of the symmetry group of the interaction vertex. An interesting
question for future work is to see whether combining different interaction vertices
could significantly alter the analysis, for example through some cancellations or mod-
ified infrared scaling. Another interesting question for future work is to examine the
extent to which the operator counting of [67] generalizes to higher melonic theories.
The construction of q-fold interaction vertices in this chapter is, by necessity,
somewhat detailed, amounting to a coloring of the complete graph of q vertices with
q − 1 colors so that all edges meeting at a vertex have different colors. The combi-
natorial problem of counting all such colorings is formidable, and it is related to the
105
problem of one-factorizations of the complete graph, for which results are available
only for modest values of q. However, as demonstrated in section 3.3 one can con-
firm the existence of colorings with all possible symmetries groups, namely Zn2 with n
bounded above by v if q = 2v and by v−1 if q = u2v with odd u greater than 1. While
this construction of interaction vertices is rooted in the natural generalizations of the
Klebanov-Tarnopolsky model [52], we cannot claim to have exhausted all possible
generalizations with melonic limits. Here is a useful way to frame the question: If we
restrict the field content to a single real field ψa0a1...aq−2 with q − 1 N -valued indices
all of which must be contracted amongst each other, is the most general interaction
vertex with a melonic limit a linear combination of the interaction vertices that can
be described in terms of colorings of a complete graph of q vertices?
106
4
Holographic Duals of Nontrivial Characters in
p-adic AdS/CFT
This chapter is based on [4] coauthored with Steven S. Gubser and Christian Jepsen.
The paper was dedicated to the memory of Peter G. O. Freund, whose discoveries
are foundational to much of this thesis. We thank M. Heydeman for substantial
input in early phases of this work, in particular for identifying the U(1) gauge field
configurations as related to Paley graphs.
Of Peter Freund’s many ideas in theoretical physics, it was clear that p-adic
string theory [7, 6, 9], also studied by Volovich in [8], was one of his favorites. The
strangeness of the p-adic numbers, the unexpectedness of Freund and Olson’s idea
to replace the reals with the p-adics on the boundary of the open string worldsheet,
and the simplicity of the resulting scattering amplitudes, all contribute to the charm
of the subject. The deep question of why the reciprocal of the Veneziano amplitude
factorizes into a product of its p-adic relatives remains mysterious. It causes one to
wonder whether, even now, we have fully plumbed the depths of perturbative string
dynamics.
In p-adic AdS/CFT [10, 11], one looks at some of the surprising features of p-
adic string theory in a new light. Recall that (see section 1.8) that in an important
107
precursor to p-adic AdS/CFT [42], Zabrodin defined a free massless scalar action over
the Bruhat-Tits tree Tp, whose boundary is the projective line P1(Qp). Integrating out
the bulk scalar was shown to result in the correlators that Freund and Olson needed to
obtain the analog of the Veneziano amplitude for p-adic strings. An updated version
of this integrating out process is the computation of holographic Green’s functions in
p-adic AdS/CFT, with the Bruhat-Tits tree playing the role of the bulk geometry.1
Free massless scalars in the bulk are just the beginning: In p-adic AdS/CFT one
wants to consider mass terms and their relation to boundary conformal dimension,
and also non-linear bulk dynamics dual to n-point boundary Green’s functions with
more interesting structure than can be obtained from Wick contractions.
In [42] as well as later works, attention focused on scalar fields in the bulk ge-
ometry, dual to scalar operators on the boundary. Likewise on the field theory side,
the study of the operator product expansion [41] focused on scalar operators. Even
the gravitational dynamics of [25] is a scalar theory, because the bulk variable is edge
length on Tp. The dual boundary operator is found to be a scalar whose scaling di-
mension equals the dimension of the boundary as a vector space over Qp. Boundary
theory correlators involving sign characters were considered in [9, 29, 28, 30, 31] in
connection with p-adic string amplitudes and supersymmetry. Work on fermionic
p-adic field theories continued in [32] with a study of a relative of the Gross-Neveu
model, and the recent work [27] investigates both fermionic and bosonic melonic the-
ories over Qp. General comments on higher spin can be found in [11]. But no bulk
dual of non-scalar operators was suggested in any of these works. Another precursor
of p-adic AdS/CFT is the stochastic cellular model eternal inflation studied in [77];
but there too the treatment was restricted to scalar operators on the boundary (best
understood as an analog of future infinity in de Sitter space). The aim of this chapter
is to present first steps in a larger project of finding bulk duals to known non-scalar
1See however [76] for a recent study of holography involving p-adic numbers in the context of acontinuous bulk geometry.
108
boundary theories on P1(Qp).
As suggested in [42, 30, 27], we consider boundary Green’s functions of the form
G(x) =C sgnx
|x|2∆p
, (4.1)
where C is a constant, | · | denotes the p-adic norm, and sgnx is a multiplicative sign
character on Qp, as defined in section 1.6. We want to inquire, when and how can we
extract a Green’s function like (4.1) from a bulk construction?
To further motivate the study of Green’s function of the type (4.1), consider the
corresponding Fourier space expression
G(k) = C(sgn k)|k|2∆−1p . (4.2)
For comparison, fermionic correlators in ordinary AdSd+1/CFTd take the form G(k) =
Cγµkµ|k|2∆−d−1 (where now | · | is the norm on Rd instead of the p-adic norm); see for
example [78]. Our assertion is that the factor sgn k is in rough analogy to the the factor
γµkµ/|k| that appears in the Archimedean case. The first point of similarity is that
(sgn k)2 = 1, just as (γµkµ/|k|)2 = 1. Furthermore, in analogy to the transformations
of γµkµ/|k| under rotations, sgn k is a representation of the rotation group on Qp,
which comes from multiplication by p-adic numbers with norm 1. This is an abelian
group, so the only finite dimensional irreducible representations are one-dimensional.
There are certainly more complicated representations than just sign characters, so
this chapter should be considered only a first foray into the potentially large subject
of p-adic AdS/CFT with spin.
In section 1.6 we claimed that for odd p, there were four distinct sign characters
on Q×p . It will be useful to us to have explicit formulas for each of them.2 First
2Sign characters over Q2 are also well known, but their relation to holographic constructions ismore intricate.
109
express any x ∈ Q×p as
x = pvx(x0 + x1p+ x2p2 + . . . ) , (4.3)
where vx ∈ Z, x0 ∈ F×p , and all other xi ∈ Fp. Here Fp is the finite field of p
elements, namely Z/pZ, which we identify with the set 0, 1, 2, . . . , p− 1. And F×p is
the non-zero elements of Fp, which form a multiplicative group. There are two sign
characters on F×p : the trivial one which maps all elements to 1, and the quadratic
residue character n→ (n|p) where (n|p), also denoted(np
), is the Legendre symbol.
It is defined so that (n|p) = 1 if n is a square in F×p and −1 otherwise.3 On Q×p , there
are four choices of sign character:
1. We can map all x ∈ Q×p to 1. This is the trivial character.
2. We can map x→ (x0|p).
3. We can map x→ (−1)vx . This means we assign p itself a sign of −1.
4. We can map x→ (−1)vx(x0|p).
This list exhausts all the sign characters on Q×p . Until we get to section 4.6, we are
going to focus exclusively on the second case: That is, we will hereafter define
sgnx =
(x0
p
). (4.4)
We will narrow our field of inquiry in two other ways. First, we will restrict attention
to nearest neighbor interactions in the bulk, expressible in terms of a classical action
either on Tp or on its line graph L(Tp). This is analogous to restricting to the lowest
non-trivial order in derivatives in Archimedean anti-de Sitter space. Second, the
boundary for us will always be Qp rather than an extension of Qp.
3The definition of the Legendre symbol is traditionally extended to all of Fp by defining (0|p) = 0,and to all integers by first reducing them modulo p. Then (n|p) = 0 or 1 precisely when n is aquadratic residue modulo p.
110
The organization of the rest of the chapter is as follows. In section 4.1, we de-
scribe the nearest neighbor actions on Tp and L(Tp) that we will need, both for bosons
and fermions. In order to obtain a factor of sgnx in the final holographic two-point
functions, we need to introduce a non-dynamical U(1) gauge field. Indeed, the factor
of sgn x in (4.1) can be thought of as a Wilson line obtained by integrating the U(1)
gauge field along the shortest path on L(Tp) between the boundary points 0 and x.
The particular gauge field configurations that we need are described in section 4.2.
The main technical steps in extracting the holographic two-point functions are out-
lined in section 4.3, which deals with bulk-to-boundary propagators, and section 4.4,
which recounts the holographic prescription. We detour briefly in section 4.5 into an
account of dynamical gauge fields in the bulk, and then in section 4.6 we summarize
how to modify the gauge fields so as to get any sign character one wants in the final
two-point function (4.1) (for p 6= 2).
4.1 Nearest neighbor actions
In this section we will formulate nearest neighbor actions, first for bosons in sec-
tion 4.1.1 and then for fermions in section 4.1.2. A key ingredient will be a non-
dynamical U(1) gauge field.
4.1.1 Bosonic actions
Starting with a complex-valued function φa on vertices a of a directed graph, we can
define the gradient of φ as the following complex-valued function on edges of the
graph:4
dφe = φt(e) − φs(e) . (4.5)
4Nothing so far privileges complex numbers: φa and dφe could be valued in any linear space V ,and then ωe as used in (4.6)-(4.7) would need to be valued in linear functions on V . For the mostpart we do not need such a general viewpoint.
111
a
b
φa
φb
e eiθe Dφe = eiθeφb − φa
Figure 4.1: The gauge covariant derivative Dφe on a small section of a directed
graph.
Here e is an oriented edge with starting point s(e) and terminus t(e).
If we start from a function ωe on directed edges, then we define
dTωa =∑t(e)=a
ωe −∑s(e)=a
ωe , (4.6)
so that ∑e
ωedφe =∑a
(dTωa)φa , (4.7)
possibly up to issues of boundary terms and/or convergence. The equality (4.7)
is an analog of integration by parts. It can be useful to think of d = dea as a
rectangular matrix with one edge-valued index e and one vertex-valued index a. Then,
for example, dφe =∑
a deaφa, and dTωa =∑
e deaωe.
A crucial ingredient in our constructions is a non-dynamical U(1) gauge field.
Because the graph is discrete, instead of a gauge-covariant derivative Dµ = ∂µ + iAµ,
we are going to consider modifying (4.5) to
Dφe = eiθeφt(e) − φs(e) , (4.8)
where θe is essentially∫Aµdx
µ across the edge e. See figure 4.1. Upon gauge trans-
formations
φa → eiλaφa θe → θe − dλe , (4.9)
112
we see that
Dφe → eiλs(e)Dφe . (4.10)
Evidently, in the absence of loops, we can use (4.9) with λe = −θe to remove the
phase from (4.8), so that D = d. On L(Tp), there are loops, so non-trivial gauge field
configurations exist.
We now consider the action
Sφ =∑e
|Dφe|2 +∑a
m2|φa|2 . (4.11)
Here and below, | · | acting on a target space field is the norm on C. Varying (4.11)
with respect to φ∗a gives
D†Dφa +m2φa = 0 , (4.12)
where D† is the adjoint of D. (Explicitly, we can write D = Dea as a rectangular
matrix, and then D∗, DT , and D† all have obvious definitions.) A helpful result for
calculations to come is
D†Dφa = oaφa −∑t(e)=a
e−iθeφs(e) −∑s(e)=a
eiθeφt(e) , (4.13)
where oa is the number of edges incident upon a. (On Tp, oa = p+ 1 for all vertices,
while on L(Tp), oa = 2p for all vertices.)
The definition (4.8) might seem asymmetrical, and one might prefer instead eiθe/2φt(e)−
e−iθe/2φs(e) on the right hand side. But for purposes of forming the action (4.11), the
overall phase of Dφe doesn’t matter because only |Dφe|2 enters. In other words, there
is U(1) gauge freedom on edges which we fix in (4.8) by locking the phase of Dφe to
φs(e).
113
4.1.2 Fermionic actions
On a graph, the natural notion of a Dirac operator has to do with the exterior deriva-
tive. See for example [79], where the Dirac operator on a graph D is essentially the
signed adjacency matrix on the clique graph of G. We will consider a simplifica-
tion of this general development, in which the only operator we need is the gradi-
ent, rendered gauge covariant as in the previous section. Explicitly, we introduce a
Grassmann-complex-valued function ψa on vertices of a directed graph, and another
such function χe on edges. We define
Dψe = eiθeψt(e) − ψs(e) (4.14)
and introduce the action
Sψ =∑e
[iχ∗eDψe + iχeD∗ψ∗e +mχ∗eχe]−
∑a
Mψ∗aψa . (4.15)
The kinetic terms in (4.15) are constructed in the spirit of b∂c lagrangians, where b
is replaced by an edge field χe and c is replaced by a vertex field ψa. The action is
real once we assume that conjugation exchanges the order of factors. We need ψa and
χe to be complex in order to make the mass terms possible. It appears that the two
mass coefficients are independently meaningful, but in fact there is a global scaling
symmetry ψa → λψa and χe → (λ∗)−1χe, where λ ∈ C is a constant, which preserves
the kinetic terms while rescaling m→ |λ|2m and M → |λ|−2M .
The action (4.15) is invariant under the gauge transformation
ψa → eiλaψa χe → eiλs(e)χe θe → θe − dλe , (4.16)
114
and the equations of motion are
iDψe +mχe = 0 iD†χa +Mψa = 0 (4.17)
(and the complex conjugates of these equations).5 From the two equations (4.17) it
follows that
d†dψa +mMψa = 0 . (4.18)
The equivalence of (4.18) and (4.12) is comparable to the way the massive Dirac
equation implies the massive Klein-Gordon equation.
4.2 The background geometries
The non-dynamical U(1) gauge fields on L(Tp) that we are going to study encode the
Legendre symbol (α|p). Consider first the case p ≡ 1 mod 4. Label the vertices of the
complete graph Kp with elements of Fp. Pick a directed structure on Kp, and define
a map
e→ α(e) = t(e)− s(e) (4.19)
from directed edges to F×p . Set eiθe = (α(e)|p) on each edge. Because (α|p) is an
even function of α ∈ F×p , the choice of eiθe doesn’t depend on the directed structure
we picked. Because eiθe is always real, the operator D†D also doesn’t depend on the
directed structure.
Now consider the case p ≡ 3 mod 4. Again label the vertices of Kp with elements
of Fp. Introduce a directed structure on Kp such that an edge runs from a vertex a
to another vertex b iff b − a is a square in F×p . This prescription uniquely specifies
the direction of every edge in Kp because for any α ∈ F×p , either (α|p) = 1 or else
5As in the scalar case, the overall phase of Dψe doesn’t matter because in (4.15) we form theproduct iχ∗eDψe, and we can adjust the phase of χe to keep the overall prefactor equal to i.
115
(−α|p) = 1, due to the fact that (α|p) is an odd function of p. Set eiθe = i on all
edges.
We will refer to the directed structures and gauge fields on Kp as Paley construc-
tions, since for p ≡ 1 mod 4 the edges with eiθe = 1 form a Paley graph (without
reference to the directed structure), while for p ≡ 3 mod 4, the directed structure that
we picked forms a Paley digraph.
There are p+1 edges incident upon each vertex A of Tp, of which one edge is below
located below A (that is, one edge lies on the path from A to the boundary point at
infinity), while p edges are located above A. The vertices in L(Tp) corresponding to
the above-lying edges we think of as forming a copy of Kp, and each of these vertices
is also connected to the vertex corresponding to the edge below A. We parametrize
A using a pair (xA, zA) as in section 1.8. Recall that xA ∈ Qp and zA is an integer
power of p, with (xA, zA) identified with (x′A, z′A) iff zA = z′A and |xA − x′A|p ≤ |zA|p.
We will parametrize elements of L(Tp) by using the same coordinates (xA, zA) to label
the vertex of L(Tp) immediately below A; usually we will write instead (xa, za) since
we use lowercase letters to label vertices of L(Tp). To fix a directed structure and
non-dynamical gauge field on L(Tp), we adopt the same Paley construction on each
Kp, and the rest of the edges are directed downward, from (x, z) to (x, z/p), with
eiθe = 1 for p ≡ 1 mod 4 and eiθe = i for p ≡ 3 mod 4. We will refer to edges inside a
copy of Kp as horizontal, and the others as vertical.
If we use a → b to denote a directed edge, then for p ≡ 1 mod 4, our choice of
gauge field is
vertical edges: θ(x,z)→(x, zp)= 0
horizontal edges: θ(x,z)→(x+α zp,z) =
π
2
[1−
(α
p
)]for α ∈ F×p .
(4.20)
In the second line of (4.20), we bear in mind that the directed edge (x, z)→(x+ α z
p, z)
116
Figure 4.2: Left: L(T5) in blue and red with T5 shown in dashed green. The blue
edges have θe = 0, while the red edges have θe = π. The pentagram
figures are the Paley constructions, and edges within them are called
horizontal. Their orientation doesn’t matter. The vertical edges are
the ones connecting each pentagram with a vertex of the pentagram
below it, and our convention is for all of them to be directed down-
ward.
Right: L(T3) in black, with T3 shown in dashed green. All the edges
have θe = π/2.
exists only for half the elements of F×p ; but which half doesn’t matter. For p ≡ 3 mod 4,
our choice of gauge field is
vertical edges: θ(x,z)→(x, zp)=π
2
horizontal edges: θ(x,z)→(x+α zp,z) =
π
2for α ∈ F×p with
(αp
)= 1 .
(4.21)
See figure 4.2 for a depiction of small subgraphs of L(Tp) showing also the choice of
gauge fields (4.20) and (4.21) for p = 5 and 3, respectively.
The assignments of eiθe in (4.20) and (4.21) are preserved under the maps (x, z)→
(rx+b, z/|r|p) for all b ∈ Qp and r ∈ (Q×p )2. This is fortunate because the correspond-
ing boundary maps, x→ rx+b, applied to two distinct points x1 and x2 in Qp, are the
ones that preserve the desired two-point function, G(x1, x2) = sgn(x1−x2)/|x1−x2|2∆p ,
up to some power of the scale factor |r|p.
117
4.3 Bulk-to-boundary propagators
In order to compute holographic two-point functions, a key ingredient is the bulk-to-
boundary propagator in momentum space. In section 4.3.1 we review the calculation
of this propagator in the case of complex scalars on Tp. Then in section 4.3.2 we work
it out for scalars and fermions on L(Tp).
4.3.1 Scalars on Tp
Consider complex scalars on Tp with action
S =∑E
|dφE|2 +∑A
m2|φA|2 , (4.22)
where vertices are labeled A = (xA, zA) and edges are labeled E. For brevity, let’s
write zA = pv. Then, as recounted in [10], a useful solution of the equations of motion
following from (4.22) is
φA(k) = fvγp(kzA)χ(−kxA) where fv = |zA|1−∆p +Q|k|2∆−1
p |zA|∆p (4.23)
where k ∈ Q×p and
Q = −p1−2∆ . (4.24)
Here χ(ξ) is the additive character on Qp, given explicitly by χ(ξ) = e2πiξ where
ξ is the fractional part of ξ ∈ Qp. Here we have used the p-adic Gaussian γp as
defined in section 1.4. The dimension ∆ is related to the mass by
m2 = − 1
ζp(∆− 1)ζp(−∆). (4.25)
118
The solution φA(k) in (4.23) can be thought of as a bulk to boundary propagator
because it is the disturbance of φA in the bulk that corresponds to deforming the
boundary field theory by a term∫dxχ(kx)O(x), where O is the operator dual to
φ.6 From the form of φA(k) given in (4.23) we can pick out the k-dependence of the
Fourier space holographic two-point function for O: G(k) ∝ |k|2∆−1.
4.3.2 Scalars and fermions on the line graph
We would now like to find solutions analogous to (4.23) on L(Tp) with the directed
structures and gauge fields as outlined in section 4.2. The invariance of the back-
ground geometry under translations xa → xa + b indicates that we should be able to
require that fields should depend on xa through a factor χ(−kxa). We immediately
encounter the need to multiply in a factor of γp(kza), because by itself, χ(−kxa) is
not single valued on L(Tp), whereas γp(kza)χ(−kxa) is (and the same logic dictated
that (4.23) must include a factor of γp(kzA)). In short, we are lead to essentially the
same ansatz as (4.23):
φa(k) = uvfvγp(kza)χ(−kxa) where fv = |za|1−∆p +Q|k|2∆−1
p |za|∆p (4.26)
and we have written za = pv. For later convenience, we have introduced the prefactor
uv where
u =
1 for p ≡ 1 mod 4
i for p ≡ 3 mod 4 .
(4.27)
The coefficient Q in (4.26) may depend on k, but we don’t expect it to depend on |k|p
since the explicit factor of |k|2∆−1p already is the dependence we expect for the k-th
Fourier mode of a holographic two-point function with dimension ∆. The aims of
6The field φa, the operator O and the deformation of the conformal field theory action, shouldin the end be real. This can be accomplished by always considering superpositions of Fourier modeswith equal and opposite k.
119
the following calculation are to verify that the ansatz (4.26) does solve (4.12) and to
determine ∆ and Q. The strategy is to plug (4.26) into (4.12) and extract a difference
equation for fv. We can handle the fermionic case by replacing φa by ψa in (4.26)
and plugging into (4.18). In the fermionic case, we assume m 6= 0 so that χe can be
determined by the first equation in (4.17).
The factor γp(kza) means that the equation of motion (4.12) is trivially satisfied
for vertices a such that vk + v < −1, where vk ∈ Z is the valuation of k: that is
k = pvk(k0 + k1p+ k2p2 + . . . ) with k0 ∈ F×p . (4.28)
Let’s first show that the equation of motion is also trivially satisfied when v = −vk−1.
In this case, the only non-zero terms in the equation of motion (4.12) are the ones
corresponding to zb = pza, i.e. in the Paley construction above the point a. Explicitly,
the equation of motion reads
−f−vk∑α∈Fp
χ(−k[xa + zaα]) = 0 , (4.29)
where we are using the fact that the p vertices in L(Tp) above (xa, za) are (xa+zaα, pza)
where α runs over Fp. Recalling that v = −vk − 1, we see that the sum in (4.29) is
proportional to ∑α∈Fp
χ(−k0α/p) = 0 . (4.30)
So (4.29) is indeed satisfied trivially and gives us no information about fv.
Let’s move on to the case v > −vk. The factor χ(−kxb) now has the same value
for all vertices b neighboring a, as well as for b = a. Also, γp(kzb) = 1 at all these
vertices. Thus we may discard the factor γp(kza)χ(−kxa) from the ansatz (4.26) and
120
work directly with φa = fv. By plugging in to (4.12) we find
(2p+m2)fv − pfv+1 − fv−1 = 0 . (4.31)
The form of (4.31) is the same for p ≡ 1 mod 4 and p ≡ 3 mod 4 because of the
overall prefactor uv in (4.26). One can view this factor as a change of gauge in
the p ≡ 3 mod 4 case which removes the factors of i from the covariant derivatives
along vertical edges while leaving them unchanged within the Paley constructions.
The only property of the Paley constructions we need in order to get (4.31) is that
contributions to D†Dφa from the p − 1 vertices connected to a by a horizontal edge
cancel out. The second order difference equation (4.31) is solved by fv = p−∆v = |za|∆p
and fv = p(∆−1)v = |za|1−∆p where
2p+m2 = p1−∆ + p∆ . (4.32)
We assume that ∆ is real, and the standard prescription is to choose it as the larger
of the two roots of (4.32), so that ∆ > 1/2. Note that 2(√p − p) < m2 < 0 when
∆ ∈ (1/2, logp(p+√p(p− 1))), and it is positive otherwise.7
In the case of fermions, the discussion up to this point proceeds unchanged, except
that m2 is replaced by mM .
To summarize progress so far: We have shown that the ansatz (4.26) trivially
satisfies the equations of motion for v < −vk, while for v > −vk we have shown that
it is consistent with the equations of motion provided we impose the mass-dimension
relation (4.32). But we have no information yet about Q. This information comes
from a boundary condition at v = −vk, and it turns out that it encodes the sign
character that we need in order to obtain two-point functions of the desired form
7Note that ∆ = 12 + is gives m2 real but violating the lower bound m2 > 2(
√p − p). A similar
result was already noted in [11] for scalars on Tp. It is tempting to think that these complex valuesof ∆ correspond to unstable actions, but they may nevertheless have some interesting role to play.
121
(4.4). The equation of motion (4.12) for v = −vk reads
(2p+m2)f−vkχ(−kxa)− f−vk+1
∑α∈Fp
χ(−k[xa + zaα])
− uf−vk∑α∈F×p
(α
p
)χ
(−k[xa + za
α
p
])= 0 .
(4.33)
There is no f−vk−1 term in (4.33) because the factor of γp(kzb) vanishes when zb =
p−vk−1, so (4.33) is only first order in differences rather than second order. Hence
it can indeed be thought of as a boundary condition for the second order equation
(4.31). In the last term of (4.33), we are using the fact that the p− 1 vertices in the
same Paley construction as xa are(xa + za
αp, za
). This last term is proportional to
the Gauss sum: ∑α∈Fp
(α
p
)χ
(−k0α
p
)=
√p
u
(k0
p
). (4.34)
(In (4.34), the α = 0 term in the sum vanishes, so including it is optional. The form
(4.34) makes it clear that we are taking a Fourier transform of the Legendre symbol
over Fp.) Simplifying, and using (4.32), we obtain
[p1−∆ + p∆ −√p
(k0
p
)]f−vk − pf−vk+1 = 0 . (4.35)
Plugging the ansatz for fv in (4.26) into (4.35), one arrives at
[p1−∆ + p∆ −√p
(k0
p
)](1 +Q)− p(p∆−1 + p−∆Q) = 0 , (4.36)
which reduces to
Q = p12−∆
(k0
p
)= p
12−∆ sgn k . (4.37)
122
4.4 Two-point functions
With the bulk-to-boundary propagators in hand, we now turn to the computation of
the holographic two-point functions, first in section 4.4.1 for real scalars on Tp and
then in section 4.4.2 for complex scalars on L(Tp), and finally in section 4.4.3 for
fermions on L(Tp).
4.4.1 Scalars on Tp
As a warmup, consider a complex scalar φA on Tp, as in (4.22)-(4.25). Implement a
cutoff by fixing the values of φA for all vertices (xA, zA) with |zA|p = |ε|p, where ε = pvε
and vε is an integer. Let Σε denote the set of vertices with |zA|p > |ε|p, together with
the edges with at least one vertex having |zA|p > |ε|p. Let ∂Σε be the edges with only
one vertex in Σε. We orient edges downward (away from the Qp boundary), so that
when E ∈ ∂Σε, t(E) ∈ Σε and s(E) 6∈ Σε. The vertices in Σε are allowed to fluctuate,
while vertices with |zA|p < |ε|p are ignored. The cutoff action is
Sε =∑E∈Σε
|dφE|2 +∑A∈Σε
m2|φA|2 . (4.38)
We now need an improvement of the partial integration formula (4.7) to include
boundary terms:8
∑E∈Σε
ωEdφE =∑A∈Σε
(dTωA)φA −∑E∈∂Σε
ωEφs(E) . (4.39)
8If ∂Σε included edges for which s(E) ∈ Σε while t(E) 6∈ Σε, then in place of (4.39) we wouldneed
∑E∈Σε
ωEdφE =∑A∈Σε
(dTωA)φA +∑
E∈∂Σεt(E)6∈Σε
ωEφt(E) −∑
E∈∂Σεs(E)6∈Σε
ωEφs(E).
123
Using (4.39) we see that
Sε =1
2
∑A∈Σε
[φ∗A(d†dφA +m2φA) + (dTd∗φ∗A +m2φ∗A)φA
]− 1
2
∑E∈∂Σε
[φ∗s(E)dφE + (dφ∗E)φs(E)
].
(4.40)
The first line of (4.40) vanishes on-shell, leaving only the boundary terms. Recalling
that −Son-shellε is the generating function of connected Green’s functions and following
the logic of [78], we see that a cutoff version of the Green’s function can be computed
as
Gε(k) =dφE(k)
φs(E)(k)for any E ∈ ∂Σε , (4.41)
where, crucially, we have plugged in the solution φA = φA(k) from (4.23). We have
to choose |kε|p < 1 in order to avoid having a vanishing denominator in (4.41). As
long as we work at fixed k, this is not a problem, since our eventual aim is to take
ε→ 0 p-adically. Straightforward calculation of the right hand side of (4.41) gives
Gε(k) =fvε−1 − fvε
fvε= − 1
ζp(∆− 1)+ |k|2∆−1
p |ε|2∆−1p
Qp∆
ζp(2∆− 1)+ . . . , (4.42)
where to obtain the first equality we used (4.23), and to obtain the second we ex-
panded in p-adically small ε. The omitted terms go to 0 more quickly than the ones
shown provided ∆ > 1/2, which is true of the larger of the two roots of the relation
(4.25). The first term in (4.42) is k-independent, so in position space it gives rise to a
pure contact term. Dropping this term, we define the Fourier space Green’s function
as
G(k) = limε→0
Gε(k)
|ε|2∆−1p
=Qp∆
ζp(2∆− 1)|k|2∆−1
p . (4.43)
124
Plugging in Q = −p1−2∆ from (4.24) and recalling the Fourier transform9
∫Qpdk χ(kx)|k|s =
ζp(1 + s)/ζp(−s)|x|1+s
p
, (4.44)
we obtain
G(x) =p∆ζp(2∆)
ζp(2∆− 1)2
1
|x|2∆p
. (4.45)
A somewhat more involved derivation of (4.45) in [10] makes it clear that the overall
normalization of G(x) is a subtle issue. Changing the location of the cutoff by one
lattice spacing results in changing G(x) by an O(1) multiplicative factor. We should
in short view (4.43) as a reasonable but non-unique prescription for normalizing the
two-point function.
4.4.2 Scalars on the line graph
For complex scalars φa on L(Tp), the extraction of a Green’s function from the bulk
to boundary propagator (4.26) proceeds almost exactly as in the warmup exercise
outlined in the previous section. Formally, in (4.38)-(4.43), one replaces d → D,
A → a, and E → e. Let’s inquire a little more closely why this works. The set
Σε comprises vertices with |za|p > |ε|p and edges with at least one vertex having
|za|p > |ε|p. The boundary ∂Σε consists of vertical edges only, and these edges all
have s(e) 6∈ Σε. Thus the partial integration formula (4.39) can indeed be carried
over to scalars on L(Tp) just by replacing d → D, A → a, and E → e. Likewise,
the subsequent manipulation of the action in (4.40) and the formula (4.41) for the
Green’s function carry over with the same alterations. The calculation (4.42) carries
over unaltered because of our careful inclusion of a factor of uv in the scalar ansatz
(4.26); a more conceptual way to say it is that this factor brings us to a gauge where
D = d on vertical edges. The result (4.43) carries over unaltered, and if we plug in
9An exposition of of Fourier integrals including (4.44) can be found in [27].
125
Q = p12−∆ sgn k, as given in (4.37), we obtain
G(k) =
√p sgn k
ζp(2∆− 1)|k|2∆−1
p . (4.46)
Using the Fourier integral
∫Qpdk χ(kx)|k|s sgn k = ups+
12
sgnx
|x|s+1p
, (4.47)
we arrive at
G(x) =up2∆
ζp(2∆− 1)
sgnx
|x|2∆p
. (4.48)
In terms of the boundary field theory, G(x) = 〈Oφ(x)Oφ(0)†〉 whereOφ is the operator
dual to φ. If we assume that translation by x is implemented in the boundary theory
by a unitary operator U(x), then Oψ(x) = U(x)†Oψ(0)U(x), and
G(x)∗ = 〈Oφ(0)Oφ(x)†〉 = 〈Oφ(−x)Oφ(0)†〉 = G(−x) . (4.49)
The relation G(x)∗ = G(−x) is indeed satisfied by (4.48): For p ≡ 1 mod 4, G(x) is
real and even under x → −x, while for p ≡ 3 mod 4, it is imaginary and odd under
x→ −x.
4.4.3 Fermions on the line graph
To derive a holographic two-point function for the operator Oψ dual to ψ, we start
from the cutoff action
Sε =∑e∈Σε
[iχ∗eDψe + iχeD∗ψ∗e +mχ∗eχe]−
∑a∈Σε
Mψ∗aψa , (4.50)
126
where Σε is defined as in section 4.1.1. After integration by parts,
Sε =∑e∈Σε
[1
2χ∗e(iDψe +mχe) +
1
2χe(iD
∗ψ∗e −mχ∗e)]
+∑a∈Σε
[1
2ψa(−iDTχ∗a +Mψ∗a) +
1
2ψ∗a(−iD†χa −Mψa)
]−∑e∈∂Σε
[i
2χ∗eψs(e) +
i
2χeψ
∗s(e)
].
(4.51)
This is an off-shell result. On shell, the first two lines vanish. By equating minus the
on-shell action with the generating functional of connected Green’s functions, we can
see by following the logic of [78] that a sensible definition of the cutoff Fourier space
Green’s function Gε(k) is
χe(k) = −iGε(k)ψs(e)(k) , (4.52)
provided Gε(k) turns out to be real. Assuming m 6= 0, we can rewrite (4.52) as
1
mDψe(k) = Gε(k)ψs(e)(k) (4.53)
Plugging in the fermionic bulk-to-boundary propagator (identical to (4.26) with φa →
ψa and Q given by (4.37)), we obtain in place of (4.42) and (4.43) the results
mGε(k) =fvε−1 − fvε
fvε= − 1
ζp(∆− 1)+ |k|2∆−1
p |ε|2∆−1p
Qp∆
ζp(2∆− 1)+ . . .
G(k) = limε→0
Gε(k)
|ε|2∆−1p
=Qp∆
mζp(2∆− 1)|k|2∆−1
p =
√p sgn k
mζp(2∆− 1)|k|2∆−1
p ,
(4.54)
where as usual we dropped the k-independent term from Gε(k) before taking the ε→ 0
limit. The presence of the factor of m in the denominator of the final expression in
127
(4.54) makes sense because with a coupling
Sint =
∫Qpdx[ψdef(x)∗Oψ(x) +Oψ(x)†ψdef(x)
](4.55)
between a renormalized version ψdef of the boundary limit of ψ and a boundary
operator Oψ, the scaling symmetry must act as ψdef → λψdef and Oψ → (λ∗)−1Oψ.
So the two-point function G(k) of Oψ with O†ψ scales as G(k) → |λ|−2G(k), which
matches the scaling of 1/m.
In short, we arrive at
G(x) =up2∆
mζp(2∆− 1)
sgnx
|x|2∆p
. (4.56)
It seems at first surprising that the symmetry of G(x) under x→ −x is the same for
bosons and fermions. The reason is that this symmetry was accompanied by complex
conjugation, which reverses the order of operator multiplication without introducing
signs related to the statistics of the operators.
4.5 Gauge field dynamics
So far we have considered only non-dynamical U(1) gauge fields. Let’s now consider
how we might add a kinetic term. On a general directed graph G, define a face f
to be any subgraph of G with three vertices only, all of which are required to be
connected, let’s say by edges e1, e2, and e3. To each face we assign (arbitrarily) a
direction, meaning a direction around which we think of the boundary of the face
circulating. If the direction of edge ei matches the direction assigned to f , then we
set s(f, ei) = 1; otherwise we set s(f, ei) = −1.10 Then, starting from a function θe
10This is of course the next step after (4.5) in constructing the incidence matrix of the cliquegraph; see [79].
128
defined on directed edges, we set
dθf =3∑i=1
s(f, ei)θei . (4.57)
Because each θe is essentially a line integral∫Aµdx
µ over the corresponding edge, the
derivative dθf is essentially the integral∫
12Fµνdx
µ ∧ dxν over the face. So an obvious
analog of the Maxwell action is
Sθ =∑f
1
2(dθf )
2 . (4.58)
Given a function Ff on faces, we can immediately read off from (4.57) the adjoint
operator (equivalently, the transpose):
d†Fe =∑∂f3e
s(f, e)Ff , (4.59)
where the sum is over all the faces whose boundary includes e. Evidently—in the
absence of couplings to other fields—the equation of motion following from (4.58) is
d†dθe = 0 . (4.60)
Although (4.58) is indeed the obvious free field action for a U(1) gauge field on a
directed graph G, it is not entirely satisfactory, since we feel that the phases eiθe
rather than the angles θe should be the fundamental variables. To say it another way,
if dθf ∈ 2πZ for some face f , we would like to say that it is equivalent to having
dθf = 0, but that is not reflected by the action (4.58).
The treatment is altogether more natural if we first define a Wilson line. The task
is not much harder for gauge group U(n). Starting therefore with a matter field φ
129
mapping vertices to Cn, we can define
Dφe = Ueφt(e) − φs(e) (4.61)
where Ue ∈ U(n). Now consider a directed path γ in our directed graph, from one
vertex, s(γ), to another, t(γ). It is simpler to rule out back-tracking and self-crossing,
but this is not really necessary. For edges e ∈ γ, let s(γ, e) = 1 if the direction of e
matches the overall direction of γ, and −1 otherwise. Now define
Uγ =∏e∈γ
U s(γ,e)e , (4.62)
where the order of factors follows the direction of γ: That is, the first edge in γ
corresponds to the leftmost factor, and the last edge corresponds to the rightmost
factor. The operator Uγ is a Wilson line for the path γ.
Let ∂af be the directed path around a face f , starting and ending on a chosen
vertex a. Then U∂af is well defined, and using it we can form the action
SU = −1
2
∑f
tr(U∂af + U−1∂af
) , (4.63)
in analogy to the Yang-Mills action. The choice of vertex a on the boundary of each
face can be made arbitrarily due to the cyclicity of the trace. To find the equation of
motion, choose an edge e and consider a face f with e ∈ ∂f . One can check that Ue
is the first factor in Us(f,e)∂s(e)f
. Therefore, upon a variation
Ue → (1 + iαH)Ue , (4.64)
where α is a small parameter and H is Hermitian, we find also Us(f,e)∂s(e)f
→ (1 +
130
iαH)Us(f,e)∂s(e)f
. We define the variation
δSU =∂SU∂α
∣∣∣∣α=0
= − i2
∑∂f3e
tr(HUs(f,e)∂s(e)f
) + c.c. , (4.65)
where the sum is over all faces whose boundary includes e. In short, the lattice
Yang-Mills equation reads
Im∑∂f3e
tr(HUs(f,e)∂s(e)f
) = 0 for all hermitian H . (4.66)
Returning to the U(1) case, we can simplify the equations of motion (4.66) to
Im∑∂f3e
Us(f,e)∂f = 0 , (4.67)
where we omit to specify the starting and ending point of ∂f because in an abelian
theory it doesn’t matter. We wish to consider a stronger condition, which we will
refer to as the complexified equation of motion:
∑∂f3e
Us(f,e)∂f = 0 . (4.68)
Imposing the condition (4.68) clearly implies the equations of motion, but the reverse
is in general not true. This is reminiscent of the situation in continuum Yang-Mills
theory where self-duality of the field strength implies the equations of motion, but
not vice versa. The point of interest for us is that the gauge field configurations we
have used throughout do satisfy (4.68). To verify this, we need the identity
∑y∈Fp
(x− yp
)(y
p
)= (−1)
p+12 , (4.69)
where x ∈ F×p is arbitrary and p is an odd prime. This convolution identity is known
as a Jacobi sum, and it can be proven starting from the Gauss sum (4.34). In the
131
following two paragraphs, we explain how (4.68) reduces to (4.69), as well as giving
some further indications of what (4.68) means physically.
For p ≡ 1 mod 4, the claim (4.68) comes down to the claim that U∂f = +1 for
half of the p − 1 faces adjoining a given edge e, and −1 for the other half. Before
we check this claim, let’s note that since all the Ue are real, the lattice Yang-Mills
equation (4.67) is satisfied trivially. It is not even clear that we should be demanding
(4.67) if we restrict the gauge fields to Ue = ±1, since (4.67) was derived on the
assumption that we could make an infinitesimal change to the gauge fields. However,
when all Ue = ±1, one can derive (4.68) as the condition that the action should
remain unaltered when just one of the Ue flips its sign. In any case, checking (4.68) is
trivial when e is a vertical edge, because it comes down to the observation that half
of the horizontal edges from any given vertex have sign +1 while the other have sign
−1. When e is a horizontal edge, say from 0 to x ∈ F×p , then we may write down
(4.68) explicitly as
(x
p
)+∑y∈F×py 6=x
(x
p
)(y − xp
)(−yp
)= 0 .
(4.70)
The first term in (4.70) comes from a face with two vertical edges plus the chosen
horizontal edge e, while the other p−2 terms come from faces with all edges horizontal.
The sum (4.70) obviously reduces to (4.69).
For p ≡ 3 mod 4, because all the Ue = i, one finds that U∂f is pure imaginary for all
faces. So in this case, (4.68) is equivalent to (4.67), and we can think of it as following
from requiring that the action is stationary under infinitesimal variations. We do not
see a useful way to understand how (4.68) arises from flipping the orientation of one
edge while preserving the property that Ue = i, since any configuration in which all
the U∂f are pure imaginary automatically has vanishing action. Once again, checking
(4.68) is trivial when e is a vertical edge because Us(f,e)∂f = i when the horizontal edge
132
in f points toward e and −i when it points away. When e is a horizontal edge, say
from 0 to x ∈ F×p , then (4.68) becomes
i
(x
p
)− i
∑y∈F×py 6=x
(x
p
)(y − xp
)(−yp
)= 0 ,
(4.71)
and this again reduces to (4.69).
4.6 Other sign characters
Let’s now inquire how we might modify our constructions on L(Tp) so as to get the
other three sign characters listed in section 4. As recounted in section 1.6, we can
identify each sign character with some non-square τ ∈ Q×p :
sgnτ x =
1 if x = a2 − τb2 for some a, b ∈ Qp
−1 otherwise .
(4.72)
Indeed, sgnτ = sgnτ ′ if τ/τ ′ is a square. There are therefore essentially only four
choices for τ : 1, ε, p, and pε, where ε is any element of F×p with (ε|p) = −1. The
characters sgnτ can be explicitly evaluated, as follows:
p ≡ 1 mod 4 p ≡ 3 mod 4
sgn1 x = 1 sgn1 x = 1
sgnε x = (−1)vx sgnε x = (−1)vx
sgnp x = (x0|p) sgnp x = (−1)vx(x0|p)
sgnεp x = (−1)vx(x0|p) sgnεp x = (x0|p)
(4.73)
where x = pvx(x0 + x1p+ x2p2 + . . . ) ∈ Q×p as in (4.3), where x0 ∈ F×p .
133
A useful intuition in the way a sign character arises is that a particle picks up a
phase Ue as it propagates across a link e. Because all vertical edges have the same
downward orientation and the same Ue, any phase that a particle picks up while
moving down from the boundary is precisely undone on its way back up. However,
on the shortest possible path on L(Tp) from one boundary point to Qp to another,
there is one horizontal edge. In a correlator 〈Oφ(x)Oφ(0)†〉, this horizontal edge runs
from (0, pvx+1) to (x0pvx , pvx+1). So it makes sense that 〈Oφ(x)Oφ(0)†〉 should include
a factor of
Ux ≡
U(0,pvx+1)→(x0pvx ,pvx+1) if (0, pvx+1)→ (x0p
vx , pvx+1) exists as an edge
U−1(x0pvx ,pvx+1)→(0,pvx+1) otherwise .
(4.74)
The factor (4.74) can be understood as a Wilson line for the path from 0 to x—where
all the factors for vertical edges canceled out. The choice of non-dynamical U(1)
gauge fields in section 4.2 can be understood as a way to get Ux = u sgnx—where the
factor of u is forced on us by the hermiticity condition U−x = U∗x . It is not entirely
clear from this discussion that the final Green’s function G(x) is proportional to Ux:
For instance, other paths exist on L(Tp) from 0 to x which do not pass through the
horizontal edge in question. However, it is a good guess that we can get the other
sign characters in (4.73) by setting
Ux = uτ sgnτ x , (4.75)
where uτ is an x-independent constant—either 1 or i—chosen so as to preserve the
condition U−x = U∗x . If we require (4.75) for all x ∈ Qp, and further require that
phases should be invariant under translations, then (4.75) amounts to a specification
of Ue for all horizontal edges. See figure 4.3 for example. As we have seen in the
discussion following (4.26), we can gauge away the phase on vertical edges.
134
Figure 4.3: Left: L(T5) in blue and red. The blue edges have θe = 0, while
the red edges have θe = π. This geometry yields the sign character
(x0|5)(−1)vx .
Center: L(T3) in black. All the edges have θe = π/2. This geometry
yields the sign character (x0|3)(−1)vx .
Right: L(T3) in blue and red. The blue edges have θe = 0, while
the red edges have θe = π. This geometry yields the sign character
(−1)vx .
135
The calculations of previous sections are for the most part straightforward to gen-
eralize to arbitrary sign characters, using the prescription (4.75) for choosing phases
on the horizontal edges of L(Tp). For an operator dual to a complex scalar on L(Tp)
with action (4.11), we find
G(x) = Csgnτ x
|x|2∆p
, (4.76)
up to divergent contact terms proportional to δ(x). We need uτ = 1 except when
p ≡ 3 mod 4 and τ = p or εp, and then uτ = i is required. The values of C and m2
can be determined from the following table:
τ C mass-dimension relation
1 p∆ζp(2∆)
ζp(2∆−1)2 m2 = − 1ζp(∆−1)ζp(−∆)
ε − 1+p2∆−2p−m2
1+p2−2∆−2p−m2
p∆ζp(4∆)
ζp(2∆)ζp(4∆−2)(2p+m2)2 = (1 + p2∆)(1 + p2−2∆)
p up2∆
ζp(2∆−1)2p+m2 = p1−∆ + p∆
εp up2∆
ζp(2∆−1)2p+m2 = p1−∆ + p∆
(4.77)
where u = 1 for p ≡ 1 mod 4 and u = i for p ≡ 3 mod 4. In all cases, we choose the
larger of the two possible values of ∆, assumed to be real.
The case τ = 1 is essentially the same as a complex scalar on Tp. The cases τ = p
and τ = εp are similar to one another, and we already presented in detail the cases
τ = p for p ≡ 1 mod 4 and τ = εp for p ≡ 3 mod 4.
Let us therefore focus on the one case with some new features: τ = ε, i.e. sgnε x =
(−1)vx . For simplicity, we consider only the complex scalar. The treatment proceeds
much as in sections 4.3.2 and 4.4.2, with a scalar ansatz
φa = fvγp(kza)χ(−kxa) . (4.78)
136
We recall the notation za = pv, and we use vk as before to denote the valuation of k.
For v > −vk, by plugging (4.78) into the scalar equation of motion (4.12), we obtain
the difference equation
(2p+m2)fv − pfv+1 − fv−1 + (−1)v(p− 1)fv = 0 . (4.79)
The last term comes from the horizontal edges. The equation (4.79) is different from
all previous difference equations we’ve encountered in that it does not have constant
coefficients, but instead coefficients that are periodic modulo 2 in v.11 Up to an overall
multiplicative scaling, the general solution to (4.79) is
fv = (1 + q(−1)v)(|za|1−∆
p +Q|k|2∆−1p |za|∆p
), (4.80)
where Q is a coefficient which at this stage is undetermined. Plugging (4.80) into
(4.79), one finds
(2p+m2)2 = (1 + p2∆)(1 + p2−2∆)
q =p∆ + p1−∆ − 2p−m2
p− 1,
(4.81)
and we assume as usual that ∆ > 1/2 is real.
For v < −vk, the equation of motion (4.12) is satisfied trivially, so the boundary
condition that determines Q comes from v = −vk, where (4.12) reads
(2p+m2)f−vk − pf−vk+1 + (−1)vkf−vk∑α∈F×p
χ
(−kza
α
p
)= 0 . (4.82)
Using the obvious identity
∑α∈F×p
χ
(−kza
α
p
)= −1 , (4.83)
11For the sign characters (−1)vx(x0|p), the difference equation obtained for v > −vk has constantcoefficients because the terms from horizontal edges cancel out.
137
We arrive at [2p+m2 − (−1)vk
]f−vk − pf−vk+1 = 0 . (4.84)
Plugging in (4.80) and using (4.81), we obtain
Q = (−1)vk+1p1−2∆ 1 + p2∆ − 2p−m2
1 + p2−2∆ − 2p−m2. (4.85)
To compute the holographic Green’s function, if we start with (4.42), we obtain
Gε(k) =1− q(−1)vε
1 + q(−1)vε
[p1−∆ + |k|2∆−1|ε|2∆−1 Qp∆
ζp(2∆− 1)+ . . .
]− 1 , (4.86)
where the omitted terms scale to 0 more quickly than the ones shown as ε→ 0 in the
p-adic norm. We now define
G(k) = limε→0
(1− q(−1)vε
1 + q(−1)vεGε(k)
|ε|2∆−1p
)=
Qp∆
ζp(2∆− 1)|k|2∆−1
p , (4.87)
where we dropped a k-independent term from Gε(k) before taking the limit. As
compared to (4.43), the definition (4.87) may seem a bit contrived. However, the extra
prefactor 1−q(−1)vε
1+q(−1)vεin (4.87) has no k-dependence, and its geometric mean between
even and odd vε is 1. So we maintain that (4.87) is the most sensible way to normalize
the Green’s function. Passing through a Fourier transform, we wind up with
G(x) = − 1 + p2∆ − 2p−m2
1 + p2−2∆ − 2p−m2
p∆ζp(4∆)
ζp(2∆)ζp(4∆− 2)
sgnε x
|x|2∆p
. (4.88)
4.7 Outlook
The main results of this chapter are summarized in (4.75)-(4.77): With a suitably
chosen configuration of a non-dynamical U(1) gauge field on the line graph L(Tp) of
the Bruhat-Tits tree, one may recover from a bulk complex scalar action on L(Tp) a
138
holographic boundary two-point function proportional to any sign character one wants
over Qp (for any odd prime p). Equally, one can work with a bulk complex fermion.
The key is not the statistics of the gauge field (or the boundary operator), but rather
the bulk gauge field, which gives rise to the desired sign character essentially as a
Wilson line between two boundary points.
While technical in nature, the results of this chapter open up many further ques-
tions. To begin with, one could try to tackle the case p = 2. There are seven
non-trivial sign characters sgnτ x over Q2 (besides the trivial one), and they depend
not only on the 2-adic norm of x, but also on its second and third non-trivial 2-adic
digits. Preliminary indications are that one can engineer elaborations of T2, including
non-dynamical gauge fields, that allow one to recover these sign characters; however,
nearest neighbor interactions are not enough. This is not too surprising given that the
second and third 2-adic digits relate to paths on the tree with at least two or three
links. Perhaps this is a hint for how to go on to more complicated multiplicative
characters over Qp, which can depend on finitely many p-adic digits and will wind up
involving finite range interactions on Tp.
It is a bit unsatisfying that we have placed so few limitations on the types of
fields that are allowed on Tp. In particular, if on L(Tp) we give each horizontal edge a
phase θe = π and leave all vertical edges with phases θe = 0, then one winds up with a
correlator of the form (4.76) with the trivial sign character, and the only effect of the
phases is to switch the sign of C. One naturally asks if there is a positivity constraint
that can fix this sign. One could observe that the choice of phases just described fails
to satisfied the complexified equation of motion (4.68); however, the same critique
can be made of the choice we made to capture the character sgnε x = (−1)vx . A more
fundamental point of view is called for explaining why particular choices of the U(1)
gauge field are natural constructions on L(Tp), perhaps analogous to the way that
the spin connection is natural on a smooth manifold.
139
Perhaps a related point is that we seem to have broken a lot of the symmetry of
Tp by introducing the distinction between horizontal and vertical edges on L(Tp). At
one level, this is not so disturbing, because the invariance we should require is dual
to the maps x → rx + b for all b ∈ Qp and r ∈ (Q×p )2. From a boundary theory
point of view, the operators Oφ and their two-point Green’s function transform with
non-trivial Jacobians under other elements of the p-adic conformal group PGL(2,Qp).
It would be very satisfying to give a full account in the bulk of how the corresponding
isometries of Tp (and L(Tp)) act on the U(1) gauge fields so as produce holographic
Green’s functions which are suitably covariant under the p-adic conformal group.
Having introduced the possibility of gauge field dynamics on L(Tp), another nat-
ural direction to explore is what the corresponding boundary operators are. If some
notion of conserved currents on the boundary is understood, perhaps the gravitational
dynamics of [25] could be refined or extended. We also hope that an enriched under-
standing of the geometry dual to p-adic conformal field theories will eventually impact
back on p-adic string theory, perhaps providing a better first-principles understanding
of Freund and Olson’s adaptation of the Veneziano amplitude and suggesting some
interesting generalizations.
140
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