Wilson loop expectations for finite gauge groups

Sky Cao

Sky Cao Wilson loop expectations for finite gauge groups

Introduction

I Lattice gauge theories are models from physics obtained bydiscretizing continuous spacetime R4 by a lattice εZ4.

I They are rigorously defined, so we can actually prove things.I The continuum counterparts to lattice gauge theories are

Euclidean Yang-Mills theories. Rigorous construction of suchtheories is of physical importance.

I One approach to rigorously defining a Euclidean Yang-Millstheory is to take a scaling limit of lattice gauge theories.

I In order to do so, various properties of lattice gauge theoriesmust be very well understood.

I This talk is about understanding one particular property.

Sky Cao Wilson loop expectations for finite gauge groups

Wilson loops

I The key objects of interest in a lattice gauge theory are theWilson loops, which are certain random variables.

I Introduced by Wilson (1974) to give a theoretical explanationof an observed phenomenon.

I The explanation was in terms of Wilson loop expectations.I Since then, there have been a number of results analyzing

Wilson loop expectations; see Chatterjee’s survey “Yang-Millsfor probabilists” for a detailed overview.

I One approach to taking a scaling limit involves being able tocalculate Wilson loop expectations. More on this later.

Sky Cao Wilson loop expectations for finite gauge groups

Notation

I Let G be a group, whose elements are d × d unitary matrices.The identity is denoted Id . We will often refer to G as thegauge group.

I Let Λ := [−N,N]4 ∩ Z4 be a large box.I Let Λ1 be the set of positively oriented edges in Λ. An edge

(x, y) is positively oriented if y = x + ei , for some standardbasis vector ei .

I Let Λ2 denote the set of plaquettes in Λ. A plaquette is a unitsquare whose four boundary edges are in Λ. Pictorially:

I Given an edge configuration σ : Λ1 → G, and a plaquette pas above, define σp := σe1σe2σ

−1e3σ−1

e4.

Sky Cao Wilson loop expectations for finite gauge groups

Definition of lattice gauge theories

I DefineSΛ(σ) :=

∑p∈Λ2

Re(Tr(Id) − Tr(σp)).

I Let GΛ1 := {σ : Λ1 → G}. Let µΛ be the product uniformmeasure on GΛ1 .

I For β ≥ 0, define the probability measure µΛ,β on GΛ1 by

dµΛ,β(σ) := Z−1Λ,β exp(−βSΛ(σ)) dµΛ(σ).

I We say that µΛ,β is the lattice gauge theory with gauge groupG, on Λ, with inverse coupling constant β.

I Examples: G = U(1),SU(2),SU(3).I For U(1), the continuum theory may be constructed directly -

Gross (1986). Convergence of the lattice U(1) theory wasestablished by Driver (1987).

Sky Cao Wilson loop expectations for finite gauge groups

Definition of Wilson loops

I Let γ be a closed loop in Λ, with directed edges e1, . . . , en.I The Wilson loop variable Wγ is defined as

Wγ(σ) := Tr(σe1 · · ·σen ).

[If e is negatively oriented, then σe := σ−1−e .]

I Let 〈Wγ〉Λ,β be the expectation of Wγ under µΛ,β. Define

〈Wγ〉β := limΛ↑Z4〈Wγ〉Λ,β.

[This limit may only exist after taking a subsequence, but I willpretend that this technical point is not present.]

Sky Cao Wilson loop expectations for finite gauge groups

A leading order computation

I Recently, Chatterjee (2018) computed Wilson loopexpectations to leading order at large β, when G = Z2 = {±1}.

I For a loop of length `, we have

〈Wγ〉β ≈ e−2`e−12β.

I Suppose we have a loop γ of length ` in R4. For ε > 0, we canobtain a discretization γε in εZ4 of length ε−1`.

I If we set βε := − 112 log ε, then as ε ↓ 0, we have

〈Wγε〉βε −→ e−2`.

I This is the first step in one approach to taking a scaling limit.I We thus want to understand the leading order of 〈Wγ〉β at

large β, for general gauge groups.

Sky Cao Wilson loop expectations for finite gauge groups

Main result

I In recent work, I’ve computed the leading order for finitegauge groups. First, some notation for the formula.

I Define∆G := min

g 6=IdRe(Tr(Id) − Tr(g)).

I

G0 := {g ∈ G : Re(Tr(Id) − Tr(g)) = ∆G}.

I

A :=1|G0|

∑g∈G0

g.

Theorem (C. 2020)

Let β ≥ ∆−1G (1000 + 14 log|G|). Let γ be a loop of length `. Let

X ∼ Poisson(`|G0|e−6β∆G ). Then

|〈Wγ〉β − Tr(EAX )| ≤ 10de−c(G)β.

Sky Cao Wilson loop expectations for finite gauge groups

Main result (cont.)

I Let −1 ≤ λ1, . . . , λd ≤ 1 be the eigenvalues of A . Then

Tr(EAX ) =d∑

i=1

e−(1−λi )`|G0 |e−6β∆G.

I There is a recent article by Forsstrom, Lenells, and Viklund(2020), which handles finite Abelian gauge groups. They areable to obtain a much better β threshold in this case.

Sky Cao Wilson loop expectations for finite gauge groups

Main result (cont.)

I Example: take K ≥ 2. Let G = {e i2πk/K , 0 ≤ k ≤ K − 1}.

I Then ∆G = 1 − cos(2π/K ), G0 = {e i2π/K , e−i2π/K }, andA = cos(2π/K ). Letting λ := `|G0|e−6β(1−cos(2π/K )), then

EAPoisson(λ) = e−λ(1−A ).

I If K = 2, then ∆G = 2, G0 = {−1}, A = −1, λ = `e−12β.Sky Cao Wilson loop expectations for finite gauge groups

Rest of the talk

I I will first outline the proof of the theorem in the Abelian case.The main probabilistic insights are already all present.

I In essence, the proof has two main steps.Use a Peierls-type argument to show 〈Wγ〉β ≈ Tr(EANγ ), whereNγ is a count of weakly dependent rare events.Show Nγ ≈ Poisson.

I This two step outline was already present in Chatterjee(2018).

I When the gauge group is non-Abelian, this still works. I willdescribe the main ideas behind showing this.

Sky Cao Wilson loop expectations for finite gauge groups

Preliminaries

I Suppose d = 1. Take some large box Λ.I Recall µΛ,β is a probability measure on GΛ1 with the form

µΛ,β(σ) = Z−1Λ,β exp

(− β

∑p∈Λ2

(1 − Re(σp))).

I Definesupp(σ) := {p ∈ Λ2 : σp 6= 1}.

Let Σ ∼ µΛ,β. Let S := supp(Σ).I When β is large, Σp = 1 for most p, and so S is typically

composed of sparsely distributed clumps.I We will see that S is easier to work with than Σ.

Sky Cao Wilson loop expectations for finite gauge groups

Picture to keep in mind

I Here’s a 2D cartoon of S:

Sky Cao Wilson loop expectations for finite gauge groups

Decomposing S

I Since S is typically made of sparsely distributed clumps, let ustry to decompose S into more elementary components.

I In general, any P ⊆ Λ2 has a unique decompositionP = V1 ∪ · · · ∪ Vn into “connected components”.

DefinitionA set V ⊆ Λ2 is called a vortex if it cannot be decomposed further.

I It turns out that if P = V1 ∪ · · · ∪ Vn, then

E[Wγ(Σ) | S = P] =n∏

i=1

E[Wγ(Σ) | S = Vi ].

I So we want to understand E[Wγ(Σ) | S = V ], for vortices V .

Sky Cao Wilson loop expectations for finite gauge groups

Understanding vortex contributions

I For a vortex V , we say that V appears in S if V is in the vortexdecomposition of S.

I For an edge e, define P(e) to be the set of plaquettes whichcontain e. Note |P(e)|= 6.

I In 3D, P(e) looks like this.I The smallest vortex which can appear in S must be P(e), for

some edge e. All other vortices must have size ≥ 7.

Sky Cao Wilson loop expectations for finite gauge groups

Understanding vortex contributions (cont.)

I For a vortex P(e), we have

E[Wγ(Σ) | S = P(e)] =

1 e /∈ γ

A e ∈ γ.

I Let Nγ be the number of edges e in γ such that P(e) appearsin S. We then have

E[Wγ(Σ) | S] = ANγY ,

where Y is the contribution from vortices of size ≥ 7.I Next, we show that with high probability, we can ignore

vortices of size ≥ 7.

Sky Cao Wilson loop expectations for finite gauge groups

Understanding vortex contributions (cont.)

I In order for a vortex V to be such that E[Wγ(Σ) | S = V ] 6= 1, itmust be close to the loop γ.

I Larger vortices are much less likely to appear in S.I So if we look in a neighborhood of γ, only P(e) vortices are

likely to appear.I We thus have

P(Y 6= 1) = lower order.

I Thus on an event of high probability,

E[Wγ(Σ) | S] = ANγ .

Sky Cao Wilson loop expectations for finite gauge groups

Poisson approximation

I It remains to show Nγ ≈ Poisson.I We apply the dependency graph approach to Stein’s method.I Given vortices V1 = P(e1), . . . ,Vn = P(en) which are not too

close to each other, we need to have

P(V1, . . . ,Vn appear in S) ≈n∏

i=1

P(Vi appears in S).

I This is done by cluster expansion. Cluster expansion is a fairlywell known tool; for example it appears in Seiler’s 1982monograph on lattice gauge theories.

I Thus we see why S is nice to work with: it has “a lot ofindependence”.

Sky Cao Wilson loop expectations for finite gauge groups

The general case

I There are some technical difficulties that appear in thenon-Abelian case.

I In the remaining time, I will present the key idea needed tohandle these difficulties.

I I will then give a toy example showing why the key idea isuseful.

Sky Cao Wilson loop expectations for finite gauge groups

The key idea

I Let us now think of Λ as a graph.I The fundamental group π1(Λ) is made of (equivalence classes

of) closed loops in Λ which begin and end at some fixedbasepoint. Every closed loop is given by some sequence ofedges e1 · · · en.

Observation (Szlachanyi and Vecsernyes (1989))

Any σ ∈ GΛ1 induces a homormophism ψσ : π1(Λ)→ G, defined by

ψσ(e1 · · · en) := σe1 · · ·σen .

Sky Cao Wilson loop expectations for finite gauge groups

Toy example

I Let T be a spanning tree of Λ. Suppose σ ∈ GΛ1 is such thatσe = Id for all e ∈ T .

I Suppose additionally that σp = Id for all p ∈ Λ2.I I then claim that in fact, σe = Id for all e ∈ Λ1.I Initial attempt:

Sky Cao Wilson loop expectations for finite gauge groups

Toy example (cont.)

I The crucial topological fact: any loop in π1(Λ) is a product of“Lasso type” loops:

I For any Lasso type loop L ∈ π1(Λ), we have ψσ(L ) = Id .I Thus ψσ is trivial.I Given e = (x, y) ∈ Λ1\T , take a loop ae ∈ π1(Λ) which uses e,

and such that every other edge of ae is in T .I Since σ = Id on T , we have ψσ(ae) = σe .I Because ψσ is trivial, ψσ(ae) = Id .

Sky Cao Wilson loop expectations for finite gauge groups

In summary

I Computing the leading order of Wilson loop expectations is afirst step in taking a scaling limit.

I The key probabilistic insight:

E[Wγ(Σ) | S] = Tr(ANγ ) w.h.p.

I The key technical tool: the components of S appearessentially independently.

I In the non-Abelian case, algebraic topology is a naturallanguage to use.

I Special thanks to Sourav Chatterjee, whose conversationsshaped this work, as well as Persi Diaconis, Hongbin Sun,and Ciprian Manolescu. I am particularly indebted to CiprianManolescu for the proof of a key technical lemma.

Sky Cao Wilson loop expectations for finite gauge groups