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
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