Wilson loop expectations for finite gauge groups Sky Cao Sky Cao - PowerPoint PPT Presentation

Wilson loop expectations for finite gauge groups Sky Cao Sky Cao Wilson loop expectations for finite gauge groups

Introduction ◮ Lattice gauge theories are models from physics obtained by discretizing continuous spacetime R 4 by a lattice ε Z 4 . ◮ They are rigorously defined, so we can actually prove things. ◮ The continuum counterparts to lattice gauge theories are Euclidean Yang-Mills theories. Rigorous construction of such theories is of physical importance. ◮ One approach to rigorously defining a Euclidean Yang-Mills theory is to take a scaling limit of lattice gauge theories. ◮ In order to do so, various properties of lattice gauge theories must be very well understood. ◮ This talk is about understanding one particular property. Sky Cao Wilson loop expectations for finite gauge groups

Wilson loops ◮ The key objects of interest in a lattice gauge theory are the Wilson loops, which are certain random variables. ◮ Introduced by Wilson (1974) to give a theoretical explanation of an observed phenomenon. ◮ The explanation was in terms of Wilson loop expectations. ◮ Since then, there have been a number of results analyzing Wilson loop expectations; see Chatterjee’s survey “Yang-Mills for probabilists” for a detailed overview. ◮ One approach to taking a scaling limit involves being able to calculate Wilson loop expectations. More on this later. Sky Cao Wilson loop expectations for finite gauge groups

Notation ◮ Let G be a group, whose elements are d × d unitary matrices. The identity is denoted I d . We will often refer to G as the gauge group. ◮ Let Λ := [ − N , N ] 4 ∩ Z 4 be a large box. ◮ Let Λ 1 be the set of positively oriented edges in Λ . An edge ( x , y ) is positively oriented if y = x + e i , for some standard basis vector e i . ◮ Let Λ 2 denote the set of plaquettes in Λ . A plaquette is a unit square whose four boundary edges are in Λ . Pictorially: ◮ Given an edge configuration σ : Λ 1 → G , and a plaquette p as above, define σ p := σ e 1 σ e 2 σ − 1 e 3 σ − 1 e 4 . Sky Cao Wilson loop expectations for finite gauge groups

Definition of lattice gauge theories ◮ Define � S Λ ( σ ) := Re( Tr ( I d ) − Tr ( σ p )) . p ∈ Λ 2 ◮ Let G Λ 1 := { σ : Λ 1 → G } . Let µ Λ be the product uniform measure on G Λ 1 . ◮ For β ≥ 0, define the probability measure µ Λ ,β on G Λ 1 by d µ Λ ,β ( σ ) := Z − 1 Λ ,β exp( − β S Λ ( σ )) d µ Λ ( σ ) . ◮ We say that µ Λ ,β is the lattice gauge theory with gauge group G , on Λ , with inverse coupling constant β . ◮ Examples: G = U (1) , SU (2) , SU (3). ◮ For U (1), the continuum theory may be constructed directly - Gross (1986). Convergence of the lattice U (1) theory was established by Driver (1987). Sky Cao Wilson loop expectations for finite gauge groups

Definition of Wilson loops ◮ Let γ be a closed loop in Λ , with directed edges e 1 , . . . , e n . ◮ The Wilson loop variable W γ is defined as W γ ( σ ) := Tr ( σ e 1 · · · σ e n ) . [If e is negatively oriented, then σ e := σ − 1 − e .] ◮ Let � W γ � Λ ,β be the expectation of W γ under µ Λ ,β . Define � W γ � β := lim Λ ↑ Z 4 � W γ � Λ ,β . [This limit may only exist after taking a subsequence, but I will pretend that this technical point is not present.] Sky Cao Wilson loop expectations for finite gauge groups

A leading order computation ◮ Recently, Chatterjee (2018) computed Wilson loop expectations to leading order at large β , when G = Z 2 = {± 1 } . ◮ For a loop of length ℓ , we have � W γ � β ≈ e − 2 ℓ e − 12 β . ◮ Suppose we have a loop γ of length ℓ in R 4 . For ε > 0, we can obtain a discretization γ ε in ε Z 4 of length ε − 1 ℓ . ◮ If we set β ε := − 1 12 log ε , then as ε ↓ 0, we have → e − 2 ℓ . � W γ ε � β ε − ◮ This is the first step in one approach to taking a scaling limit. ◮ We thus want to understand the leading order of � W γ � β at large β , for general gauge groups. Sky Cao Wilson loop expectations for finite gauge groups

Main result ◮ In recent work, I’ve computed the leading order for finite gauge groups. First, some notation for the formula. ◮ Define ∆ G := min Re( Tr ( I d ) − Tr ( g )) . g � = I d ◮ G 0 := { g ∈ G : Re( Tr ( I d ) − Tr ( g )) = ∆ G } . ◮ 1 � A := g . | G 0 | g ∈ G 0 Theorem (C. 2020) Let β ≥ ∆ − 1 G (1000 + 14 log | G | ). Let γ be a loop of length ℓ . Let X ∼ Poisson ( ℓ | G 0 | e − 6 β ∆ G ). Then |� W γ � β − Tr ( E A X ) | ≤ 10 de − c ( G ) β . Sky Cao Wilson loop expectations for finite gauge groups

Main result (cont.) ◮ Let − 1 ≤ λ 1 , . . . , λ d ≤ 1 be the eigenvalues of A . Then d e − (1 − λ i ) ℓ | G 0 | e − 6 β ∆ G . � Tr ( E A X ) = i =1 ◮ There is a recent article by Forsstr¨ om, Lenells, and Viklund (2020), which handles finite Abelian gauge groups. They are able to obtain a much better β threshold in this case. Sky Cao Wilson loop expectations for finite gauge groups

Main result (cont.) ◮ Example: take K ≥ 2. Let G = { e i 2 π k / K , 0 ≤ k ≤ K − 1 } . ◮ Then ∆ G = 1 − cos(2 π/ K ), G 0 = { e i 2 π/ K , e − i 2 π/ K } , and A = cos(2 π/ K ). Letting λ := ℓ | G 0 | e − 6 β (1 − cos(2 π/ K )) , then E A Poisson ( λ ) = e − λ (1 − A ) . ◮ If K = 2, then ∆ G = 2, G 0 = {− 1 } , A = − 1, λ = ℓ e − 12 β . Sky Cao Wilson loop expectations for finite gauge groups

Rest of the talk ◮ I will first outline the proof of the theorem in the Abelian case. The main probabilistic insights are already all present. ◮ In essence, the proof has two main steps. Use a Peierls-type argument to show � W γ � β ≈ Tr ( E A N γ ), where N γ is a count of weakly dependent rare events. Show N γ ≈ Poisson . ◮ This two step outline was already present in Chatterjee (2018). ◮ When the gauge group is non-Abelian, this still works. I will describe the main ideas behind showing this. Sky Cao Wilson loop expectations for finite gauge groups

Preliminaries ◮ Suppose d = 1. Take some large box Λ . ◮ Recall µ Λ ,β is a probability measure on G Λ 1 with the form � � � µ Λ ,β ( σ ) = Z − 1 Λ ,β exp − β (1 − Re( σ p )) . p ∈ Λ 2 ◮ Define supp ( σ ) := { p ∈ Λ 2 : σ p � = 1 } . Let Σ ∼ µ Λ ,β . Let S := supp ( Σ ). ◮ When β is large, Σ p = 1 for most p , and so S is typically composed of sparsely distributed clumps. ◮ 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 ◮ Here’s a 2D cartoon of S : Sky Cao Wilson loop expectations for finite gauge groups

Decomposing S ◮ Since S is typically made of sparsely distributed clumps, let us try to decompose S into more elementary components. ◮ In general, any P ⊆ Λ 2 has a unique decomposition P = V 1 ∪ · · · ∪ V n into “connected components”. Definition A set V ⊆ Λ 2 is called a vortex if it cannot be decomposed further. ◮ It turns out that if P = V 1 ∪ · · · ∪ V n , then n � E [ W γ ( Σ ) | S = P ] = E [ W γ ( Σ ) | S = V i ] . i =1 ◮ So we want to understand E [ W γ ( Σ ) | S = V ], for vortices V . Sky Cao Wilson loop expectations for finite gauge groups

Understanding vortex contributions ◮ For a vortex V , we say that V appears in S if V is in the vortex decomposition of S . ◮ For an edge e , define P ( e ) to be the set of plaquettes which contain e . Note | P ( e ) | = 6. ◮ In 3D, P ( e ) looks like this. ◮ 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.) ◮ For a vortex P ( e ), we have  1 e / ∈ γ   E [ W γ ( Σ ) | S = P ( e )] = e ∈ γ .   A   ◮ Let N γ be the number of edges e in γ such that P ( e ) appears in S . We then have E [ W γ ( Σ ) | S ] = A N γ Y , where Y is the contribution from vortices of size ≥ 7. ◮ 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.) ◮ In order for a vortex V to be such that E [ W γ ( Σ ) | S = V ] � = 1, it must be close to the loop γ . ◮ Larger vortices are much less likely to appear in S . ◮ So if we look in a neighborhood of γ , only P ( e ) vortices are likely to appear. ◮ We thus have P ( Y � = 1) = lower order . ◮ Thus on an event of high probability, E [ W γ ( Σ ) | S ] = A N γ . Sky Cao Wilson loop expectations for finite gauge groups

Poisson approximation ◮ It remains to show N γ ≈ Poisson . ◮ We apply the dependency graph approach to Stein’s method. ◮ Given vortices V 1 = P ( e 1 ) , . . . , V n = P ( e n ) which are not too close to each other, we need to have n � P ( V 1 , . . . , V n appear in S ) ≈ P ( V i appears in S ) . i =1 ◮ This is done by cluster expansion. Cluster expansion is a fairly well known tool; for example it appears in Seiler’s 1982 monograph on lattice gauge theories. ◮ Thus we see why S is nice to work with: it has “a lot of independence”. Sky Cao Wilson loop expectations for finite gauge groups