# Questions tagged [wilson-loop]

In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given (closed) loop $C$. It is the trace of a path-ordered exponential of the gauge field $A_\mu$ transported along $C$, $W_C := \mathrm{Tr}(\mathcal{P}\exp i \oint_C A_\mu dx^\mu)$, where $\mathcal{P}$ is the path-ordering operator.

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### Exactly what value does the Wilson line take?

Let $G$ be the Lie group of a given theory with the Lie algebra $\mathfrak{g}$. According to the Wikipedia article, a Wilson line is of the form W[x_i,x_f]= P e^{i \int_{x_i}^{x_f} A} ...
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### Extracting a gauge-invariant variable from a given Wilson line? (NOT Wilson loop)

Let $W[x_i,x_f]$ be the Wilson line as defined here. Under a local gauge transform $g(x)$, it transforms as $$W[x_i,x_f] \to g(x_f)W[x_i,x_f] g^{-1}(x_i)$$ which is shown ...
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### Definition of four-potential in lattice gauge theory

In Wen's book 'Quantum Field Theory of Many Body Systems' at chapter 6.4, he defines scalar potential on lattice sites while vector potential at lattice links in two dimensional square lattice. What ...
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### Wilson loop is not an element of $\mathrm{SU}(3)$ in color deconfinement

The center symmetry in QCD comes from the $$a\ \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right) a^{-1} = \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right),$$ where $C$ ...
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### Problems about "boundary conditions and topology"

In the book Field Theories of Condensed Matter Physics by Fradkin In Page 311, when discussing the effects of boundary conditions on $Z_2$ lattice gauge theory, in the weak coupling phase, Fradkin ...
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### Physical meaning of the Wilson Loops as spin impurities

This is in reference to the paper of David Tong here. In this paper in section 2, it says In this first section, we explain how spin impurities, coupled to bulk gauge fields, can be thought of as ...
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### Gauge theories, boundaries and Wilson lines

My understanding of Wilson loops Let's work with classical electromagnetism. The 4-potential $A_\mu$ determines the electric and magnetic fields, which are the physical entities responsible for the ...
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### Relationship between holonomy and fundamental group

In my notes of topological QFT we demonstrated that the holonomy associated with a path in $\mathbb{R}^3$ is invariant under smooth deformation of the path if the connection is flat. Then I wrote: If ...
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### De Rham current associated with knot in abelian CS theory on a generic manifold

I'm studying TQFT and I'm stucked on this part of the paper of my teacher: My teacher didn't explain a lot about it and I've never followed an advanced course on differential geometry or algebraic ...
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### How can I find the third spectrum of lines in $SU(2)$ gauge theory?

In the article https://arxiv.org/abs/1305.0318, they take a gauge theory based on the algebra $su(2)$ as a first example of how to determine the allowed line operators. Once different lines can be ...
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### Vacuum expectation value of the Wilson loop in pure QED

In Chapter 57 of Srednicki's QFT book he derives the generating function of pure QED and finds \begin{align*} Z(J) = \exp\left[\frac{i}{2}\int\mathrm{d}^4x\,\mathrm{d}^4y\ J_\mu(x) \Delta^{\mu\nu}(x-y)...
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### Wilson loops as representations of the Lorentz group

Wilson loops in lattice $4d$ Yang-Mills theory are used to build various glueball states of different spins when they are applied to the vacuum. The spin dependence of such states is related with the ...
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### Area and perimeter

Apparently (?), a line operator over a very large loop with length $L$ can obey either perimeter law or area law, $-\log\langle U\rangle\sim L^a$ with $a=1,2$, respectively. We call these options &...
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### Expand an infinitesimal Wilson loop

I have a question about expanding an infinitesimal Wilson loop operator to get the field tensor $F_{\mu \nu}$ in chapter 3 of Fradkin's notes Classical Symmetries and Conservation Laws. For a ...
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### Wilson loop shapes and glueball operators

In the AdS/QCD correspondence, glueballs operators are given, for example, by $\text{Tr}[F_{\mu \nu}F^{\mu \nu}]$ for $0^{++}$ or $\text{Tr}[F_{\mu \nu}\widetilde{F}^{\mu \nu}]$ for $0^{-+}$. However, ...
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### Deriving the comparator to second order in Peskin and Schroeder

In section 15.1 of Peskin and Schroeder, expression (15.9) is given for the comparator $U(y,x)$ in an infinitesimal expansion to second order: U(x+\epsilon n, x)=\exp \left[-i e \epsilon n^{\mu} A_{\...