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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 \begin{equation} 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 \begin{equation} W[x_i,x_f] \to g(x_f)W[x_i,x_f] g^{-1}(x_i) \end{equation} 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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Wilson lines with Chan-Paton factors in string theory

In the context of compactifying the open string with Chan-Paton factors, Polchinski (Volume I Section 8.6) considers a toy example with a point particle of charge $q$ which has the action $$ S = \int ...
Adrien Martina's user avatar
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Are pseudo Riemannian manifolds with identical Wilson loops isometric?

It is well established that in gauge theory, the Wilson loops of the theory determine the gauge potential up a gauge transformation. That is, two gauge potentials $A_\mu$ and $B_\mu $ produce the same ...
Trevor Scheopner's user avatar
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What is a non-linear space of connections

In the book "Loops Knots Gauge Theory and Quantum Gravity" when trying to define a loop representation, one needs to integrate over the space of connections (modulo Gauge transformations). ...
Confuse-ray30's user avatar
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Singularity of free energy in $\mathbb{Z}_2$ lattice gauge theory

I'm currently reading Kardar's Statistical physics of Fields. In the book, the $\mathbb{Z}_2$ lattice gauge theory is constructed as the dual of the 3d Ising model. (Note: the Hamiltonian is $H = \...
Jason Chen's user avatar
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How does Witten's path integral know about changing crossings?

At a crossing of a knot, if I change the crossing by swapping the two lines, the knot is changed, along with its Jones Polynomial. Witten's path integral $$ \int {D \mathcal{A}\ e^{i\mathcal{L}}\ W_R(...
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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 ...
Dr. user44690's user avatar
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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 ...
P. C. Spaniel's user avatar
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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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Holonomy expansion for path deformation

A path deformation by $\epsilon^{\mu}(s)$ induces a variation of the connection $A'(s)=A(s)+\Delta A(s)$. I'm trying to obtain the first-order expansion of the holonomy $H_{\gamma}(A)=Pe^{i\int_{\...
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Representation of nonabelian Wilson line in terms of fermionic fields

Context: The coupling action of a particle of charge $q$ to a $U(1)$ gauge field is given by \begin{equation} S = q \int d \tau A_\mu \left( X \right) \frac{dX^\mu(\tau)}{d \tau} = -i \ln W_q, \tag{...
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Dressing an operator by Wilson line in Quantum Electrodynamic

I am reading a paper arXiv:1507.07921 which introduce gravitational dressing. The paper compare it to dressing in QED. Consider the scalar QED lagrangian $$\mathcal{L}=-\frac{1}{4}(F^{\mu\nu})^2-|D_\...
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Non-Abelian Chern-Simons Theory References

I am studying Chern-Simons theories and am fairly familiar with the usual Abelian $U(1)$ Chern-Simons theory. I am now looking to extend my knowledge to non-Abelian Chern-Simons and am having a hard ...
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Wilson loops in Gauge theories in statistical mechanics

In this discussion @Seth Whitsitt mentioned in a comment that for gauge theories in statistical mechanics the Wilson lines are extremely important objects for the thermodynamic and phase properties of ...
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Can one build Wilson lines in general relativity?

This question has two parts: Firstly, I am curious if one can build Wilson lines as a 'parallel transport operator' in general relativity in direct analogy with what is done in gauge theory. For a ...
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AdS-CFT on a torus

A gauge theory at finite volume and temperature on a space like $S^{n-1}\times S^1$ is supposed to be dual to a string theory on either $AdS_{n+1}$ or a Schwarzschild black hole in $AdS_{n+1}$, with a ...
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How do I numerically compute the interquark potential from the correlation function of Polyakov Loops?

I know that the potential can be calculated in the following way: $$ aV(r) =-\ln(<\sum_{\textbf{x}} (P(\textbf{x}+R)P^{\dagger}(\textbf{x}))>)/N_T. $$ Now, suppone I have some procudure to ...
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Calculating Diagrams with Wilson Lines to One Loop

I have a question concerning the calculation of amplitudes containing Wilson lines. I want to calculate the jet function defined by equation (3.3) from this paper. $$\mathcal{J}(\text{arguments})u_s(p)...
George Smyridis's user avatar
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How to couple the Higgs field with $SU(3)$ (and/or $SU(2)$) Yang-Mills theory in numerical simulations?

I'm trying to couple the Higgs field to numerical simulations of pure gauge theory with heatbath and overrelaxation update of link variables. I don't know how to insert the Higgs field into the ...
Gennaro's user avatar
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One-form symmetry and its spontaneously breaking

In the paper "Generalized global symmetry" by Gaiotto, Kapustin, Seiberg and Willett: https://arxiv.org/abs/1412.5148, in section 5 they describe the spontaneously breaking of 1-form ...
JQ Skywalker's user avatar
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Calculating a rectangular Wilson loop for the free photon

I'm studying Creutz's Quarks, gluons and lattices, in chapter 6 on page 33, we have the following exercise Calculate a rectangular Wilson loop for the field theory of free photons. Using any ...
Simplyorange's user avatar
11 votes
1 answer
913 views

Berry phase and Wilson loop

According to the definition, the Wilson loop is \begin{equation} W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right] \end{equation} ...
Fang Lyu's user avatar
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Relation between sources and Wilson loops in Tong's gauge theory

In Euclidean quantum Yang-Mills with compact gauge group $G$, the VEV of a Wilson loop is: $$\tag{$\star$} \langle W[C]\rangle \equiv \int_{\mathcal{A}/G}DA e^{-S_{YM}[A]}tr\mathcal{P}\exp\left(i\...
nodumbquestions's user avatar
3 votes
1 answer
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Linking of a sphere with a Wilson line

In the papers such as Ref.[I] and Ref.[II], they have introduced the operator, $$ U_\alpha (M_{d-2}) = e^{\frac{i\alpha}{g^2}\int_{M_{d-2}}*F} . $$ They said that the Wilson loop: $$W_n(\gamma)=e^{in\...
Tuhin Subhra Mukherjee's user avatar
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1 answer
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Question on ordered exponential explanation in Wikipedia

Let $\gamma:[0,1] \to \mathbb R^2$ be a path describing a rectangle with vertices $x$, $x+u$, $x+u+v$, $x+v$, where $x, u, v \in \mathbb R^2$ ($u, v$ linearly independent). Let $J:\mathbb R^2 \to \...
Overflowian's user avatar
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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 ...
Lucas Queiroz's user avatar
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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 ...
Fra's user avatar
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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 &...
AccidentalFourierTransform's user avatar
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1 answer
351 views

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 ...
Hao's user avatar
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Knots in 3d pure gravity

Three-dimensional pure gravity on $\mathrm{AdS}_3$ can be described as a Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$,with action $$ S[A,\bar{A}] = \...
ɪdɪət strəʊlə's user avatar
-1 votes
1 answer
245 views

Modified non-abelian Wilson loop

Consider a $\mathrm{SU}(2)$ Wilson loop in some representation $R$, $$W_R= \mathrm{Tr}_R\left\{\mathcal P\exp\left(i \oint \mathrm dx^{\mu} A^a_{\mu} T^a\right)\right\}$$ where $T^a= \sigma^a/2$ is ...
user34104's user avatar
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Fourier Transforrm of a Wilson Line

Lots of times it is helpful to perform computations of Feynman Diagrams in momentum space as opposed to position space. This however appears a little tricky when it comes to Wilson Lines. Ignoring ...
user3166083's user avatar
2 votes
2 answers
393 views

Variation of a time-ordered exponential

Consider the time-ordered exponential (Wilson line): $$ U(t_{f},t_{i}) = \mathcal{T}\text{exp}\left(-i\int_{t_{i}}^{t_{f}}\mathcal{A}(t)dt\right)\tag{1} $$ Where $\mathcal{A}(t)$ is some matrix-valued ...
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Book for lattice field theory for somebody with basic understanding of Quantum Field Theory [duplicate]

I have finished a first course in Quantum Field theory and I'm looking to get into lattice field theory (mostly QCD on the lattice). What are some resources for somebody wanting to learn how things ...
3 votes
1 answer
257 views

References for prerequisite material for understanding papers on Generalized Global Symmetries

I want to understand the papers https://arxiv.org/abs/1412.5148 and https://arxiv.org/abs/1703.00501. Assuming that I understand basics of gauge theories, could someone suggest some references ...
2 votes
1 answer
207 views

How to prove $\Lambda_\mathrm{weight}(\mathfrak{g})/ \Lambda_\mathrm{root}(\mathfrak{g}) = \mathbb{Z}_N$ for $\mathrm {SU}(N)$?

I have a question about $\mathrm {SU}(N)$ Lie group from David Tong's Gauge Theory notes (p. 92). He considers $\Lambda_w(\mathfrak{g})$ and $\Lambda_\mathrm{root}(\mathfrak{g})$ as the weight lattice ...
Kitchen's user avatar
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Expansion of Wilson loop operator to obtain the field tensor in non-Abelian gauge theory

I am reading Prof. Eduardo Fradkin's QFT lecture note (the link to the lecture note is gone). In his note, he considered the Wilson loop operator \begin{equation} \hat{W}_{\Gamma(x,x)}=\hat{P} \left( \...
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Missing factor 1/2 when using generalized Stokes theorem

I'm doing the following homework question: By invoking Stokes' theorem, according to which the integral of a vector field (which equals the field strength) over any two-dimension surface S that is ...
AccidentalTaylorExpansion's user avatar
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1 answer
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How to calculate the chiral condensate from wilson fermions in lattice qcd?

In lattice qcd, respective more specifically in (1+1)-dimensional massless schwinger model on a lattice, iam trying to reproduce the chiral condensate by using correct constructed wilson overlap ...
Sebastian P's user avatar
2 votes
0 answers
74 views

Wilson action equations of motion

Let $S_W$ be a Wilson action of $1\times 1$ plaquettes for a gauge group $G$: \begin{equation*} S_W = \beta a^4 \sum_P \left( 1-\frac{1}{N_G} \text{Re Tr}(U_P) \right), \end{equation*} where $\beta$ ...
Jeanbaptiste Roux's user avatar
2 votes
1 answer
221 views

QFT/QM on loop space

I am reading "Gauging What's Real" by Richard Healey and the author argues for formulating electrodynamics/QED on loop space/the holonomy group, so that the real objects described by the ...
NicAG's user avatar
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1 answer
699 views

Wilson line and external source

Let's consider free Maxwell theory: $$ L = -\frac{1}{4g^2} F^{\mu\nu}F_{\mu\nu} $$ As I understand, one can describe external particles with help of Wilson lines: $$ W(q,l) = e^{iq\int_l dx^\mu A_\mu} ...
Nikita's user avatar
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3 votes
1 answer
209 views

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, ...
Jeanbaptiste Roux's user avatar
3 votes
1 answer
512 views

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_{\...
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