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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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.
$$
Now, suppone I have some procudure to ...
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How to compute the susceptibility of the Polyakov loop in Monte Carlo lattice field theory?
I am having troubles understanding the definition of the susceptibility of the Polyakov loop give, for example, in the book by Gattringer, Lang "Quantum Chromodynamics on the Lattice", page ...
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Calculating Diagrams with Wilson Lines to One Loop
I have a question concerning the calculation of amplitudes containing Wilson lines.
I want to calculate the jet function defined by equation (3.3) from this paper.
$$\mathcal{J}(\text{arguments})u_s(p)...
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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 ...
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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 ...
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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 ...
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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}
...
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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:
$$\tag{$\star$} \langle W[C]\rangle \equiv \int_{\mathcal{A}/G}DA e^{-S_{YM}[A]}tr\mathcal{P}\exp\left(i\...
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Linking of a sphere with a Wilson line
In the papers such as Ref.[I] and Ref.[II], they have introduced the operator,
$$
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\...
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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 \...
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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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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
$$ S[A,\bar{A}] = \...
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Modified non-abelian Wilson loop
Consider a $\mathrm{SU}(2)$ Wilson loop in some representation $R$,
$$W_R= \mathrm{Tr}_R\left\{\mathcal P\exp\left(i \oint \mathrm dx^{\mu} A^a_{\mu} T^a\right)\right\}$$
where $T^a= \sigma^a/2$ is ...
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Fourier Transforrm of a Wilson Line
Lots of times it is helpful to perform computations of Feynman Diagrams in momentum space as opposed to position space. This however appears a little tricky when it comes to Wilson Lines.
Ignoring ...
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Variation of a time-ordered exponential
Consider the time-ordered exponential (Wilson line):
$$
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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$ ...
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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 ...
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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}
...
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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:
$$U(x+\epsilon n, x)=\exp \left[-i e \epsilon n^{\mu} A_{\...
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What does it mean for an extended operator to possess "local excitations"?
In the context of defect conformal field theory, we consider in operator product expansions "local excitations" of the defect (see e.g. text between eq. $(1.1)$ and $(1.2)$ in the paper ...
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Wilson loop expectation value in $RP^3$ using Dehn surgery
I am currently reading Guadagnini's The link invariants of Chern-Simons field theory, the part where he computes some examples of expectation values for different spaces.
For $S^2 \times S^1$, he ...
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Expectation value of Wilson loop in Chern-Simons theory
I have read Witten's paper, and I am interested on computing the expectation value of a Wilson loop with a representation $R$ on Chern-Simons theory in $d=3$. I am especially interested in cases for $...
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Expectation value of Polyakov loop
Assuming a pure Yang-Mills theory, how exactly does one get that, for appropriate $\beta$ for confinement, the expectation value of the Polyakov loop $<\Phi>$ equals zero? I do not seem to get ...
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How do I show that a given Wilson loop satisfies the loop equation?
In the book Methods of Contemporary Gauge Theory by Yuri Makeenko, the loop equation in the large-$N$ limit is given by
$$\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu \nu}} W(C) = \lambda \oint_C ...
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Connection between Wilson loops and fusion rules in $Z_2$ topological order
I'm looking for references (reviews, original articles, lecture notes, etc.) that discuss the connection between the expectation value of Wilson loops (the "disorder parameter" of the system) and ...
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Wilson loop as path integral of parallel transport action
I am trying to get that the path integral of the parallel transport action is the Wilson loop. Here is the setting:
Let $w$ be a complex vector dimension $N$, and $A_{\mu}$ a fixed Yang-Mills ...
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Path ordering in the expansion of a Maldacena-Wilson line
In $4$d Euclidean space, the Maldacena-Wilson line is defined as:
$$\mathcal{W}(C) = \frac{1}{N} \text{Tr} \left\lbrace \mathcal{P} \exp \int_C d\tau \left( i \dot{x}_\mu A_\mu^a(x) + \left| \dot{x} \...
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Is there an analogy for Wilson loops/lines in statistical mechanics?
When reformulated in Euclidean space, quantum field theory bears some strong resemblance to statistical mechanics: for example a scalar field $\phi$ can be seen as a spin $s$ in Landau theory, and the ...
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Does the path of a Wilson line in a quark-correlator matter?
Consider a gauge-invariant quark correlation function nested inside an arbitrary state $|p\rangle$
$$\langle p |\bar \psi(z)_{\alpha,a}\left( W_{\Gamma}(z,0)\right)_{ab}\psi(0)_{\beta,b}|p\rangle \...
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Derivative of a Wilson Line
It's my first post on this website so please excuse any breaches of protocol that I'm unaware of.
I've come across a formula for taking derivatives of Wilson Lines with respect to points on the path,...
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Evaluating Wilson loop in Abelian theory (Srednicki)
In chapter 82 https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
Srednicki comes to the following form for the Wilson loop for free electromagnetic theory:
$$\langle 0|W_C|0\rangle=\exp\left[-\frac{...
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Wilson loop operator in electrodynamics
I'm trying to prove that the Wilson loop operator is well-defined in non-interacting quantum electrodynamics without matter, that is, $\hat{W}(\gamma)$ is a bounded operator on the Hilbert space.
...
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