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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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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Confinement in 3+1 dimensions from confinement in 2+1 dimensions

It is well known that Yang-Mills theories in $2+1$ dimensions exhibit the color confinement property. This property is characterized by the average of a Wilson loop that is the exponential of a term ...
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Is a Wilson line evaluated in zero cosmological constant equal to correlation functions in 2D CFT's?

The path Integral defined by the Wilson lines over some connection $A\subset Hol$ for correlation functions dominated by the vacuum block is $e^{2iL_0}$ evacuated at <0|$e^{2iL_0}$|0>. Does this ...
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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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Can an instanton be a Wilson loop?

I was wondering if one could write an $\mathrm{SU}(N)$-instanton as a Wilson loop. Since in temporal gauge, I can interpret a YM-instanton on $M\times\mathbb{R}$ as a one-parameter family of gauge ...
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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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6 votes
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Kinds of Wilson Loops in a $U(1)$ Chern-Simons Theory

Pardon the potentially easy question, but I am currently reading through a paper by Seiberg and Witten [1], and while reading appendix C I'm not sure where some of these results are coming from. The ...
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How do we derive the Hamiltonian of the Wilson Loop Action?

I'm reading Fradkin and Susskind's 1978 paper "Order and disorder in gauge systems and magnets" to try and understand how they derive the Hamiltonian for the U(1) compact lattice gauge theory. The ...
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Time ordering identities in integration over gluon fields

My question arised when trying to compute the Wilson Loop of a hybrid meson. When calculating the loop one has to keep in mind the path ordering and time ordering respectively. I have the following ...
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1 answer
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Is Loop Quantum Gravity related with loops?

I read this article on wikipedia on loops. And I wondered if the loops of loop quantum gravity have the algebraic structure of loops or it's just a coincidence.
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BPS Wilson loop operators and supersymmetries

In recent papers the circular Wilson loop in $\mathcal{N}=4$ SYM is always called a 1/2 BPS operator. So, my initial idea was that a 1/2-BPS operator was an operator that preserves half of the ...
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2 votes
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Evaluation of Wilson loop in QED

I'am trying to figure out the evaluation of the expectation value of the Wilson loop for QED. (Its actually the problem 15.3 in Peskin and Shroeder) Lets say the Wilson loop is $W(x) = \exp(-ie\oint_P ...
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Why don't we add Wilson loops to the SM Lagrangian?

As the title says: why don't we add Wilson loops to common Lagrangians such as the Standard Model? They're gauge invariant and (correct me if I'm wrong, not sure on that) are renormalizable. Suppose ...
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Minimal area for circular Wilson loops in these coordinates

In all references you can see that the Poincare coordinates are used to get the minimal area for the circular wilson loop. I want to use the metric that is used also for the D3-brane (e.g. see ...
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A question on supersymmetry variation of the Wilson loop in $\mathcal{N}=4$ SYM

The Wilson loop in $\mathcal{N}=4$ SYM is $$W=\frac{1}{N}tr P \exp \int ds (i A_\mu(x) \dot{x}^\mu+\Phi_i(x)\theta^i|\dot{x}|).\tag{2.3}$$ In order to check whether this operator is supersymmetric I ...
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Wilson loop and Polyakov loop

As I understand, the Wilson line is the operator $W(x) = P\exp(i\int_{xi}^{xf} A.dx)$, where $P$ is path ordering. The Polyakov loop $P(x)$ on the other hand is the trace of the Wilson loop $W(x)$ ...
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False solution of Landau Hamiltonian

The Landau Hamiltonian in 2D is given (in natural units $q=c=2m=1$) by $$ \hat{H} = (\hat{\vec{p}}-\vec{A}(\hat{\vec{x}}))^2 \,,$$ where $\vec{A}$ is the magnetic vector potential field. We know that ...
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Generalization of Chern-Simons Wilson Line

The Wilson line in Abelian Chern-Simons theory is $$\langle\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}\right)\rangle=\int\mathcal{D}A\exp\left(i\int dt\dot{x}^{\mu}A_{\mu}+\frac{i}{4\pi}\int A\wedge F\...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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Expectation value of a path-ordered exponential

Let us define our path-ordered operator $\overrightarrow{U}\left(t_1,t_2\right)$: $$ \overrightarrow{U}\left(t_1,t_2\right)=\overrightarrow{\mathcal{P}}\exp\int_{t_1}^{t_2}dt\,\mathcal{O}\left(t\...
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2 votes
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Questions on how Wilson loops relate to field & charge conservation, and lattice QFT

The path-ordered exponential from which the Wilson loop is traced is, crudely, $$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$ which returns a matrix $\...
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9 votes
2 answers
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What is the physical meaning of Wilson loops?

I'm a mathematician trying to get some very basic physical intuition on gauge theories, so I apologize if what follows is really naive. My first super elementary question is: Am I right to think ...
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Are these the only gauge-invariant functions of $A_\mu$?

I know off course that $F_{\mu\nu}$ is a gauge invariant function of $A_\mu$ in the abelian case. Also we have $\epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}F_{\mu\nu}$ in that case. Are there any ...
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