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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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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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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Does gauge invariance of scalars/fermions in the adjoint representation induce the existence of Wilson loop and (then) covariant derivative?

First, for the known case of $U(N)$ gauge invariance we have scalars (it works for fermions too) transforming as (fundamental representation) $$ \phi(x)\to V(x)\phi(x), \ \ V(x)\in U(N) $$ So then we ...
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Coordinate space of loops in loop quantum gravity

I have a basic and perhaps admittedly misguided question about the loops in LQG. Broadly speaking I think I understand the logic that lead to them, namely that working in the ADM formalism (using 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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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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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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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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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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Linear Potential from Wilson Loop

It is well known that if you compute the expectation value of the Wilson loop along a suitable rectangle you can get the Coulomb potential $$\langle W(\mathcal{C})\rangle\sim e^{TV(R)} , \ \ V(R)\sim ...
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Wilson Loops and Confinement in QED

In [1] Kenneth G. Wilson proposed a mechanism for confinement using lattice paths what leds him to the concept of Wilson loop. It seems to me that he is using mainly a single abelian field. He says ...
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Polyakov Loop and Chemical Potential

I have read in a paper (http://arxiv.org/abs/1203.3556) that in a thermal field theory, the chemical potential is $\mu=T \ln P$ where $$T^{-1}=\int_{0}^{\beta} \sqrt{-\xi^2}dt,$$ $\xi$ is $\partial_t$,...
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Path-Integral of Charged Particle in Chern-Simons Gauge Fields

From the paper "Fermi-Bose Transmutations Induced by Gauge Fields" by Polyakov, http://inspirehep.net/record/22956 http://dx.doi.org/10.1142/S0217732388000398 the theory in 3D, $$\mathcal{L}=\sum_{...
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Simple explanation for what a torsor is

I am studying Chris Elliott's notes on Line and Surface Operators in Gauge Theories (available here). In the notes, there's a mention of the fact that (for $G = U(1)$), $$W_{\gamma, n}(A) = e^{in\...
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Partial derivatives of Wilson like integrals

I have a one-form field on Euclidean space. Suppose we integrate it over a loop around the specific point $x$. $$I(x)=\int_xU.$$ I want to calculate the partial derivatives of this integral respect to ...
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Wilson loops, confinement and chiral symmetry relations

As per my limited understanding of confinement, I understood it as phenomenon between fermions of a gauge theory, such as $SU(3)$ gauge theory in the low energy limit (where the coupling constant ...
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Wilson loop in AdS/CFT : string interpretation

It is well known that Wilson loop is a quite hard observable to compute. In the case in which the QFT is dual to a gravitation theory in AdS space, we can use holography to compute the Wilson loop, ...
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In what types of QFTs are the Wilson loops of interest?

I have a very basic question about Wilson lines (WL). This is what I know about the WL: WL help us to learn about the important properties of gauge fields (treated as connections on the space of ...
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272 views

A Wilson line property

Consider a path, say $P$, and three points on it like $x,y,z$. If there were infinitesmally closed, then the following relation would be true. $$W_{P}(x,y) W_{P}(y,z)=W_{P}(x,z)$$ If the Lie algebra ...
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The gauge transfomation of the Wilson line

I have a question in Capter 15 of Peskin & Schroeder. The gauge transformation here in its infinitesimal form: \begin{cases} \psi(x) \to V(x)\psi(x) \quad \quad \quad \quad \quad \quad \,\,\,\, \...
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PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2', '...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix ...
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A derivative about chiral current in Peskin's book

In Peskin's book (an introduction to QFT), Page 655, the axial vector current is defined as follows, \begin{eqnarray*} j^{\mu5} & = & \text{symm }\lim_{\epsilon\rightarrow0}\bigg\{\bar{\psi}(x+...
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Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The Berry phase accumulated on a path can be described by a matrix when we look at adiabatic time evolution with a Hamiltonian with degenerate energy levels. The Berry phase matrix is given by $$ \...
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Circular Wilson Loop in AdS/CFT

I'm trying to get the AdS solution to the circular wilson loop. The standard AdS metric is: $ds^2 = \frac{L^2}{z^2}(\eta_{\mu \nu} dx^{\mu} dx^{\nu} + dz^2)$ If I take the circle of radius R at the ...
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$\mathcal{N} = 4$ Super-Yang Mills propagators

In $\mathcal{N} = 4$ Super-Yang mills there are only massless particles. If one wishes to obtain a heavy quark one can see the SYM theory as a stack of (N+1)-branes in AdS$_5 \times$S$^5$ where one ...
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Wilson Loop in AdS/CFT

In AdS/CFT correspondence one can compare results in $\mathcal{N}=4$ SYM with string theory type IIB in $AdS_5 \times S^5$. One of the observables that it's possible to get non-perturbative results is ...
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Supertrace of holonomy of commutator

On page 47 of Surface operators in four-dimensional topological gauge theory and Langlands duality by Kapustin et al., the following expression is given \begin{equation} \delta\mathcal{N}=d(\omega_\...