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
$$ S = \int ...
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Are pseudo Riemannian manifolds with identical Wilson loops isometric?
It is well established that in gauge theory, the Wilson loops of the theory determine the gauge potential up a gauge transformation. That is, two gauge potentials $A_\mu$ and $B_\mu $ produce the same ...
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What is a non-linear space of connections
In the book "Loops Knots Gauge Theory and Quantum Gravity" when trying to define a loop representation, one needs to integrate over the space of connections (modulo Gauge transformations). ...
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Singularity of free energy in $\mathbb{Z}_2$ lattice gauge theory
I'm currently reading Kardar's Statistical physics of Fields. In the book, the $\mathbb{Z}_2$ lattice gauge theory is constructed as the dual of the 3d Ising model.
(Note: the Hamiltonian is $H = \...
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How does Witten's path integral know about changing crossings?
At a crossing of a knot, if I change the crossing by swapping the two lines, the knot is changed, along with its Jones Polynomial. Witten's path integral
$$
\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 ...
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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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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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