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Questions tagged [topological-field-theory]

Use this tag for topological field theory (Tft) and topological string theory (tst) questions.

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Non-abelian Yang-Mills in 1+1 dimensions

Abelian electrodynamics in 1+1 dimensions is solvable, in the sense that we can find the space of solutions for the equation of motions $\partial_\mu F^{\mu\nu}=0$. To see this, one first notice that ...
BVquantization's user avatar
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Non-invertible symmetries: Half gauging and 't Hooft lines

In (2.27) of https://arxiv.org/abs/2205.05086, when performing a gauge transformation of the background gauge field $B \to B +d \Lambda $, the 't Hooft line $H(\gamma)$ transforms as \begin{equation} ...
superyangmills's user avatar
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Abelian Chern-Simons large gauge transform

My question concerns the $U(1)$ Chern-Simons theory with the action $$S = \frac{k}{2\pi}\int A\wedge \mathrm{d}A.$$ In my lecture, it is stated that: A large gauge transformation involves taking $A\...
shamwowexcitante's user avatar
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How is classical Chern-Simons theory topological?

Note: I am using "global" and "topological" somewhat interchangable. This seems to be the case in texts and papers, but please point out if this is inappropriate. Classical Chern-...
Silly Goose's user avatar
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Fermions coupled to BF theory and asymptotic freedom

Suppose we couple $N$ colors of fermions to an $SU(N)$ gauge field $A$, but instead of a Yang-Mills action, there is a BF theory that restricts the gauge field to be flat $dA+A\wedge A\equiv F=0$ (by ...
octonion's user avatar
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What is the boundary action need for topological massive gravity (TMG)?

For pure Einstein gravity with Dirichilet boundary conditions, Gibbons-Hawking-York boundary action is needed to make the variational principle well defined. I am considering the case for topological ...
miranda li's user avatar
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What is the meaning of the statistical gauge field in the fractional quantum Hall effect

I'm a grad student studying the fractional quantum Hall effect. To get started, I read chapter 9.5.1 of A. Altland and B. Simons' Condensed Matter Field Theory. They use the composite fermion (CF) ...
Steffen Bollmann's user avatar
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Obtaining the topological charge

I want to obtain the topological charge or winding number of the map $$ f_n(\mathbf{r})=(\sin \theta \cos (n \varphi), \sin \theta \sin (n \varphi), \cos \theta) $$ and my lecture notes say that it is ...
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Mathematical references for gauge theory in condensed matter physics

I am currently trying to go through some literature on the classification of symmetry protected topological phases. Primarily, I am interested in the classical of topological phases using mathematical ...
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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(...
Alex's user avatar
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Resources: Tensor Categories and Topological Phases of Matter

For a mathematician with knowledge of tensor categories who is interested in the growing application of categorical techniques in topological phases of matter and topological order, along with their ...
3 votes
2 answers
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Is there a Majorana representation for toric code

Kitaev's toric code is known to be the Z2 gauge field theory, which suggests that there might exists a Majorana representation for the toric code, e.g., Majorana + Z2 gauge field. Hence, I wonder if ...
Richard's user avatar
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What's the Newtonian potential in 2+1 gravity?

I understand that there are no propagating degrees of freedom (i.e. gravitational waves) in 2+1 dimensions. There are a couple of arguments to show this. One is to count degrees of freedom of general ...
P. C. Spaniel's user avatar
3 votes
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Holonomies for BF theory

I am trying to understand the basic properties of BF theory, but I am unable to derive the holonomies of the fields. In my case I want to study a more general version of $BF$ theory defined on a $4d$ ...
Truth and Beauty and Hatred's user avatar
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Can Toric Code have a gapless boundary?

The toric code model is known to have two types of "gapped" boundaries, namely, the rough boundary and the smooth boundary. See, for example, Chap. 4.1 of this beautiful review https://arxiv....
Quasiphysics's user avatar
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Gauge connection on topological 3-manifold

To describe a gauge theory on a topological nontrivial 3-manifold we need to consider a good cover of the manifold in contractible open sets with associated set of 3-connections: $A=\{v_a,\lambda_{ab},...
polology's user avatar
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Higher category's consistency relations

I have been reading on higher category and symTFTs. It appears to me that, for higher categories, people seldom mention the consistency relations (like the MacLane coherence theorem in the category ...
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What am I summing over in the Lagrangian of a BF theory

I'm reading the article A pure Dirac's canonical analysis for four-dimensional BF theories. But when I get to the action, written as $$S[\text{A},\textbf{B}]=\int_M \textbf{B}^{IJ}\wedge\textbf{F}_{IJ}...
iakidesantos's user avatar
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1 answer
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Understanding Chern-Simons on non-trivial manifold

I am studying abelian Chern-Simons theory on a non-trivial manifold. Could you let me know how accurate my understanding is? Here's what I figured out: The action of $U(1)$ leaves the action invariant ...
polology's user avatar
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Why is the correlation function of local operators in topological field theory independent of the position they are inserted?

I am reading "A mini course on topological strings"(hep-th/0504147). In the last paragraph of section 3.1, author mentioned that if a topological field theory have general coordinate ...
Yutai Zhang's user avatar
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How to obtain the relation of eta invariant of the trivial gauge field and Chern-Simons invariant of the flat connection?

In Quantum Field Theory and the Jones Polynomial by Edward Witten(1989), how does $\eta(0)$ come from in this equation? $$\frac{1}{2}(\eta(A^{(\alpha)})-\eta(0))=\frac{c_2}{2\pi}I(A^{(\alpha)})$$ $c_2(...
Jimi's user avatar
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Why can Principal $G$ Bundles be Trivialized when $G = SU(N)$?

Reading about TQFT one usually comes about the fact that over 3-manifolds, Simply Connect Lie Group-bundles can be trivialized, yet it is a bit hard to find a clear answer online. Why is that the case?...
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Basic Question on Differential Forms (Chern-Simons Level Quantization)

I came across the following post regarding the boundary term in Chern-Simons theory (specifically the level quantization of the theory). I am new to differential forms so the following questions may ...
tumm's user avatar
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Quantization of string via topological twist

Polyakov action of a bosonic string propagating in Minkowskian spacetime is: $$S[\gamma, X] = \frac{T}{2}\int \mathrm{d}^{2}\sigma{\sqrt{-\gamma}}\gamma^{ab}\partial _{a}X^{\mu}(\sigma)\partial_{b}X^{\...
user avatar
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2 answers
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If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition

Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g. If $(\nabla\times\nabla\Phi)_i = \...
Cuntista's user avatar
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Looking for a source to explain the Process of Topological twisting

as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
2 votes
0 answers
45 views

Normalization in the Abelian Chern-Simons action

In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
Tuhin Subhra Mukherjee's user avatar
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Integrating out a Lagrange multiplies field enforcing a differential form to be "$2\pi$-integral periodic"

In the appendix A (page: 26-27) of the paper https://arxiv.org/abs/hep-th/0108152 by Juan Maldacena, Gregory Moore and Nathan Seiberg, the author writes the following "action" $$e^{-i\int_{...
Tuhin Subhra Mukherjee's user avatar
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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 ...
1 vote
1 answer
126 views

What is a symmetry defect?

I found that it is a normal concept appearing in condensed matter physics and especially topological order field. I have been aware of the topological defect. But what is a symmetry defect? Could ...
user35734's user avatar
3 votes
0 answers
106 views

Fusion 2-categories for string-like excitations: a more concrete description?

I'm familiar with how fusion categories describe the fusion of point-like excitations, and how braided fusion categories describe the fusion of anyons in 2+1D topological order. Concretely, a fusion ...
Lagrenge's user avatar
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Dimensionality of state space of TQFTs

As the title suggests, I am wondering about the dimensionality of state spaces in $d$-dimensional TQFTs. As of yet I have mostly been concerned with the mathematical, functorial definition of TQFTs as ...
Topological Sigma Grindset's user avatar
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Topological charges and universal gravity

After reading two days ago a paper, https://arxiv.org/abs/2307.10365, I read that topological charges DO NOT gravitate. Is that statement right? Why I have doubts: Topological charges have "...
riemannium's user avatar
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1 vote
1 answer
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Can anyons exist on a torus without any additional conditions?

While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
devoted4gainz's user avatar
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0 answers
62 views

What effect does the topological term have in string theory?

In string theory we may have an Polyakov action like: $$S=\int\sqrt{g} (g^{ij}\eta_{\mu\nu}\partial_i X^\mu \partial_j X^\nu + \kappa R) d\sigma^2$$ The curvature $R$ gives a topological term (it is ...
user avatar
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0 answers
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Link complement states in Chern Simons theory

I have been trying to understand how to get an explicit quantum state from a given link in Chern-Simons. Lets say the compact gauge group being SU(2) (this seems to be the most widely studied). I ...
Bizwhiz's user avatar
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2 votes
0 answers
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QED theta term as pullback of a volume form

The well-known QED theta term is $S_\theta = \frac{\theta}{4\pi} \int d^4 x F \wedge F $ where $F=dA$ is the field strength 2-form. Altland and Simons (p.547) introduce a more general category of $\...
catalogue_number's user avatar
2 votes
1 answer
107 views

Sigma models as topological quantum field theories

I'm wondering how sigma models are supposed to define TQFTs. Suppose I want to consider a 2D TQFT with target $X$ (see page 15 of https://www.ams.org/bookstore/pspdf/ulect-72-intro.pdf)*. According ...
JLA's user avatar
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2 votes
1 answer
281 views

Is it actually true that Chern-Simons theory is topological?

Chern-Simons theory has action $$\tag{1} S = \frac{k}{4\pi}\int_X tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$ Here, $X$ is some compact 3-manifold, perhaps with boundary, and $A$ is a connection ...
nodumbquestions's user avatar
1 vote
0 answers
103 views

How do you calculate the partition function on a manifold-with-corners in extended TQFT?

In Atiyah's formulation, a Topological Quantum Field Theory (TQFT), is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns: \begin{align} \text{Closed compact $(d-1)$-manifolds} &\to \...
nodumbquestions's user avatar
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1+1D simple vacuum EFE solution

Can there be any solutions for simple vacuum Einstein Field Equations in 1+1D (1 space and 1 time dimension) i.e $R_{\mu\nu} = 0$ except for flat space? I tried different combinations of random ...
Nayeem1's user avatar
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2 votes
1 answer
323 views

Why does additional term to electromagnetic Lagrangian leave Maxwell's equations unchanged?

The addition of $$\mathcal{L}' = \epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \vec{E}\cdot\vec{B}$$ to the electromagnetic Lagrangian density leaves Maxwell's equations unchanged (shown ...
Aiden's user avatar
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1 vote
0 answers
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What are non-propagating fields?

I have read at different places that in 3 spacetime dimensions, there are NO propagating gravitational degrees of freedom. This seems to imply that we have only "non-propagating" degrees of ...
baba26's user avatar
  • 503
4 votes
0 answers
73 views

Fundamental group of configuration space of gauge theories

If I consider the space of all the gauge fields $A_{\mu}$ (call this $A$) and a proper gauge group $\Omega_*$, I know that the fundamental group $\pi_1(A)=0$ and the for the gauge group, for example $...
nabla_quadro's user avatar
1 vote
0 answers
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How do equations of motion in BF theory imply triviality of powers of observables?

Following the lectures of Nathan Seiberg at PiTP in 2015 https://www.youtube.com/watch?v=pqgNrVTQ4yM&t=666s, consider $U(1)$ BF theory in 2D $$S(B,A)=\frac{n}{2\pi}\int_\Sigma B\text{d}A,$$ and ...
Ivan Burbano's user avatar
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2 votes
1 answer
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Simplicity constraints from $SO(4)$ Plebanski action

The $SO(4)$ Plebanski action yields a first order formulation of Euclidean General Relativity as a constrained (topological) BF-theory. It depends on a $so(4)$ connection 1-form $\omega^{IJ} = \omega_{...
juli073's user avatar
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1 vote
0 answers
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Can any useful physical theories other than TQFTs be formulated on a smooth manifold without a metric structure?

The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The ...
tparker's user avatar
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2 votes
0 answers
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Duality and corrections to second-order gravity without and with torsion terms

Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like: $$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{...
riemannium's user avatar
  • 6,561
0 votes
2 answers
111 views

What is the direct sum of 1d domain walls in Toric code model?

I have read this paper:"An invitation to topological orders and category theory" (https://arxiv.org/abs/2205.05565v2). In page 93 and page 109, they show the result of fusion of simple 1d ...
ph3's user avatar
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2 votes
0 answers
52 views

Topological classification of (classical, Abelian) vortices on a lattice

Consider the XY model on the square lattice. A field configuration $\theta$ is specified by an element of the Abelian group $\mathbb{R}/2\pi \simeq U(1)$ at each vertex of the lattice. The gradient of ...
Kai's user avatar
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