Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
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Why can't there be an infinite number of simple objects in an anyon model?
It is a well-established fact that topological excitations (anyons) in 2D topologically-ordered systems are described by unitary modular tensor categories, see, e.g., Appendix E in Kitaev (2006). One ...
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Question about Path Integrals and Exchange Statistics in Steve Simon's "Topological Quantum"
In the introduction to the path integral approach leading to exchange statistics for many particles, Steve Simon breaks up the sum of paths into two types: paths where particles do not exchange (type ...
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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 ...
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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}
...
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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\...
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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-...
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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 ...
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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 ...
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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) ...
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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(...
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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 ...
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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 ...
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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 ...
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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$ ...
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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....
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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},...
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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}...
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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 ...
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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 ...
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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(...
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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 ...
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Quantization of string via topological twist
Polyakov action of a bosonic string propagating in Minkowskian spacetime is:
$$S[\gamma, X] = \frac{T}{2}\int \mathrm{d}^{2}\sigma{\sqrt{-\gamma}}\gamma^{ab}\partial _{a}X^{\mu}(\sigma)\partial_{b}X^{\...
user361492
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If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition
Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g.
If $(\nabla\times\nabla\Phi)_i = \...
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Looking for a source to explain the Process of Topological twisting
as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
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Normalization in the Abelian Chern-Simons action
In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
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Integrating out a Lagrange multiplies field enforcing a differential form to be "$2\pi$-integral periodic"
In the appendix A (page: 26-27) of the paper https://arxiv.org/abs/hep-th/0108152 by Juan Maldacena, Gregory Moore and Nathan Seiberg, the author writes the following "action"
$$e^{-i\int_{...
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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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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 ...
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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 ...
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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 ...
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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....
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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 ...
user84158
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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 $\...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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_{...
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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 ...
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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_{...
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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 ...
- 3
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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 ...
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