Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
512
questions
0
votes
0
answers
15
views
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 ...
2
votes
0
answers
33
views
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) ...
1
vote
0
answers
45
views
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 ...
- 123
2
votes
0
answers
32
views
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 ...
1
vote
1
answer
64
views
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(...
- 143
1
vote
1
answer
93
views
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
1
answer
115
views
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 ...
- 81
1
vote
1
answer
59
views
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 ...
- 4,689
3
votes
0
answers
129
views
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$ ...
1
vote
1
answer
143
views
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....
1
vote
0
answers
47
views
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},...
- 144
4
votes
1
answer
128
views
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 ...
- 498
1
vote
0
answers
61
views
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}...
- 11
1
vote
1
answer
169
views
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 ...
- 144
0
votes
0
answers
17
views
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 ...
2
votes
1
answer
58
views
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(...
- 23
3
votes
1
answer
134
views
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?...
- 921
1
vote
0
answers
85
views
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 ...
- 25
5
votes
1
answer
115
views
Quantization of string via topological twist
Polyakov action of a bosonic string propagating in Minkowskian spacetime is:
$$S[\gamma, X] = \frac{T}{2}\int \mathrm{d}^{2}\sigma{\sqrt{-\gamma}}\gamma^{ab}\partial _{a}X^{\mu}(\sigma)\partial_{b}X^{\...
user361492
1
vote
2
answers
133
views
If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition
Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g.
If $(\nabla\times\nabla\Phi)_i = \...
- 310
0
votes
0
answers
23
views
Looking for a source to explain the Process of Topological twisting
as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
2
votes
0
answers
43
views
Normalization in the Abelian Chern-Simons action
In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
2
votes
0
answers
59
views
Integrating out a Lagrange multiplies field enforcing a differential form to be "$2\pi$-integral periodic"
In the appendix A (page: 26-27) of the paper https://arxiv.org/abs/hep-th/0108152 by Juan Maldacena, Gregory Moore and Nathan Seiberg, the author writes the following "action"
$$e^{-i\int_{...
1
vote
0
answers
38
views
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
93
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 ...
- 81
3
votes
0
answers
101
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 ...
- 853
0
votes
1
answer
159
views
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 ...
0
votes
0
answers
29
views
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 "...
- 6,491
1
vote
1
answer
81
views
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....
0
votes
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 ...
user84158
0
votes
0
answers
16
views
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 ...
- 103
2
votes
0
answers
65
views
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 $\...
- 2,829
2
votes
1
answer
105
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 ...
- 1,687
2
votes
1
answer
219
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 ...
- 1,405
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 \...
- 1,405
0
votes
1
answer
78
views
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 ...
- 997
2
votes
1
answer
311
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 ...
- 1,661
1
vote
0
answers
117
views
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 ...
- 431
4
votes
0
answers
69
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 $...
- 159
1
vote
0
answers
65
views
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 ...
- 3,875
1
vote
1
answer
80
views
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_{...
- 11
1
vote
0
answers
34
views
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 ...
- 47.5k
2
votes
0
answers
65
views
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_{...
- 6,491
0
votes
2
answers
110
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 ...
- 3
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 ...
- 3,700
1
vote
0
answers
60
views
Topological Insulators with different spin band
To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
- 137
2
votes
0
answers
116
views
Gauging the one form symmetries in the GKSW paper
Following is few of the many questions I have regarding the paper "Generalized Global Symmetry" by Gaitto et al. (https://arxiv.org/abs/1412.5148).
In Sec. 4 of the paper they considered the ...
3
votes
0
answers
47
views
Topology active subgroup for the QCD vacuum
I've been reading about the nontrivial topological structure of the QCD vacuum and, when studying the different equivalence classes created by the pure gauge fields, all papers say that it is possible ...
- 96
2
votes
1
answer
132
views
Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
- 669
1
vote
1
answer
139
views
Why topology indicates edge state?
I have learned some models with topology like Kitaev chain and SSH. All of them possess edge states if the bulk is topologically non-trivial. The reading materials did the calculation and verified ...
- 159