Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
0
votes
1answer
21 views
Integer quantum Hall conductance and time-reversal symmetry
If we have a (2+1)-dimensional electronic gapped system with a unique ground state and it has a nonzero integer quantum Hall conductance, then the system (or its ground state) must break the time-...
3
votes
0answers
53 views
What is knotted in EM and GR?
I found this paper with beautiful illustrations: Helicity, Topology and Kelvin Waves in reconnecting quantum knots, and this one which seems to describe something closely analogous: New knotted ...
0
votes
0answers
22 views
Non-abelian anyons, relation between representation of braid group and fusion rules
As far as I understand, anyons correspond to fields that live in the representation space of some (unitary?) representation of the braid group. One-dimensional representations commute and give rise to ...
1
vote
1answer
60 views
One-dimensional $SU(3)$ Heisenberg Model, the non-linear sigma model, $\theta$-term
Let's consider a one dimensional $SU(N)$ antiferromagnetic Heisenberg Model with an irreducible representation and its conjugate on alternating sites, such that they correspond to a Young tableaux ...
1
vote
0answers
27 views
Aharony-Bernman-Jafferis-Maldacena (ABJM) and k=1 Chern Simons matter
I have read recently that the partition function / half-BPS wilson vev (w/ NG probe) of a Chern-Simons matter theory with N=6 U(N)k x U(N)-k super-conformal symmetry (ABJM) on S3 is proportional to ...
2
votes
1answer
41 views
Is a constant electric field CP violating?
Consider, for instance, a fundamental massless three-form field $C_{\alpha\beta\gamma}$ in the Coulomb phase:
$$
\mathcal L = E_{\mu\alpha\beta\gamma}E^{\mu\alpha\beta\gamma} + C_{\alpha\beta\gamma}J^...
2
votes
0answers
75 views
A good instruction on Symmetry enriched Topological phases
I am looking for a good introduction to SETs, and topologically ordered phaeses it should be something describing first principles and gives a good explanation on the basics and the logic of this ...
0
votes
0answers
34 views
$\theta$ Term In 2d for $U(1)$ and $SO(2)$
For the $U(1)$ gauge theory in 2d, there can be a theta term
$$\frac{\theta}{2\pi}\int_{M} dA$$
where $A$ is the $U(1)$ gauge field and $\theta\sim \theta+2\pi$.
However, it is known that $U(1)=SO(2)...
5
votes
2answers
77 views
Braiding matrix from CFT first principles
Various CFT models are known to produce representations of braid groups.
A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps ...
1
vote
1answer
44 views
Is the quantum Hall state a topological insulating state?
I am confused about the quantum Hall state and topological insulating states.
Following are the points (according to my naive understanding of this field) which confuse me:
Topological insulator has ...
2
votes
1answer
60 views
Supercharge transformation rules
Consider ${\cal N}=2$ supersymmetry with $SU(2)$ global symmetry group. Then both supercharges $Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$ transform by 2 dimensional representation of $SU(2)$. Denote $SU(2)_I$ ...
1
vote
0answers
70 views
On string-like excitations in (3+1)d discrete gauge theory
(3+1)d discrete $G$-gauge theory (untwisted Dijkgraaf-Witten theory) has both point-like and loop-like excitations;
Point-like excitation is an electric charge labeled by an irreducible ...
2
votes
1answer
87 views
Chern-Simons and framing dependence$.$
According to ref.1, the correlation functions of a Chern-Simons theory are topological invariants, up to the so-called framing, that is, the trivialisation of $TM\oplus TM$. The origin of this framing ...
5
votes
1answer
73 views
How does extending a Chern-Simons theory to the bulk fix potential singularities?
According to ref.1 (§A.3), the naive definition of Chern-Simons
$$
S[A]=k\int_M \mathrm{CS}[A]\tag{A.17}
$$
is ill-defined, because $A$ may have "Dirac string singularities". The solution is to extend ...
5
votes
0answers
104 views
A question about Witten's paper on QFT and Jones polynomial
So I have been reading the famous paper on quantum field theory and Jones polynomial and have the following questions:
On P.31 (381), it was said that the eigenvalues of $B$ are $$λ_i =
±\exp(i\...
1
vote
0answers
50 views
How exactly does a spin TQFT depend on the spin structure?
Take a spin Chern-Simons TQFT, such as $U(N)$ or $SO(N)$ with odd level. Such system depends on the spin structure of the underlying manifold.
But how exactly does the theory depend on the spin ...
3
votes
0answers
50 views
What is the topological data for $(\mathbb Z_n)_p$ theories?
Consider the 3d TQFT described by the Lagrangian (Dijkgraaf-Witten with gauge group $\mathbb Z_n$ at level $p$):
$$
\mathcal L=\frac{n}{2\pi} B\wedge\mathrm dA+\frac{p}{4\pi}A\wedge\mathrm dA
$$
with $...
2
votes
0answers
125 views
Deriving Chern-Simons term from path integral representation of the first quantized non interacting many body Hamiltonian
This is an exercise from condensed matter filed theory book of altland and simons.
Exercise
Subject the first quantized many particle hamiltonian $H=\sum_{i=1}^{N}\frac{p^{i}{^{2}}}{2m}+V(x^i)$ to ...
5
votes
1answer
104 views
APS $\eta$-invariant and spin-Ising TQFT
I am interested in the relation between the Atiyah-Patodi-Singer-$\eta$ invariant and spin topological quantum field theory. In the paper Gapped Boundary Phases of Topological Insulators via Weak ...
4
votes
0answers
103 views
Is $U(2)_{2, 1}$ Chern Simons Theory Completely Trivial?
I am using the method outlined in appendix C4 of a paper by Seiberg and Witten [1] to calculate the statistics of lines in $U(2)_{2, 1}$. However, this method shows that all lines are trivial.
...
8
votes
0answers
100 views
Lattice QFT of the Jones Polynomial
Start with a gauge theory with Chern-Simons action
$$ S[A] = \frac{k}{4 \pi} \text{Tr} \intop_{M} \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $$
and a Wilson loop observable in the ...
1
vote
0answers
41 views
Is it possible to directly derive the $K$ matrix for a topological order described by a gauge-theory Hamiltonian?
To be concrete, Let us consider a $Z_2$ gauge theory in the deconfined phase coupled to matter field,
\begin{align}
S_{Z_2}=\beta\sum_{\vec r\mu}\phi(\vec r)U_\mu(\vec r)\phi(\vec r+\vec e_{\mu}) + K \...
5
votes
1answer
111 views
Composite Operators in Topological/Conformal Field Theory
In virtually any reference on conformal or cohomological field theory, you'll eventually see a formula like: $$\frac{\delta}{\delta h^{\alpha \beta}}\langle\mathcal{O}_1(x_1)...\mathcal{O}_n(x_n)\...
1
vote
1answer
70 views
Winding number and area formula
I need to show some properties of the topological expression involving a map $\vec{n}(x): S^2 \rightarrow S^2$
$$W=\frac{1}{4\pi}\int \vec{n} \cdot (d\vec{n} \wedge d\vec{n}),$$
but I am not very ...
1
vote
0answers
42 views
Is Yang-Mills theory as a topological quantum field theory a good alternative?
Suppose we know a classical solution for gauge Boson fields $A_{\mu,c}$ and Fermion fields $\psi_c$ and now we want to consider ist Quantum fluctuations. These fluctuations arise from loop corrections,...
0
votes
0answers
26 views
Metric-dependent operator and its variation
In the proof that the metric variation of correlation functions for certain operator ($\mathcal{O}$) vanishes, do we always need to assume that operator itself is metric independent?
In other words, ...
2
votes
0answers
30 views
Are twisted versions of supersymmetric field theories supersymmetric on any general four manifold?
Witten in his famous paper (Topological Quantum Field Theory) writes -- " I do not know whether twisted versions of other $\mathcal{N} \ge 2$ supersymmetric field theories will similarly be ...
2
votes
0answers
79 views
Understanding the fractional quantum Hall effect in Chern-Simons formalism described in Wen's book
So I study fractional quantum hall effect with Chern-Simons formalism by using Wen's book, this is an excellent book, but it assumes that you know field theory very well thus it has gaps between steps....
3
votes
0answers
81 views
Twisted Chern-Simons, and Twisted Wess-Zumino Term
I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten.
Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
12
votes
1answer
256 views
Faddeev-Popov Determinant of Chern-Simons Theory
I am asking this question in order to figure out the expression of the Faddeev-Popov determinant given by Edward Witten is his paper "Quantum Field Theory and Jones Polynomial".
Starting from the ...
0
votes
0answers
53 views
3d TQFT for Fibonacci (Yang -Lee) anyons
What is the 3d TQFT whose Wilson line produces Fibonacci (Yang -Lee) anyons?
I heard that 3d $SO(3)_3$ Chern-Simons theory produces the correct physics for Fibonacci anyon ($e$). How to show it?
If ...
6
votes
1answer
190 views
The Hilbert space of Chern-Simons on a torus, part one$.$
There is a key result in 2+1 dimensional Chern-Simons theory, which was first discussed in ref.1.: the Hilbert space of the theory, when quantised on $T^2\times\mathbb R$, is isomorphic to
$$
\frac{\...
0
votes
0answers
30 views
Sign on Particle-Vortex Duality from 3d Bosonization
This is most likely a question with a very simple answer but well, on the paper "Particle-Vortex Duality from 3d Bosonization" by Karch and Tong, on page 11 it is claimed that the first three terms on ...
3
votes
0answers
59 views
Does topological mass imply preservation of global symmetry whose current is topological?
This question is general but the motivation for it lies within the paper "A Duality Web in 2+1 Dimensions and Condensed Matter Physics". On pages 20-23, they consider a system which has four phases ...
6
votes
1answer
186 views
Why does a monopole operator break the global symmetry with topological current?
I am currently reading the paper "A Duality Web in 2+ 1 Dimensions and Condensed Matter Physics" by Seiberg et al, and on page 22 they add to the Lagrangian a monopole operator of the form $\phi^{\...
0
votes
0answers
54 views
Relationship between TQFTs and gauge theories
The motivation for this question comes from observing that the toric code model has the properties of a TQFT (robust ground state degeneracy) and of a $\mathbb Z_2$ gauge theory (local spin flips don'...
2
votes
0answers
69 views
Formalization of the concept of a topological charge
I want to write precisely in mathematical terms what a topological charge is.
This is what I have, but I am not sure of how correct it is. Let $M$ be spacetime. Quantization of $M$ in some QFT will ...
2
votes
2answers
127 views
Braiding matrix in Chern-Simons theories$.$
Consider a Chern-Simons system with gauge group $G$ and level $k$. Such a system can be used to model anyons, where the latter are identified with the integrable representations of $G$.
One of the ...
3
votes
0answers
140 views
Which Chern-Simons TQFTs are spin?
Refs.1&2 prove several level/rank dualities among different 3d Chern-Simons theories. An important point is that some dualities involve, on one side, a theory that depends on the spin structure, ...
3
votes
0answers
164 views
Wess-Zumino-Witten vs. Yang-Mills-Chern-Simons and Kac-Moody$.$
There is a really nice (holographic) duality between 2d Wess-Zumino-Witten and 3d Yang-Mills-Chern-Simons models (cf. Ref.1). For example, for a given gauge group $G$, the spectrum of both theories is ...
1
vote
1answer
127 views
What makes spin 1/2 anyons special?
If the spectrum of a TQFT contains a fermion, the theory becomes a spin-TQFT, and it depends on the spin structure of the manifold (cf. 1505.05856). On the other hand, if no such anyon exists, the ...
8
votes
1answer
196 views
Anyon condensation – what's the precise definition?
Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete ...
2
votes
0answers
55 views
Why is the semiclassical approximation of the abelian Chern-Simons theory exact?
I was told that in abelian Chern-Simons theory (say, with a general level matrix $K$), semiclassical approximation is exact because there is no trivalent vertex, which in non-abelian case makes the ...
2
votes
0answers
57 views
Can topological degrees of freedoms interact?
Question
Can topological degrees of freedom interact with:
other topological dof's
other local dof's
Premise
For concreteness, I can't understand why the following term will not be allowed in a ...
2
votes
0answers
54 views
Let's put a $\mathrm{Pin}$ in it$.$
As it turns out, the low-energy behaviour of (gapped) systems with fermionic degrees of freedom can be described by a so-called Spin TQFT. Such models require the introduction of a (S)pin structure. ...
4
votes
0answers
108 views
Precise justification for quantization of Chern-Simons level
Consider $U(1)$ Chern-Simons theory on some three-manifold M:
$$S = \frac{k}{4\pi}\int_M A \wedge dA.$$
The standard argument for why we require $k\in \mathbb{Z}$ comes from demanding invariance under ...
2
votes
0answers
39 views
Relation of SPT phases with different boundary conditions
Using the definition that two SPT phases are distinct if they can't be connected by a symmetric finite depth local unitary, how does one relate systems with different boundary conditions?
For example,...
3
votes
1answer
85 views
Why is the flux quantized in 4D quantum Hall effect?
I am reading "Topological Field Theory of Time-Reversal Invariant Insulators" by Qi, Hughes, and Zhang (https://arxiv.org/abs/0802.3537). It argues that time reversal invariant (TRI) insulators in 2+1 ...
8
votes
1answer
185 views
Allowed anyons for Chern-Simons at level $k.$
Ref.1. proves that the allowed representations of Chern-Simons $\mathrm{SU}(2)_k$ are those with dimension
$$
\dim(R)\le k+1\tag{7.53}
$$
Question: Is the generalisation of $(7.53)$ to arbitrary $N$ ...
5
votes
1answer
174 views
Meaning of Yang-Baxter equation for classical $r$-matrix
I'm reading this [math/9802054] paper on the structure of the phase space of Chern-Simons TQFT. I'm stuck at the definition of the classical $r$-matrix, which goes as follows:
This might sound dumb, ...
