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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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$\theta$ 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)...
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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 ...
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1answer
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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 ...
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1answer
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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 $SU(2)_I$ ...
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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 ...
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1answer
70 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 ...
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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 ...
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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\...
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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 ...
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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 $...
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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 ...
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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 ...
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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. ...
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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 ...
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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 \...
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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)\...
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1answer
66 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 ...
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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,...
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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, ...
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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 ...
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What Exactly is a “Supersymmetric Cycle” in String Theory?

In a lot of physics papers I hear talk of "wrapping a stack of D-branes on a supersymmetric cycle in a Calabi-Yau threefold $X$" and I'm wondering if I understand this correctly from a mathematical ...
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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....
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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 $P\rightarrow M$ ...
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1answer
220 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 ...
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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 ...
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1answer
172 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{\...
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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 ...
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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 ...
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1answer
161 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^{\...
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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'...
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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 ...
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2answers
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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 ...
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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, ...
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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 ...
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1answer
113 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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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,...
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1answer
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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 ...
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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 $N$ ...
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160 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, ...
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1answer
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When is a Wess-Zumino term well-defined?

According to wikipedia, a Wess-Zumino term is well-defined when the Lie group (target space) $G$ is compact and simply connected, because that implies that $\pi_2(G)$ is trivial. But there are Lie ...
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1answer
518 views

The system is topological; so what?

Lately I've been studying topological quantum field theory, mainly from mathematically oriented sources and, while I understand why a mathematician would care about TQFT's, I'm not yet quite sure why ...
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2-nd quantized TQFT formalism?

Suppose that we have a certain TQFT in the Atiyah-Singer sense. It is given by a functor $Z$ which associates: To connected oriented $n-1$-manifolds $a, b, \dots$ (in what follows called compact ...
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Loop corrections to string scattering via gluing axiom

The usual method of calculating higher-loop corrections to string scattering is to consider the stringy CFT on Riemann surfaces with increasing genus, and take the integral over the Teichmuller space. ...
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How is BRST symmetry related to local integrals of motion?

I'm hoping someone can confirm or check my reasoning below: In this wiki, they describe caos in a classical system as the spontaneous symmetry breaking of a BRST. In this stackexchange, they clarify ...
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1answer
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Worldline in very basic TQFT question

Can somebody explain this diagram to me. You have pair production and pair annihilation, and the worldlines of the particles in between the two events. Why are the worldlines a loop, rather than a ...