Questions tagged [topological-field-theory]

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

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Magnetization and Polarization in an electromagnetic field theory

I am currently reading through a paper by Hughes and Ramamurthy (ref: https://arxiv.org/abs/1508.01205), which describes the electromagnetic response of a line-node semimetal by the action $$S[A,B] = \...
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1answer
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Why is $T^*S^3$ a conifold?

So, I was reading the famous Gopakumar Vafa paper, and they mention that $T^*S^3$ is a conifold. Why is this the case? I would naively expect $T^*S^3$ to be basically the same everywhere ($S^3$ is a ...
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lecture notes about the relation between algebraic topology, topological quantum field theory, condensed matter physics [closed]

I am an undergraduate student and I am very interested in topology with its application in physics. So last year I've read some books about this field, mainly about topological soliton, some ...
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1answer
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A-brane boundary conditions

This question concerns the boundary conditions that A-branes solve. Consider the following problem: Suppose that an A-brane wraps a submanifold $Y$ of $X$. Let $\mathcal{L} \rightarrow Y$ be a rank ...
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Holomorphic instantons in target torus

For computing instantons contributions from worldsheet torus to target torus, one can evaluate zero modes contribution of genus 1 partition function given by following expression: $$Tr(-1)^FF_LF_Rq^{...
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1answer
286 views

Is it possible to bound a single $0$-brane to a $4$-brane?

I'm studying the Jafferis solution for twisted $N=4$ Yang-Mills theory in four dimensions from the paper Crystals and intersecting branes. Consider the problem of computing the charges of the allowed ...
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1answer
138 views

Holomorphic anomaly at genus 1

Partition function on torus can be defined using a generalized Witten like index as given below: $$F_1=\int_\mathbb{T}\frac{d^2\tau}{\tau_2} Tr(-1)^F F_LF_R \;q^{L_0} \bar{q}^{\bar{L_0}},$$ where $\...
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1answer
140 views

References on mathematical stacks for a string theory student

This question was posted on mathoverflow (here) without too much success. I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" ...
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Why string amplitude could be computed by path integral of string field theory?

I am trying to understand BCOV's paper: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. In this paper, it was shown that the higher genus string amplitude for B-...
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3answers
173 views

How is cohomology theory used in quantum field theory?

Quantum field theory uses a large amount of mathematics and I was wondering about some applications of cohomology theory in QFT, I understand it has applications in string theory but I was wondering ...
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Staggered Zeeman field in topological magnetic insulators

I was reading the following paper. However, I do not understand a crucial part of their argumentation. They add a parity (P) and time (T) symmetry breaking term to the Hamiltonian in eq (2). Then they ...
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Topological Descent Equation

Assume that we have a cohomological field theory, with an odd symmetry generated by an odd operator $Q$ and an exact energy momentum tensor $T_{\mu\nu}=[Q,G_{\mu\nu}]$. Then by integrating over an ...
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1answer
52 views

One question about topological excitation in quantum many body system

I attended a lecture given by Professor Wen Xiaogang. In the lecture, Prof.Wen gave an example of topological excitation: For a state $$(\uparrow\downarrow)(\uparrow\downarrow)(\uparrow\downarrow)(\...
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Quantum Hall effects with an additional uniform unit flux on a compact manifold

I have two questions: Let us imagine that we have an integer quantum Hall system with electric Hall conductance as $\sigma_\text{H}$ on a two-dimensional (spatial) torus with size $L_1\times L_2$. If ...
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63 views

Topological Quantum Field Theory with Symmetries and Knot Quandles

It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
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D-branes as the atoms of Calabi-Yau crystals

I'm studying the Nishinaka-Yoshida crystal models that encode the generating function of $D4$-$D2$-$D0$ BPS bound states on a Calabi-Yau divisor. The case of conifold at its singular point is ...
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1answer
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3. Topological field theories in two-dimensions$.$

As in my previous two posts (1 & 2), a unitary two-dimensional TQFT is specified by a set of real numbers $\{\lambda_i\}$ such that the partition function is $$ Z(\lambda)=\sum_{i=1}^n\lambda_i^{g-...
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Are Chern-Simons theories classified by bordism groups?

For a long time it was thought that anomalies for a group $G$ were classified by $H^n(BG)$, although it is now understood that they are in fact classified by $\Omega^n(BG)$. On the other hand, ...
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60 views

Topology in quantum materials

So far I have learned about topological quantum material, my understanding is that topological order in a quantum material is the way the eigenvectors of the Hamiltonian of the system aligned. So if I ...
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References for topological strings on supermanifolds

This question concerns topological string theory. It was known sice its outset, that the BRST-cohomology ("the ring of observables") of the weakly coupled B-model topological string on a ...
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2. Topological field theories in two-dimensions$.$

It is known that 3d TQFTs are classified by modular tensor categories, and 2d TQFTs by Frobenius algebras. A 3d TQFT on a manifold of the form $S^1\times M_2$ induces a 2d TQFT on $M_2$. So there must ...
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Deducing fusion rules of non-abelian fluxons

I have been reading about non-abelian fluxons in John Preskill's lectures notes on topological quantum computing and I do not understand how he deduced the fusion rules for fluxons in the example he ...
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1. Topological field theories in two-dimensions$.$

The paper arXiv:hep-th/9308043 proves that the partition function of an arbitrary (unitary) two-dimensional topological theory is given by $$ Z(\lambda)=\sum_{i=1}^n\lambda_i^{g-1}\tag1 $$ where $g$ ...
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Physical interpretation of TFTs

1. Defining TFTs Let $n$ be a positive integer and $\mathbb k$ be a field. In my lecture I was introduced to TFTs using the following definition going back to Atiyah (around 1988): A $n$-dimensional, ...
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Why is short-range entanglement defined in terms of its possible deformations?

After reading the question and answers in Definition of short range entanglement I wonder why the definition of a short-range entangled state is given in terms of its possible deformations - A SRE ...
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Two dimensional conformal field theories with changing central charge

For two-dimensional conformal field theories it is usually assumed that the cental charge is fixed (for simplicity let's assume that $c=\bar c$). Is there a generalization or a concept that uses the ...
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1answer
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Chern-Simons (CS) theory

I have a question about Constructuion of Chern-Simon Action. In its paper "Non-commutative geometry and string field theory", Witten construct the Action of the String Field Theory inspiring ...
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$F$-symbols for compact Lie groups

Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices $$ B\begin{bmatrix}j_1&j_2\\i&k \end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
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Expectation value of Wilson loop in Chern-Simons theory

I have read Witten's paper, and I am interested on computing the expectation value of a Wilson loop with a representation $R$ on Chern-Simons theory in $d=3$. I am especially interested in cases for $...
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Lost reference: Kähler gravity in six dimensions and three dimensional $SL(2,\mathbb{C})$ Chern-Simons theory

I've noticed that several references take for a fact that by studying Kähler gravity on a Calabi-Yau threefold one can demostrate that any lagrangian submanifold embedded in the threefold posees three ...
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1answer
60 views

Definition for long range entanglement (LRE) by generalized local unitary (gLU) and generalized stochastic local (gSL) transformations

I am studying this book: Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems (https://arxiv.org/abs/1508.02595). In chapter 7, it introduces ...
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1answer
113 views

Winding number is an integer

In computing the variation of the action in Chern-Simons, and in other contexts, we get the following expression that is named the winding number, where $U$ comes from a gauge transformation: $$ W[U] =...
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1answer
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What is the Topologically Twisted Index?

I know that one can take a supersymmetric theory defined on $\mathbb{R}^n$ and topologically twist it by redefining the rotation group of the theory into a mixture of the (spacetime) rotation group ...
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1answer
136 views

What is a Topological Twist?

I have come across topological twists on numerous occasions but I have never actually seen them explained in an understandable way. So, I was wondering What does it physically mean to topologically ...
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How does anomaly inflow work in terms of the eta invariant?

I'm trying to understand the non-perturbative picture of anomaly inflow, mainly following these two articles by Witten and Yonekura: [1] - https://arxiv.org/pdf/1909.08775.pdf , [2] - https://arxiv....
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What is an NS-2 brane?

This question is about topological string theory and it was also posted in MathOverflow. The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph in the page ...
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1answer
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Integrating over non-trivial fiber bundles - Chern-Simons Theory

I have been reading Tong's notes on QHE and Gauge Theories, specifically the part about quantizing the Abelian U(1) Chern-Simons level at finite temperature in the presence of a monopole (These ...
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1answer
108 views

Can a symmetry of a topological field theory be spontaneously broken?

There are examples of topological "terms" causing spontaneous symmetry breaking. One that comes to mind is the $\theta$ term in $4d$ $SU(N)$ Yang-Mills, which at $\theta=\pi$ spontaneously breaks time ...
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Chiral anomaly in Weyl semimental

In Weyl semimetal, there is an analog of ABJ anomaly, which is a $E \cdot B$ term. The ABJ anomaly can be viewed as winding number because of the homotopy group of sphere $\pi_3(S^3)= \mathbb{Z}$ for ...
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1answer
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On TQFT and theories without propagating degrees of freedom

Maybe not a very sensible question, but I would like to know, whether there exist topological field theories (TQFT) with propagating degrees of freedom, or, conversely, theories without propagating ...
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Can Berry phase been carried by bulk electrons in TIs?

I'm studying 3D topological insulators and more in particular, weak antilocalization (WAL) effects, so I know that they are characterized by a $\pi$ Berry phase that gives rise to destructive ...
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Question about the proof of different contractions for the effective Brillouin zone differing by an even Chern number

I am reading Moore and Balents'paper (DOI: 10.1103/PhysRevB.75.121306) which proves the Z2 invariant is the parity of the Chern number for the effective Brillouin zone. I am confused about some ...
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1answer
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Difference between TCFTs and 2D TQFTs

I have been reading a lot lately on Topological String Theory and general TQFTs and as I noticed, in most contexts the terms "2-dimensional TQFT" and "Topological Conformal Field Theory" (TCFT) seem ...
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free fermion- monopole operator in scalar $QED_3+$ Chern simons term equivalence proof?

In most papers discussing 3D Abelian bosonization duality, they say that monopole operator in scalar $QED_3+CS$ is dual to free fermions. How do they know it, because I have never seen an actual ...
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The value of Gravitational Chern Simons theory integration on some three manifolds

Consider the 3d gravitational Chern Simons theory $$S= \frac{k}{192 \pi} \int_{M_3} \mathrm{Tr}\left(\omega\; \mathrm{d} \omega + \frac{2}{3}\omega^3\right)$$ where $\omega$ is the spin-connection on ...
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1answer
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Chern-Simons term in Coulomb or radiation gauge

In some of the literature (for example, below Eq. (A3) of this paper), the following is claimed to be the Chern-Simons term in the Coulomb gauge: \begin{equation} 2a_0(\partial_1a_2-\partial_2a_1) \...
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Winding number of QCD vacuum

We know that QCD vacuum has instantons, which corresponds to tunneling process. Consider $SU(2)$ gauge theory without matter. We say that in the classical configuration of vacuum state $F^a_{\mu\nu}=0$...
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Mutual statistics between dyons (charge-monopole composite)

I am asking for some intuitive understanding between two dyons with $(e,m)$ in 3-dimensional space. Here the magnetic charge $m$ is normalized as \begin{eqnarray} m=\int_{S^2}\frac{B}{2\pi}\in\mathbb{...
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1answer
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Why do not we consider the topological term in Abelian gauge theory?

The second Chern form $\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}$ is topological in 4-dimensional spacetime. However, we usually only consider this term in non-Abelian gauge theory, but not ...
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1answer
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Normalization of the Chern-Simons action in the Dijkgraaf-Witten paper

I am trying to understand the seminal paper "Topological gauge theories and group cohomology" by Dijkgraaf and Witten. They consider an oriented three-manifold $M$, compact Lie group $G$ and a $G$-...

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