Questions tagged [topological-field-theory]

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

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31 views

Singularity and charts-problems met in Dirac quantization condition

Context: 45'23'' in a lecture given by Professor Wu, https://www.koushare.com/video/videodetail/4619. Consider a vector field $\vec{A}(\vec{x})$, with $\nabla\times\vec{A}(\vec{x})=\vec{B}(\vec{x})=g\...
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How to get the generators of $\mathfrak{so}(3)$ in the paper by Fidkowski and Kitaev?

In the paper by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana ...
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32 views

Why stacking two $p+ip$ superconductors or superconductors with Chern number 1 ($C=1$) is a quantum hall state?

My question is based on the lecture by Bernevig in PiTP 2015 on "Category Theory and the Kitaev 16 Fold Way"41:00. Why by stacking two superconductors with Chern number $C^{(1)}=1$ we have ...
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78 views

Geometric intuition for $\mathcal{L}^{-m} \oplus \mathcal{L}^{m} \rightarrow T^{2}$ Calabi-Yau threefolds [migrated]

I'm having some problems to understand the geometry involved in the section 3.1 of the paper Two Dimensional Yang-Mills, Black Holes and Topological Strings. Concretely, I was wondering to know if it ...
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1answer
74 views

How we can get the “Fermion Parity” and “Ground states” for Majorana fermions in Bernevig's talk PiTP 2015?

I have two questions regarding the talk, Topological Superconductors, Majorana...and Interactions, by Bernevig in PiTP 2015. How he gets the "Fermion Parity" for the ground states in the ...
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1answer
58 views

Evaluating the $A \land A \land A$ in the Chern-Simons action

I am trying to evaluate $A \land A \land A$, but I am a bit confused on how exactly to do it and produce the usual notation used in physics. I am trying to use the definition of the wedge product of ...
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2answers
56 views

Chern-Simon level quantization and quantum Hall effect

It is well-known that integer and fractional quantum Hall effect can be effectively described by $U(1)$ abelian Chern-Simon theory. In both cases, quantization(fractionalization) of Hall resistance is ...
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42 views

Topological Field Theory for Physicists [duplicate]

I was wondering if anyone knows good resources for Topological Field Theories aimed at physicists. In particular, I am looking for references which are full of examples, starting with simple toy ...
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1answer
36 views

What is the difference between topological theta term and Wess-Zumino-Witten term?

It seems that they both proportional to some thing like $\vec{n}\cdot \partial_{\tau}\vec{n}\, \times\,\partial_{s}\vec{n}$. References: Fradkin, Quantum field theory: an integrated approach
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2answers
128 views

Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory

Is there a duality between (2+1)D gravity and Chern-Simons Theory? Or they merely have related features? If so, of which type and why?
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Visualizing topological deformation and quantum mechanical interference

In section VI.1 of Zee's QFT, he says that for indistinguishable hard core particles in 2D, when comparing trajectories with different winding numbers: Since the classes cannot be deformed into each ...
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Excitations of the string-net Hamiltonian

A quite general 2D topological order can be constructed through the string-net theory. Here, if the input data is some braided fusion category $C$ (i.e. the $F$-symbols), the elementary excitations ...
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What symmetry breaks in exciton condensation?

Recently the possibility of exciton condensates have shown up in some lattice TQFT models we've been studying, and I've been trying to learn more about them. Suppose that we have a band insulator with ...
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1answer
56 views

What is the role of the dilaton in Jackiw-Teitelboim 2D gravity?

I read that the Einstein Hilbert action is topological in 2 dimensions. (What does that mean?). To write down a non-trivial action one introduces the dilaton field in JT gravity. Does this field have ...
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Physics behind the Kobayashi-Hitchin correspondence

Let $X$ be a $d$-dimensional Kähler manifold with Kähler matric $\omega$. Let's consider the following setups: Suppose $E \rightarrow X$ is hermitian vector bundle with hermitian connection $A$. In ...
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1answer
39 views

Classification of topolgical phases when eigenstates belong to complex Grassmannian

I want to understand the paper which belongs to Ludwig (I put it below). I do not understand why exactly he got the new space $U(m+n)/U(m) \times U(n)$. My understanding from Grassmannian Manifold is ...
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40 views

Physical origin of coisotropic branes

The paper "Remarks on A-branes, Mirror Symmetry, and the Fukaya category" develops the possibility of A-model open strings ending on a coisotropic submanifold equipped with a holomorphic ...
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Mass deformations in D-brane systems

It is well known that the worldvolume theory of $N$ coincident D$p$-branes is given by the $U(N)$ Yang-Mills theory in $(p+1)$-dimensions. One important feature of this setup is the possibility of ...
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24 views

Rigorous definition of “condensation” in Topological Lattice Gauge theories?

I have encountered the term "string-net condensation" in the string-net paper by Levin-Wen, and in Kitaev's Toric Code - where we say that the ground state is a string net condensate and (I ...
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1answer
129 views

Deriving the Topological Descent Equations

I am trying to show that in a cohomological TQFT, given a physical operator $\phi^{(0)}$, one can construct a chain of non-local physical operators. In doing so, I need to show that a certain set of ...
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160 views

How to calculate a TQFT Gaussian path integral from Seiberg's “fun with free field theory”?

In his talk "Fun with Free Field Theory", Seiberg discusses a topological quantum field theory in $d+1$ dimensions with the action $$ S = \frac{n}{2\pi} \int \phi\, \mathrm{d} a \tag{1}$$ ...
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S-Matrix of Modular Tensor Category and physical S-matrix

Let $\mathscr{C}$ be a Modular Tensor Category (MTC). Then it has a finite set of simple objects $\{X_i\}$. Moreover, we define the S-matrix as $$S_{ij} = \text{Tr}(B_{X_i,X_j}^{-1}B_{X_j,X_i})$$ ...
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1answer
54 views

A pedagogical semi-rigorous review of topological phases, topological order, and related subjects

I'm looking for a pedagogical review or book about topological phases, topological order, TQFTs, and related subjects. The ideal thing would be a mix of rigorous definitions and physical examples, ...
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1answer
110 views

Why is Kitaev's toric code a $Z_2$ gauge theory?

I am reading Kitaev's 2003 paper. In the literature, it is often said that the model proposed in this paper is a $Z_2$ gauge theory. I don't quite see why it is the case. Where is the $Z_2$ gauge ...
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23 views

Topological defects in general and Chern-Simons in particular

I'm trying to gain intuition on some physical concepts that I cannot yet fully understand, and I think many of you can help me. Is it correct to think of of a topological defect as the addition ad hoc ...
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What is the physical importance of topological quantum field theory?

Apart from the fascinating mathematics of TQFTs, is there any reason that can convince a theoretical physicist to invest time and energy in it? What are/would be the implications of TQFTs? I mean is ...
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1answer
129 views

Why do we call fracton by its name?

I am reading on fractons. In the literature, it is said that factons are fractionalized excitations. My understanding about fractons is that it is energetically costly to move fractons, and in this ...
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21 views

Partition functions of descendent SPTs of the Haldane chain

The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\...
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1answer
193 views

Criteria to Define a (Classical) Topological Field Lagrangian? + Conjecture

I have a question concerning topological field theories. I'd rather keep the discussion at the classical level, so as to concentrate on the feature of topological evolution, which is what interests me ...
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1answer
141 views

The argument for a mass gap for the $O(3)$ Heisenberg ferromagnet

One possible argument for asymptotic freedom in the 2D $O(3)$ ferromagnetic Heisenberg model is the existence of so-called instantons, discovered in the 1975 paper of Belavin and Polyakov. This is ...
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72 views

Excitations & Pentagon axiom in algebraic theory for anyons

I have been reading the anyon theory by Kitaev and Wang. I have two possibly related questions: Why is the Pentagon equation/axiom sufficient for characterizing associative relations? Are there anyon ...
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1answer
45 views

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
112 views

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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78 views

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
88 views

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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255 views

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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2answers
343 views

Is it possible to bound a single D0-brane to a D4-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
151 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
231 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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48 views

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
214 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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47 views

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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50 views

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
58 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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71 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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1answer
112 views

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
84 views

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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86 views

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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67 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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