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

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Two definitions of topological terms in field theory

I've seen two distinct definitions for "topological" terms in the context of quantum field theory. Topological terms don't depend on the metric $g_{\mu\nu}$. This makes sense since topology is '...
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49 views

Physical side of TQFT

How would one go about understanding the physical side of TQFTs? What are the best introductory resources? I know Atiyah axioms but I don't know any QFT.
3
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90 views

Are there more recent developments on connections between QFT and knot theory than Witten's initial 1989 paper?

What has happened in the field of QFT and knot theory since Witten's paper "Quantum field theory and the Jones polynomial" from 1989? What resources should one read to get up-to-date on this story? ...
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44 views

Why RR cohomology is important in string theory?

I want to know the RR cohomology in string theory or topological field theory in detail. (RR stands for Ramond Ramond). In following papers they compute the nilpotency of differential operator for RR ...
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1answer
121 views

Partition functions for a (3+1)-d TQFT

It is well known that for a Chern-Simons theory defined on an arbitrary (2+1)-d oriented manifold, its partition function can be evaluated based on Witten's surgery method. My question is: is there a ...
3
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1answer
73 views

Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
5
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1answer
111 views

What do we exactly mean by a “topological object” in physics?

I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have ...
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72 views

How is a half twist of a diagram equivalent to exchanging anyons?

Consider the fusion of $a$ and $b$ to form $c$, with the exchange of anyons $a$ and $b$, as per Pachos' book Introduction to Topological Quantum Computation. I'm confused how the half twist of $c$ ...
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1answer
58 views

Supertrace of holonomy of commutator

On page 47 of Surface operators in four-dimensional topological gauge theory and Langlands duality by Kapustin et al., the following expression is given \begin{equation} \delta\mathcal{N}=d(\omega_\...
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88 views

TQFT's as effective theories of the groundstate subspace

I often hear: "The degenerate groundstate subspace of a QFT is often a TQFT". I'm trying to work out an example of this for, say, superconductors: In the context of condensed matter physics, the ...
2
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1answer
86 views

Few basic questions about instantons

For the $SU(2)$ Yang-Mill's theory, (1) how can one understand that the finite action solutions of the Euclidean equations of motion (called Instantons) exhibit tunneling effects? (2) Since, this ...
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1answer
87 views

Why are there $F$-symbols in the splitting in anyon theory?

I am learning some basic knowledge of anyon theory by reading P. Bonderson's thesis: http://thesis.library.caltech.edu/2447/2/thesis.pdf. $F$-symbols and $R$-symbols are two basic operations on ...
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143 views

1/m Laughlin state and $U(1)_M$ chiral CFT

I am a little confused that people claim that the edge theory of a 1/m Laughlin state corresponds to a $U(1)_m$ chiral CFT. This indicates there should be $m$ primary field operators in $U(1)_m$ ...
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1answer
96 views

TQFT lectures with exercises

I'm interested in topological quantum field theory and study works of Witten and others by myself. Does anybody know where can I find any good exercises in this subject? By "good" I mean something ...
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25 views

What exactly is the “diagonal embedding” in the supersymmetric topological twist?

Consider $\mathcal{N}=2$ pure SYM theory. If we want to put the theory in a 4-manifold we take its topological twist. The global symmetry group $$G= SU(2)_{+} \times SU(2)_{-} \times SU(2)_I \times U(...
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1answer
61 views

Equivariant cohomology formula

I'm studying equivariant cohomology on three references: Szabo's review about equivariant localization (S); Libine's note on equivariant cohomology (L); Berline, Getzler, Vigne's book "Heat Kernels ...
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91 views

Topology and Quantum Field Theory

I am interested in finding any one particle state $\left| \Psi \right>$, mostly possibly topological in nature like a kink, such that $$ \left< VAC | R \widetilde{R} | \Psi \right> \neq 0$...
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2answers
103 views

Orbifold with discrete torsion

I'm trying to understand some of the early works of Vafa and Witten [1-3]. The way I look at orbifolds is they are the quotient space $M/G$. This is simply a quotient manifold when the action of $G$ ...
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117 views

Basic questions about fusion of two anyons

Suppose we have two anyons $a$ and $b$ on a manifold, and we use $|a\otimes b\rangle$ to label the corresponding wavefunction. Based on the fusion rule: $a\otimes b=\oplus_c N_{ab}^c c$, we may ...
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267 views

Topological susceptibility

In QCD we have strong CP violation (and hence a $\theta$-dependence of the theory) only if the topological susceptibility of the vacuum is nonzero: $<F\tilde{F},F\tilde{F}>_{q \rightarrow 0} =...
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104 views

R-matrix for $SU(N)_k$ anyon model

Does anyone know the $R$-move or $R$-matrix for $SU(N)_k$ anyon model? Thanks! For the definition of $R$-move or $R$-matrix, please see the definition in Eq.(2.30) of this paper: http://arxiv.org/abs/...
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1answer
99 views

What makes a system topological?

As I understand, if the Chern number which is obtained by integrating Berry curvature over a surface with a boundary is an integer, then the Chern number is a topological invariant. So when does Chern ...
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1answer
118 views

Quantum field theory with constraint: energy-momentum conservation?

Suppose I have a 2-form field $B$ and a Lagrange multiplier field $\lambda$, then the Lagrangian $S = \int (B \wedge \delta B + \lambda \delta B \wedge \delta B)$ with a Lie derivative operator $\...
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1answer
104 views

Total quantum dimension of excitations in the Toric code

In the Toric code, the excitations are e, m, fermion $\epsilon$ and vacuum. Thus, the total quantum dimension is $D= \sqrt{\sum{d_{a}^{2}}} = 2$. It seems one takes into account all sorts of possible ...
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45 views

Have Witten-type TQFT's nonconservation of energy and momentum in interactions?

Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be $G$ a geometric field. Then every observable expectation value ...
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55 views

How to derive the effective action or Vertex generating functional of complex scalar field?

I am studying-self QFT. Recently, I am studying and trying to follow the calculation of the Ginzburg-Landau free energy functional of superconductor in this paper:http://arxiv.org/abs/hep-ph/0108256. ...
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36 views

What are the global symmetries of $\mathcal{N}=2$ SYM before twisting and after twisting?

I am confused with the global symmetries of $\mathcal{N}=2$ SYM. On one hand I know that the theory has a $U(2)_R = SU(2)_R \times U(1)_R$ symmetry. Now, there exists also a $U(1)_B$ global symmetry. ...
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105 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols:...
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46 views

On open Gromov-Witten invariants of the projective line

apologies in advance in case this is a stupid question. I'm a mathematician interested in mathematical physics, but, again, penetrating the physics literature is not so easy for me. In the ...
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17 views

Correlation length during phase transitions in early Universe

During phase transitions of the second kind topological defects may form on the bounds of two areas separated by correlation length. In early Universe during phase transitions correlation length $l_{...
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67 views

Representation theory and the Nekrasov partition function

Is there any review or lecture notes on the Nekrasov partition function which particularly thinks of this from a representation theorist's point of view? Some possibly related references I know of ...
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44 views

Topological configurations and phase transitions

It is known that topological defects might appear only during phase transitions of the first kind, while continuous transitions of the second kind and crossovers don't product them. How to explain ...
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55 views

Axion domain walls and QCD phase transition

Now it is known that QCD phase transition corresponds to crossover. This it seems that no topological defects is produced during phase transition. Do axion domain walls arise during QCD phase ...
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64 views

The bounds of axion domain walls are axion strings?

There are two phase transitions which are important for the axion physics. The first one is Peccei-Quinn phase transition, during which axions arise. The second one is QCD phase transition, at which ...
2
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1answer
68 views

Kalb-Ramond action and topological string radiation

Let's have simple scalar $\Phi$ action involves spontaneously symmetry breaking in a form $$ \tag 1 S = \int d^{4}x\left( |\partial_{\mu}\psi|^{2} + \psi^{2}|\partial_{\mu}\theta |^{2} - \frac{\lambda}...
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43 views

Global cosmic strings evolution

Recently I've read about axion string. It can be shown that the energy per unit length of the string located along $z$ axis is $$ \mu = 2 \pi f_{a}^{2}\ln\left( \frac{L}{\delta}\right), $$ where $L$ ...
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93 views

Axion strings and spontaneously broken symmetry

I have two question about axion strings: Why their appearance is connected with spontaneously broken symmetry? How to demonstrate that? Why they are stable topological configurations (look to the "...
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121 views

Choice of framing in Gravitational Chern-Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
2
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1answer
106 views

Fermion version of Gauss-Milgram sum?

For Bosonic topological order, a very useful formula was proved to be true: $\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $ (for more detail: $d_a$ is the quantum dimension of anyon ...
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1answer
475 views

Generalization of De Rham cohomology for spinor fields

I am interested in possible generalizations of The De Rham cohomology for spinor fields. I am also interested in applications to physics such as in the construction of topological charges I can see ...
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2answers
90 views

Topological strings: Why is the complex structure for $T^2$ denoted as $\tau$ in string theory?

In these notes by Vafa on topological string theory he says in page 7 that the moduli of the 2-torus can be repackaged into two quantities: $$A=iR_1/R_2 \,\,\,\,\,\,\,\,\, \tau=iR_2/R_1$$ where $A$ ...
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1answer
178 views

What does it mean physically if pentagon identity or hexagon identity doesn't have any answers?

Imagine I write a fusion rule for some anyons on a paper. Then, I try to solve Pentagon identity and Hexagon identity, imagine finally I find out for example the Hexagonal equation doesn't have any ...
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2answers
345 views

Real World application of Topological Quantum Field Theory

What is a "killer-app" for the formalism of topological quantum field theory in "established real world physics"? To be more precise, I'm looking for an actual physical experiment, state of matter or ...
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79 views

A question about genus one string amplitude

In BCOV's paper http://arxiv.org/abs/hep-th/9309140 the genus one string amplitude of a Calabi-Yau 3-fold was explained in the B-model as the Ray-Singer torsion (there is a similar discussion in the ...
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2answers
108 views

Dilemma: Fusion space from a direct sum of anyons or NOT

In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that: The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have ...
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58 views

Topological term under electron-electron interaction

By integrating out fermions in gapped Dirac Hamiltonian, one can obtain a topological term for topological insulator. Why there is no further correction to this term when electron-electron interaction ...
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1answer
162 views

How the Vortex containing majorana bound state is non-abelian statistics

Recently,I read some papers about non-abelian statistics of majorana fermion, such as: Majorana Returns F. Wilczek http://www.nature.com/nphys/journal/v5/n9/full/nphys1380.html and Non-Abelian ...
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231 views

Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
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1answer
48 views

transformations between 1st and 2nd order formalism in pure gravity

I am currently studying about 1st order formalism and I was wandering if the gauge transformation in the vielbein can be mapped to the coordinate transformation of the metric ( pure 2+1 gravity), ...
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2answers
280 views

Chern Simons action in 4 dimensions

I can not understand why we do not have a Chern-Cimons action for 4 or even forms? And why it not good theory for (3+1) dim?