Questions tagged [topological-field-theory]

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

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1answer
47 views

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
55 views

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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1answer
76 views

Anomalies on boundary and bulk physics

Few times I faced with such statements: The gravitational anomaly of the 1+1d boundary system is known to be proportional to the thermal Hall conductivity of the 2+1 dimensional bulk How ...
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1answer
49 views

Describing Majorana fermions with operations

I'm reading a book on topological quantum theory and one of the exercises says that Majorana fermions $\gamma_j$ are such that $\{\gamma_j,\gamma_i\}=\delta_{ij}$ and that $\gamma_j=\gamma_j^\dagger$, ...
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33 views

Why “we can use the compactness of the Brillouin zone to contract the sphere back onto itself on the other side”?

This is from David Tong's Gauge Theory Notes Sec 4.3.3 page 224(book) or page 26(pdf) http://www.damtp.cam.ac.uk/user/tong/gaugetheory/4lattice.pdf Consider Hamiltonian $H=v_i(\vec{k})\sigma_i+\...
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19 views

How can the localization property of the edge mode in topological insulator/quantum hall system be manifested through the effective action?

To be more specific, we can write down the Chern-Simons term from coupling the system to EM to describe the 2d quantum hall system and its derivative respect to the EM field gives the current. How can ...
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18 views

How does the effective action describing the EM response of (3+1)d topological insulator become the (2+1)d Chern-Simons term?

Mathematically, it seems to be resulting from Stokes theorom once the 3d manifold has a 2d boundary. However, the EM response described by (2+1)d CS term requires the system to break time-reversal ...
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78 views

Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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33 views

Experimental progress: Topological phases of matter

It's been: 43 years since Leinaas & Myrheim's seminal paper 38 years since Wilczek coined the term anyon 29 years since Moore & Read's paper on non-Abelions in the Fractional Quantum Hall ...
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117 views

Trivial vs nontrivial TQFT

This question is inspired by Examples of "gauging a global symmetry" and answer to that question. I list main statements from answer: 1) We start from free scalar field $\phi$ in d+1 ...
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36 views

Anyons under weaker assumptions?

I. Duality assumption In Anyons in an exactly solved model and beyond p.74, Kitaev says, "We will see that for theories with particle-antiparticle duality, condition 3 can be dispensed with." Of ...
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1answer
70 views

Theta-terms in 3+1D QCD and 1+1D QED / $CP^1$ models

It is well known that topological $\theta$-terms in gauge theories are total derivatives and vanish after integration over the Lagrangian (or Hamiltonian) density, unless there are nontrivial boundary ...
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81 views

Equivalence of 2d quantum and topological gravity

I have heard of the statement that 2d quantum gravity (defined as minimal models of CFT coupled to Liouville theory) and 2d topological gravity are equivalent. The former is described by the continuum ...
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36 views

Every theory with global SUSY is a free boson?

I'm reading this (review?) article https://www.sciencedirect.com/science/article/pii/0370157391901175 on topological field theories. In section 3.2 (page 142) they seem to be claiming that for any ...
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2answers
168 views

Worldline formulation of QFT

Exist formulation of bosonic scalar field theory using action: $$ S= \frac{1}{2}\int dt\; \sqrt{g}\left(g^{tt} G_{\mu\nu} \partial_t X^\mu \partial_t X^\nu - m^2\right) $$ Also exist formulation of ...
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20 views

Semi-classical exactness of the topological QFT

There are two types of topological quantum field theories (TQFT): of the Schwarz type and of the Witten type. While the latter is, per definition$^1$, required to be semiclassically exact, the former ...
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1answer
176 views

Kinds of Wilson Loops in a $U(1)$ Chern-Simons Theory

Pardon the potentially easy question, but I am currently reading through a paper by Seiberg and Witten [1], and while reading appendix C I'm not sure where some of these results are coming from. The ...
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39 views

Gravitational correction in index theorem for 3+1 time-reversal invariant TI

In Witten's review paper: Fermion path integrals and topological phases, the index theorem for 3+1 $\mathcal{T}$-conserving TI is given by $$e^{\mp i\pi \eta/2}e^{\pm i\pi(P-\hat{A}(R))}=(-1)^{\...
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1answer
75 views

Alternate proposed differential form of Maxwell action

The electromagnetic action is defined by the wedge product of the electromagnetic tensor F and its hodge star *F What would be the explicit form of the action if this would be substituted with the ...
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84 views

Principal bundles of Lie groups in a short exact sequence

Consider a short exact sequence of Lie groups $$1 \rightarrow G \rightarrow H \rightarrow L \rightarrow 1.$$ What can we say about the principal bundles with the above groups as structure groups (...
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1answer
85 views

Topological photon mass and QED in (2+1)-D

My question was: how to compute topological photon mass in (2+1) QED and why it comes?
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32 views

Spin structure, Intersection form and 2D TQFT

I am reading this (page 20) and watching this Anton Kaputins' talk (33:24). Here, he tried to explain how to define a spin structure on a lattice in a closed oriented (1+1)-D manifold $M$ (or at ...
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42 views

Gauged DIII superconductor and Z_2 topological order

I keep hearing that a gauged 2D topological superconductor with preserved time-reversal symmetry that belongs to the DIII class is equivalent to the Z_2 gauge theory (Z_2 topological order with ...
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33 views

Relation between a partition function and Hilbert spaces

I'm studying the Jones Polynomial paper by Witten, and a bulk of the analysis involves computing various partition functions. He says, in section 4, that calculating the partition function over a sub ...
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37 views

Why does the cross term disappear?

At 1:02:00 in Marino's video on topological quantum field theories (link: https://m.youtube.com/channel/UCBR8-60-B28hp2BmDPdntcQ ) he wrote that $S_{YM} = \int \text{tr} (F\wedge F) = \int \text{...
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2answers
104 views

How to discretize a Hamiltonian?

In the famous article Physical Review B 92, 064520 (2015), a theoretical model was proposed to realize the chiral Majorana zero mode. $$H_{\mathrm{BdG}}=\left(\begin{array}{cc}{H_{0}(\mathbf{k})-\mu} ...
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38 views

How the variation of Gauss Bonnet term is done?

I am trying to do the variation of Gauss Bonnet Invariant which is $G=R^2+R_{abcd}R^{abcd}-4R_{ab}R^{ab}$ and having problem in doing the variation of $δ(R_{abcd}R^{abcd})$. Can anyone please give ...
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1answer
117 views

Fusion of anyons

I have been studying anyons and I have found the algebraic approach rather abstract and I am struggling to understand it as it seems quite different to the usual procedure of quantum mechanics. I do ...
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0answers
48 views

Factor of 2 issue in the non-gauge invariance of Chern-Simons theory with a boundary

It is well known that the Chern-Simons (CS) theory by itself is not gauge invariant in the presence of a spacetime boundary. Concretely, suppose the flat half space $\mathcal{M}$ with $x\in \mathbb{R},...
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31 views

How build a topological charge from of the mapping between physical and inner space?

How build a topological charge from the mapping between physical and inner space? When we make a mapping between two coordinates system, we normally relate both systems by coordinate transformation as,...
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1answer
53 views

Number of states in Z2 gauge theory on a finite square lattice

In Wen's Quantum Field theory of many body systems, on page 254, it discusses Z2 gauge theory, and states that Count the number of states in the Z2 gauge theory on a finite square lattice. We ...
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36 views

Questions about unitary condensate wave-function and $p_x+ip_y$ superconductor

I read about the unitary and non-unitary order parameter states here https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.75.657 and https://arxiv.org/abs/1512.01151 The form of the order ...
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1answer
40 views

Exchange Non-abelian Anyons results in rotation in Vacuum degenerate space?

A basic notion when studying non-abelian anyons is that the system's groundstate is degenerate. Not only that, but exchanging two anyons' position rotates the state in this degenerate subspace. I'm ...
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1answer
184 views

What are emergent gauge fields in condensed matter physics?

My background: I have a very little knowledge about topological insulators. Medium level knowledge of Quantum mechanics and linear algebra. Almost no knowledge about Field Theories. I have studied ...
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0answers
60 views

Is there a difference between topological defects and topological soliton?

Is there a difference between topological defects and topological soliton? Or are these objects the same thing? I ask this because it very common find some papers whose the authors itself refer, for ...
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0answers
25 views

Chern-Simons theory with a discrete gauge symmetry

Let us consider a Chern-Simons theory on a $3$-manifold $M$ (can be a spin manifold with a given spin structure if needed) with a discrete-symmetry gauge field e.g. $\mathbb{Z}_n$ symmetry. It can be ...
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0answers
106 views

What is difference between $U(1)$ symmetry and $U(1)$ gauge invariance

According to Wen's description if two states $|a\rangle$ and $|b\rangle$ with $\langle a|b\rangle=0$ have same physical properties, they are symmetric. On the the other hand if we label same ...
4
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1answer
94 views

Chern-Simons theory on a plane/sphere with a single charge insertion

Consider the pure Chern-Simons theory on the plane $\mathbb{R}^2$ with a single charge insertion in some representation $\rho$ of the group $G$. What does the Hilbert space look like? Is it null or ...
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51 views

Why does the Holst term not affect gravitational dynamics?

The general first-order Palatini action in four dimensions is given by $$S[e,\omega]=\frac{1}{2\kappa}\int_{\mathcal{M}} F_{IJ}[\omega]\wedge\left(\star+\frac{1}{\gamma}\right)\left(e^I\wedge e^J\...
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0answers
29 views

What “manifold in band parameters ” means?

I was reading an article https://arxiv.org/abs/0907.0500 in which they write about manifold in band parameter ,like in first line in my picture , and then they call it band parameter . can some ...
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1answer
40 views

EM Dual Lagrangian

I am working on the dual Lagrangian as given by $$\mathcal{L}_D=F_{\mu\nu}\tilde{F}^{\mu\nu}$$ In literature, this term is often written as $$\boxed{\mathcal{L}_D=2\partial^\mu(\varepsilon_{\mu\nu\rho\...
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48 views

Spacetime manifolds of 1+1 d systems for writing TQFT partition functions

Are the spacetime manifolds of two unentangled systems disconnected? This arises in the context of thinking of an operator whose expectation value we wish to take by writing this quantity in terms ...
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0answers
114 views

Non-abelian higher symmetries

I am reading about generalised global symmetries or higher-form symmetries (for example, in Generalized Global Symmetries, Gaiotto, Kapustin, Seiberg, and Willett), and came across this question. One ...
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122 views

How are local observables encoded in this formulation of quantum field theory as a functor?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]....
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0answers
48 views

Lattice construction of 2D topological field theory and Frobenius algebras vs. associative algebras

I have a basic confusion about 2D topological field theories (TFTs). In the lattice construction of 2D TFTs introduced by Fukuma et al (https://arxiv.org/abs/hep-th/9212154) only associative algebras ...
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75 views

String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops ...
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0answers
189 views

What's the origin of the vortex's ansatz $\phi\big(\vec{x}\big)=f\big(r\big)e^{-in\theta}$?

What's the origin of the vortex's ansatz $\phi\big(\vec{x}\big)=f\big(r\big)e^{-in\theta}$ in the de Vega and Schaposnik paper? In their article Classical vortex solution of the Abelian Higgs model,...
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53 views

What does it mean for gravity in $(2+1)$ dimensions to be topological?

I have been studying gravity in $(2+1)$-dimensions and I have come across the idea that gravity in this lower-dimensional spacetime is topological but I haven’t been able to find a simple explanation ...
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1answer
51 views

How Does A $\theta$ Angle Shift Affect the Wilsonian Effective Lagrangrian?

Say we have some quantum field theory which includes a gauge field, and some matter, and a topological $\theta$ term so that the Lagrangian reads $$L=(stuff)+\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\...
3
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1answer
179 views

Why doesn't the $\theta$ Angle Renormalize?

The $\theta$ term for Yang-Mills takes the form $$L_{\theta}=\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{{\mu\nu}}F^a_{\rho\sigma}$$ A fact that I have heard is that $\theta$ does not ...

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