Questions tagged [topological-field-theory]

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

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32
votes
7answers
7k views

Reading list in topological QFT

I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm ...
47
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2answers
20k views

What is a $p_x + i p_y$ superconductor? Relation to topological superconductors

I often read about s-wave and p-wave superconductors. In particular a $p_x + i p_y$ superconductor - often mentioned in combination with topological superconductors. I understand that the overall ...
14
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1answer
626 views

geometric quantization of the moduli space of abelian Chern-Simons theory

I wish to understand the statement in this paper more precisely: (1). Any 3d Topological quantum field theories(TQFT) associates an inner-product vector space $H_{\Sigma}$ to a Riemann surface $\...
26
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3answers
2k views

Quantum field theory variants

Wikipedia describes many variants of quantum field theory: conformal quantum field theory topological quantum field theory axiomatic/constructive quantum field theory algebraic quantum field theory ...
10
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0answers
247 views

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

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 ...
24
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2answers
2k views

Do topological superconductors exhibit symmetry-enriched topological order?

Gapped Hamiltonians with a ground-state having long-range entanglement (LRE), are said to have topological order (TO), while if the ground state is short-range entangled (SRE) they are in the trivial ...
10
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2answers
1k views

Topological ground state degeneracy of SU(N), SO(N), Sp(N) Chern-Simons theory

We know that level-k Abelian 2+1D Chern-Simons theory on the $T^2$ spatial torus gives ground state degeneracy($GSD$): $$GSD=k$$ How about $GSD$ on $T^2$ spatial torus of: SU(N)$_k$ level-k Chern-...
8
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2answers
662 views

No monopoles in the Weinberg-Salam model

I'm reading Chapter 10.4 on the 't Hooft-Polyakov monopoles in Ryder's Quantum Field Theory. On page 412 he explains why magnetic monopoles cannot appear in the Weinberg-Salam model. I'm I right by ...
18
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2answers
910 views

Normalization of the Chern-Simons level in $SO(N)$ gauge theory

In a 3d SU(N) gauge theory with action $\frac{k}{4\pi} \int \mathrm{Tr} (A \wedge dA + \frac{2}{3} A \wedge A \wedge A)$, where the generators are normalized to $\mathrm{Tr}(T^a T^b)=\frac{1}{2}\delta^...
3
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1answer
1k views

difference between classical vacuum solutions and instantons

What does the classical vacuum of the $SU(2)$ Yang-Mills action correspond to? Does it correspond to $F_{\mu\nu}=0$ everywhere or just at the spatial infinity? In Srednicki’s book, he has shown that, ...
33
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1answer
3k views

Zero modes ~ zero eigenvalue modes ~ zero energy modes?

There have been several Phys.SE questions on the topic of zero modes. Such as, e.g., zero-modes (What are zero modes?, Can massive fermions have zero modes?), majorana-zero-modes (Majorana zero ...
12
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2answers
3k views

Definition of short range entanglement

When studying Symmetry Protected Topological phases, one needs to define what a short range entangled (SRE) states means. But there appears to be different definitions that are not equivalent to each ...
17
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3answers
1k views

about the Atiyah-Segal axioms on topological quantum field theory

Trying to go through the page on Topological quantum field theory - The original Atiyah-Segal axioms - "Let $\Lambda$ be a commutative ring with 1, Atiyah originally proposed the axioms of a ...
9
votes
2answers
6k views

Is band-inversion a 'necessary and sufficient' condition for Topological Insulators?

According to my naive understanding of topological insulators, an inverted band strucure in the bulk (inverted with respect to the vaccum/trivial insulator surrounding it) implies the existence of a ...
20
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1answer
2k views

Why is there no dynamics in 3D general relativity?

I heard a couple of times that there is no dynamics in 3D (2+1) GR, that it's something like a topological theory. I got the argument in the 2D case (the metric is conformally flat, Einstein equations ...
18
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1answer
1k 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 ...
10
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4answers
2k views

Is gravitational Chern-Simons action “topological” or not?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection: $$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$ $$ S=...
7
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1answer
390 views

What do we learn from gravity in three spacetime dimensions?

The last decades there has been a lot of research going on in the the area of three dimensional gravity. The motivation, I understand, is threefold: Whereas gravity is not perturbatively ...
11
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3answers
670 views

TQFT associates a category to a manifold

Any 3d TQFT (topological-quantum-field-theory) associates a number to a closed oriented 3-manifold, a vector space to a Riemann surface, a category to a circle, and a 2-category to a point. This is ...
9
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2answers
760 views

Understanding Cherns-Simons-Witten Theory

I want to read about Wittens work, on Cherns-Simons theory, and relations to knots and jones polynomials. I am extremely motivated to read his paper: Quantum Field Theory and Jones polynomial. What ...
14
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1answer
660 views

How is the Chern-Simons action well-defined?

The Chern-Simons action $$ S = \int_M A \wedge \mathrm{d} A + \frac{2}{3}A \wedge A \wedge A $$ is not obviously gauge invariant. It is usually stated that under a gauge transformation, the action ...
7
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1answer
261 views

Can one quantize systems with local (non-gauge!) symmetries?

Is it inherently problematic to quantize classical theories with local symmetries? For example, consider the action of EM but now interpret $A_\mu$ as physical. At a classical level, there is nothing ...
12
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3answers
941 views

Chern-Simons degrees of freedom

I'm currently reading the paper http://arxiv.org/abs/hep-th/9405171 by Banados. I am just getting acquainted with the details of Chern-Simons theory, and I'm hoping that someone can explain/elaborate ...
3
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2answers
1k 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 ...
3
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1answer
2k views

Topological phase

Can anybody tell me, if generically any system, which is solely described by a topological field theory, resides in a topological phase? I cant find any clear notion of topological phase. Only ...
8
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2answers
388 views

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 $g$ ...
6
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0answers
371 views

Reference on Chern-Simons theory [duplicate]

I have recently been trying to refresh my memory on the Quantum Field Theory I learned 25 years ago while getting my Ph. D. At the time I did not study Chern-Simons modifications to QFT Lagrangians. ...
1
vote
2answers
172 views

How does this Gaussian integral over the auxiliary field in 2D topological gauge theory work?

In "Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques" by Thompson and Blau equation (2.2) reads $$ \int [DA] \exp\left( \int Tr(F\star F \right) = \int [DA \, D\phi] \...
21
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1answer
1k views

Phase Structure of (Quantum) Gauge Theory

Question: How to classify/characterize the phase structure of (quantum) gauge theory? Gauge Theory (say with a gauge group $G_g$) is a powerful quantum field theoretic(QFT) tool to describe many-body ...
19
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1answer
1k views

Global Chern-Simons forms and topological gauge theories

I am reading the classic Dijkgraaf and Witten paper on topological gauge theories and something struck me that I didn't understand. For a trivial bundle $E$ on smooth 3-manifold $M$ with compact ...
24
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1answer
2k views

How does Haldane conjecture follow from the topological $\Theta$ term

The one dimensional SU(2) Heisenberg quantum spin chain is known to be described by the 1+1d O(3) nonlinear $\sigma$ model with a $\Theta$ term, following the action $$S=\int\mathrm{d}^2x\frac{1}{g}(\...
24
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3answers
1k views

How to understand topological order at finite temperature?

I have heard that in 2+1D, there are no topological order in finite temperature. Topological entanglement entropy $\gamma$ is zero except in zero temperature. However, we still observe some features ...
16
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2answers
1k views

Braiding statistics of anyons from a Non-Abelian Chern-Simon theory

Given a 2+1D Abelian K matrix Chern-Simon theory (with multiplet of internal gauge field $a_I$) partition function: $$ Z=\exp\left[i\int\big( \frac{1}{4\pi} K_{IJ} a_I \wedge d a_J + a \wedge * j(\...
23
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1answer
590 views

Anyons as particles?

I'm trying to understand the basics of anyons physics. I understand there is neither a Fock space they live in (because Fock is just the space of (anti-)symmetrized tensor product state, see e.g. ...
22
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0answers
959 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring ...
7
votes
1answer
795 views

“Topological” notions in physics

I've been trying to make sense recently of the usage of 'topological' in various fields of physics, and get sort of an intuition for what this means in context. This all boils down to my main question ...
11
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1answer
1k views

Chern-Simons Energy-Momentum Tensor

I'm assuming the following statement is true. I'm not finding any reference which shows that explicitly. Statement: Chern-Simons term is a topological one and does not contribute to the Energy-...
10
votes
2answers
753 views

Wilson Loops as raising operators

Consider a U(1) Chern Simons theory on a torus $\mathbb{T}$: \begin{align} L &= \frac{k}{4\pi} \int_{\mathbb{T}} a \partial a \end{align} where a is some U(1) gauge field, $k\in\mathbb{Z}$ and we ...
16
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5answers
2k views

Applications of Geometric Topology to Theoretical Physics

Geometric topology is the study of manifolds, maps between manifolds, and embeddings of manifolds in one another. Included in this sub-branch of Pure Mathematics; knot theory, homotopy, manifold ...
9
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1answer
2k views

Difference between instantons and sphalerons

What is the difference between instantons and sphalerons? If they are different, how do they violate baryon and lepton number in the standard electroweak theory?
8
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1answer
717 views

Intersecting Wilson loops in 2D Yang-Mills

I am currently trying to understand 2D Yang-Mills theory, and I cannot seem to find an explanation for calculation of the expectation value of intersecting Wilson loops. In his On Quantum Gauge ...
8
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1answer
839 views

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 '...
7
votes
1answer
386 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{\...
9
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0answers
281 views

Intuition for Homological Mirror Symmetry

first of all, I need to confess my ignorance with respect to any physics since I'm a mathematician. I'm interested in the physical intuition of the Langlands program, therefore I need to understand ...
13
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1answer
497 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 ...
12
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3answers
813 views

Why are topological properties described by surface terms?

An example are the anomalies in abelian and non-abelian gauge quantum field theories. For example, the abelian anomaly is $\tilde {F}_{\mu\nu}F^{\mu\nu}$ and the integral over this quantity is a ...
9
votes
1answer
430 views

Theta Vacuum of Yang-Mills theory and Baryon number violation

Background 1. In classical SU(N) Yang-Mills theories, there are a countably infinite number of homotopically inequivalent gauge field configurations of zero energy labelled by a winding number $n\in \...
8
votes
1answer
383 views

Gravitational Chern-Simons theory for bosons and fermions

Q1: What is the difference of boson and fermions for their Gravitational Chern-Simons theory? I suppose in general if the metric is not flat, we have vierbein ${e_{\hat{b}}}^{\nu}$, with $$ g_{\mu\...
8
votes
2answers
3k views

Einstein Field Equations in other space-time dimensions than 3+1?

This question is apparently quite simple but I can't seem to find an answer to it, so I was hopping anyone could clarify me. Are the Einstein field equations (EFE) only valid for a 3+1 dimensional ...
7
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0answers
270 views

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 ...