Questions tagged [topological-field-theory]

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

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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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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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Who can give metaphors to differentiate fermion string from bosonic string?

From my rough elementary understanding the diff between fermion and bosonic particle/strings are fermion is more materialistic, and bosonic somehow relates to the energy/function part of the common ...
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Instantons in theoretical and mathematical physics [duplicate]

Could any of you give me just an overview on the open problems involving instantons in theoretical and mathematical physics? In particular TQFT and GR. I would appreciate very much both answers from ...
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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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About 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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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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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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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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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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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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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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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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Physical and mathematical significance of the 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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.&...

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