Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
405
questions
0
votes
0answers
18 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 ...
1
vote
1answer
34 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 $$(\uparrow\downarrow)(\uparrow\downarrow)(\uparrow\downarrow)(\...
1
vote
0answers
14 views
Quantum Hall effects with an additional uniform unit flux on a compact manifold
I have two questions:
Let us imagine that we have an integer quantum Hall system with electric Hall conductance as $\sigma_\text{H}$ on a two-dimensional (spatial) torus with size $L_1\times L_2$. If ...
0
votes
1answer
57 views
Who can give metaphors to differentiate fermion string from bosonic string?
From my rough elementary understanding the diff between fermion and bosonic particle/strings are fermion is more materialistic, and bosonic somehow relates to the energy/function part of the common ...
-2
votes
0answers
41 views
Instantons in theoretical and mathematical physics [duplicate]
Could any of you give me just an overview on the open problems involving instantons in theoretical and mathematical physics? In particular TQFT and GR.
I would appreciate very much both answers from ...
1
vote
0answers
48 views
Topological Quantum Field Theory with Symmetries and Knot Quandles
It is well known that Chern-Simons theory provides an intrinsically three dimensional way to compute knot invariants like the Jones Polynomial. 3D TQFTs also have an algebraic description in terms of ...
3
votes
0answers
43 views
About the “atoms” of Calabi-Yau crystals
I'm studying the Nishinaka-Yoshida crystal models that encode the generating function of $D4$-$D2$-$D0$ BPS bound states on a Calabi-Yau divisor.
The case of conifold at its singular point is ...
3
votes
0answers
46 views
3. Topological field theories in two-dimensions$.$
As in my previous two posts (1 & 2), a unitary two-dimensional TQFT is specified by a set of real numbers $\{\lambda_i\}$ such that the partition function is
$$
Z(\lambda)=\sum_{i=1}^n\lambda_i^{g-...
asked Sep 7 at 22:17
AccidentalFourierTransform
36.5k14 gold badges88 silver badges165 bronze badges
5
votes
0answers
57 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, ...
asked Sep 5 at 15:27
AccidentalFourierTransform
36.5k14 gold badges88 silver badges165 bronze badges
0
votes
1answer
42 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 ...
2
votes
0answers
36 views
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 ...
8
votes
0answers
66 views
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 ...
asked Aug 28 at 17:56
AccidentalFourierTransform
36.5k14 gold badges88 silver badges165 bronze badges
2
votes
0answers
31 views
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 ...
8
votes
2answers
320 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 $g$ ...
asked Aug 22 at 17:47
AccidentalFourierTransform
36.5k14 gold badges88 silver badges165 bronze badges
0
votes
0answers
66 views
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, ...
1
vote
0answers
27 views
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 ...
1
vote
0answers
24 views
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 ...
4
votes
1answer
111 views
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 ...
6
votes
0answers
59 views
$F$-symbols for compact Lie groups
Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices
$$
B\begin{bmatrix}j_1&j_2\\i&k
\end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
2
votes
0answers
53 views
Expectation value of Wilson loop in Chern-Simons theory
I have read Witten's paper, and I am interested on computing the expectation value of a Wilson loop with a representation $R$ on Chern-Simons theory in $d=3$. I am especially interested in cases for $...
5
votes
0answers
65 views
Lost reference: Kähler gravity in six dimensions and three dimensional $SL(2,\mathbb{C})$ Chern-Simons theory
I've noticed that several references take for a fact that by studying Kähler gravity on a Calabi-Yau threefold one can demostrate that any lagrangian submanifold embedded in the threefold posees three ...
3
votes
1answer
44 views
Definition for long range entanglement (LRE) by generalized local unitary (gLU) and generalized stochastic local (gSL) transformations
I am studying this book: Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems (https://arxiv.org/abs/1508.02595).
In chapter 7, it introduces ...
3
votes
1answer
83 views
Winding number is an integer
In computing the variation of the action in Chern-Simons, and in other contexts, we get the following expression that is named the winding number, where $U$ comes from a gauge transformation:
$$ W[U] =...
2
votes
1answer
89 views
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 ...
2
votes
1answer
105 views
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 ...
4
votes
0answers
52 views
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....
3
votes
0answers
78 views
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 ...
5
votes
1answer
237 views
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 ...
7
votes
1answer
103 views
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 ...
2
votes
0answers
34 views
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 ...
6
votes
1answer
191 views
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 ...
0
votes
0answers
21 views
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 ...
0
votes
0answers
61 views
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 ...
5
votes
1answer
82 views
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 ...
2
votes
0answers
37 views
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 ...
4
votes
0answers
73 views
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 ...
1
vote
0answers
28 views
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)
\...
3
votes
0answers
57 views
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 $F^a_{\mu\nu}=0$...
2
votes
0answers
18 views
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{...
3
votes
1answer
57 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 ...
4
votes
1answer
74 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 $G$-...
2
votes
1answer
105 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 ...
0
votes
1answer
110 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 $\gamma_j=\gamma_j^\dagger$, ...
1
vote
0answers
35 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+\...
1
vote
0answers
24 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 ...
0
votes
0answers
21 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 ...
5
votes
0answers
86 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 ...
2
votes
0answers
41 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 ...
7
votes
0answers
188 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 ...
1
vote
0answers
45 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.&...
