Questions tagged [topological-field-theory]
Use this tag for topological field theory (Tft) and topological string theory (tst) questions.
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Why is the correlation function of local operators in topological field theory independent of the position they are inserted?
I am reading "A mini course on topological strings"(hep-th/0504147).
In the last paragraph of section 3.1, author mentioned that if a topological field theory have general coordinate ...
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How to obtain the relation of eta invariant of the trivial gauge field and Chern-Simons invariant of the flat connection?
In Quantum Field Theory and the Jones Polynomial by Edward Witten(1989),
how does $\eta(0)$ come from in this equation?
$$\frac{1}{2}(\eta(A^{(\alpha)})-\eta(0))=\frac{c_2}{2\pi}I(A^{(\alpha)})$$
$c_2(...
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Why can Principal $G$ Bundles be Trivialized when $G = SU(N)$?
Reading about TQFT one usually comes about the fact that over 3-manifolds, Simply Connect Lie Group-bundles can be trivialized, yet it is a bit hard to find a clear answer online. Why is that the case?...
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Basic Question on Differential Forms (Chern-Simons Level Quantization)
I came across the following post regarding the boundary term in Chern-Simons theory (specifically the level quantization of the theory). I am new to differential forms so the following questions may ...
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Quantization of string via topological twist
Polyakov action of a bosonic string propagating in Minkowskian spacetime is:
$$S[\gamma, X] = \frac{T}{2}\int \mathrm{d}^{2}\sigma{\sqrt{-\gamma}}\gamma^{ab}\partial _{a}X^{\mu}(\sigma)\partial_{b}X^{\...
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If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition
Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g.
If $(\nabla\times\nabla\Phi)_i = \...
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Examples of (3d or 4d) Chern Simons in the real world
In theoretical physics, people define:
$3d$ Chern Simons theory attached to a compact Lie group $G$, e.g. its fields on a $2d$ time slice is the space of flat $G$-connections on that slice. This ...
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Looking for a source to explain the Process of Topological twisting
as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
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Normalization in the Abelian Chern-Simons action
In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
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Integrating out a Lagrange multiplies field enforcing a differential form to be "$2\pi$-integral periodic"
In the appendix A (page: 26-27) of the paper https://arxiv.org/abs/hep-th/0108152 by Juan Maldacena, Gregory Moore and Nathan Seiberg, the author writes the following "action"
$$e^{-i\int_{...
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Non-Abelian Chern-Simons Theory References
I am studying Chern-Simons theories and am fairly familiar with the usual Abelian $U(1)$ Chern-Simons theory. I am now looking to extend my knowledge to non-Abelian Chern-Simons and am having a hard ...
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What is a symmetry defect?
I found that it is a normal concept appearing in condensed matter physics and especially topological order field. I have been aware of the topological defect. But what is a symmetry defect? Could ...
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Fusion 2-categories for string-like excitations: a more concrete description?
I'm familiar with how fusion categories describe the fusion of point-like excitations, and how braided fusion categories describe the fusion of anyons in 2+1D topological order. Concretely, a fusion ...
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Dimensionality of state space of TQFTs
As the title suggests, I am wondering about the dimensionality of state spaces in $d$-dimensional TQFTs. As of yet I have mostly been concerned with the mathematical, functorial definition of TQFTs as ...
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Topological charges and universal gravity
After reading two days ago a paper, https://arxiv.org/abs/2307.10365, I read that topological charges DO NOT gravitate. Is that statement right? Why I have doubts:
Topological charges have "...
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Can anyons exist on a torus without any additional conditions?
While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
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What effect does the topological term have in string theory?
In string theory we may have an Polyakov action like:
$$S=\int\sqrt{g} (g^{ij}\eta_{\mu\nu}\partial_i X^\mu \partial_j X^\nu + \kappa R) d\sigma^2$$
The curvature $R$ gives a topological term (it is ...
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Link complement states in Chern Simons theory
I have been trying to understand how to get an explicit quantum state from a given link in Chern-Simons. Lets say the compact gauge group being SU(2) (this seems to be the most widely studied). I ...
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QED theta term as pullback of a volume form
The well-known QED theta term is
$S_\theta = \frac{\theta}{4\pi} \int d^4 x F \wedge F $
where $F=dA$ is the field strength 2-form. Altland and Simons (p.547) introduce a more general category of $\...
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Sigma models as topological quantum field theories
I'm wondering how sigma models are supposed to define TQFTs. Suppose I want to consider a 2D TQFT with target $X$ (see page 15 of https://www.ams.org/bookstore/pspdf/ulect-72-intro.pdf)*. According ...
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Is it actually true that Chern-Simons theory is topological?
Chern-Simons theory has action $$\tag{1} S = \frac{k}{4\pi}\int_X tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$
Here, $X$ is some compact 3-manifold, perhaps with boundary, and $A$ is a connection ...
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How do you calculate the partition function on a manifold-with-corners in extended TQFT?
In Atiyah's formulation, a Topological Quantum Field Theory (TQFT), is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns:
\begin{align}
\text{Closed compact $(d-1)$-manifolds} &\to \...
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1+1D simple vacuum EFE solution
Can there be any solutions for simple vacuum Einstein Field Equations in 1+1D (1 space and 1 time dimension) i.e $R_{\mu\nu} = 0$ except for flat space?
I tried different combinations of random ...
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Why does additional term to electromagnetic Lagrangian leave Maxwell's equations unchanged?
The addition of
$$\mathcal{L}' = \epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \vec{E}\cdot\vec{B}$$
to the electromagnetic Lagrangian density leaves Maxwell's equations unchanged (shown ...
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What are non-propagating fields?
I have read at different places that in 3 spacetime dimensions, there are NO propagating gravitational degrees of freedom. This seems to imply that we have only "non-propagating" degrees of ...
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Fundamental group of configuration space of gauge theories
If I consider the space of all the gauge fields $A_{\mu}$ (call this $A$) and a proper gauge group $\Omega_*$, I know that the fundamental group $\pi_1(A)=0$ and the for the gauge group, for example $...
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How do equations of motion in BF theory imply triviality of powers of observables?
Following the lectures of Nathan Seiberg at PiTP in 2015 https://www.youtube.com/watch?v=pqgNrVTQ4yM&t=666s, consider $U(1)$ BF theory in 2D
$$S(B,A)=\frac{n}{2\pi}\int_\Sigma B\text{d}A,$$
and ...
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Simplicity constraints from $SO(4)$ Plebanski action
The $SO(4)$ Plebanski action yields a first order formulation of Euclidean General Relativity as a constrained (topological) BF-theory. It depends on a $so(4)$ connection 1-form $\omega^{IJ} = \omega_{...
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Can we expect some nice "conservation laws" to be hold at each vertex of any Feynman diagram?
I'm reading two textbooks: A. Zee's QFT book and Bruce Bartlett's TQFT book.
In Zee's book, chapter 1 & 2 introduces Feynman diagram smoothly.
Although notations are slightly different, I'll ...
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Can any useful physical theories other than TQFTs be formulated on a smooth manifold without a metric structure?
The vast majority of physical theories are formulated on a spacetime that is mathematically represented by a pseudo-Riemannian manifold, i.e. a smooth manifold with a metric tensor structure. The ...
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Duality and corrections to second-order gravity without and with torsion terms
Recently, there appeared a paper by Giacomo Pollari, A Nieh-Yan-like topological invariant in General Relativity, where the action for gravity looks like:
$$S_g=S_{EHP}+S_{HO}+S_{PO}+S_{GB}+S_{NY}+S_{...
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What is the direct sum of 1d domain walls in Toric code model?
I have read this paper:"An invitation to topological orders and category theory" (https://arxiv.org/abs/2205.05565v2). In page 93 and page 109, they show the result of fusion of simple 1d ...
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Topological classification of (classical, Abelian) vortices on a lattice
Consider the XY model on the square lattice. A field configuration $\theta$ is specified by an element of the Abelian group $\mathbb{R}/2\pi \simeq U(1)$ at each vertex of the lattice. The gradient of ...
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Topological Insulators with different spin band
To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
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Gauging the one form symmetries in the GKSW paper
Following is few of the many questions I have regarding the paper "Generalized Global Symmetry" by Gaitto et al. (https://arxiv.org/abs/1412.5148).
In Sec. 4 of the paper they considered the ...
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Topology active subgroup for the QCD vacuum
I've been reading about the nontrivial topological structure of the QCD vacuum and, when studying the different equivalence classes created by the pure gauge fields, all papers say that it is possible ...
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Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
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Why topology indicates edge state?
I have learned some models with topology like Kitaev chain and SSH. All of them possess edge states if the bulk is topologically non-trivial. The reading materials did the calculation and verified ...
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Does Fibonacci anyon come from a representation category of Hopf algebra?
I have heard that the UMTC(unitary modular tensor category) of Fibonacci anyon comes from a quantum group, but the representation category of Hopf algebra is equipped with a forgetful functor to $\...
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Is this a suitable topological quantum field theory?
I thought the last days on how to construct topological quantum field theories (TQFTs) and now I have the following idea:
Suppose I have a 4-dimensional manifold $M$ with a 2-form bosonic field $B$ (...
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Dijkgraaf-Witten for a 2-Group
As a natural extension of my previous question Higher Dijkgraaf-Witten Theory on DW Theory for a 1-form symmetry, we can extend now to 2-groups.
How can we generalize the notion of gauging to a 2-...
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Hilbert Space in Categorical Gauged TQFT
I am trying to understand how gauge theory interacts with the categorical formulation of TQFT. I will formulate my doubts in two different questions.
I have understood gauging a TQFT in different ...
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Higher Dijkgraaf-Witten Theory
I am trying to understand higher-form symmetries in TQFT. In particular the higher-form version of Dijkgraaf-Witten Theory.
It is known that for a 0-form symmetry we can specify the principal G-bundle ...
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Role of fusion/splitting spaces in TQFT
In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They ...
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Energy of a flux tube
I was reading the paper Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of
Gauge Theories by Gukov and Kapustin. I don't understand many of the things in there. But, I think ...
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References for undergraduate Classical Mechanics [duplicate]
If you know of any russian text books that are available in English On Mechanics, the undergrad level please let me know. I have Kleppner and Kolenkow but I wish to read from a Russian Author
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Linking of a sphere with a Wilson line
In the papers such as Ref.[I] and Ref.[II], they have introduced the operator,
$$
U_\alpha (M_{d-2}) = e^{\frac{i\alpha}{g^2}\int_{M_{d-2}}*F} .
$$
They said that the Wilson loop:
$$W_n(\gamma)=e^{in\...
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Topological and non-topological defects?
The meaning of topological defect is only known intuitively to me. One explanation is it is some discontinuity in a system that cannot be removed.
But I would like to know the precise mathematical ...
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$Z_n$ gauge theory from $U(1)$
In Appendix A of the paper, "Generalized Global Symmetries" by Gaiotto et al., they have considered an action, which for the purpose of the question, can be taken to be
$$S=\frac{n}{2\pi}\...
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What does it mean a theory is the gauging of a current?
What does it mean when people say that the Chern-Simons theory $$\mathcal{L}\sim\operatorname{Tr} \left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right)$$
is the "gauging of the the topological ...
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