Questions tagged [chern-simons-theory]

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Knot invariants in WZW CFT via Holographic Principle

In the physics literature the Holographic Principle relates theories in the bulk and the theories in the asymptotic boundary. While the bulk theory is the 3D Chern-Simons theory, the corresponding ...
Student's user avatar
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1 answer
352 views

Defining extended TQFTs *with point, line, surface, … operators*

$\newcommand\Cob{\mathrm{Cob}}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\Rep{Rep}$The ordinary definition of a TQFT is: Defnition: A $d$-dimensional TQFT is a symmetric monoidal functor $\Cob^...
Pulcinella's user avatar
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3 votes
0 answers
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Discreteness of volumes of boundary-parabolic representations

Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
Calvin McPhail-Snyder's user avatar
3 votes
1 answer
84 views

Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy

I am looking for non-trivial examples of flat $U(2)$ connections over the complement of a torus link $\mathcal{S}^3-L$ i.e. $\mathcal{A}:\mathcal{S}^3-L \longrightarrow \mathfrak{U}(2)$ such that $F_{\...
Kilmuntad's user avatar
8 votes
0 answers
240 views

What does the Chern-Simons invariant of a hyperbolic $3$-manifold mean?

Let $M$ be a closed $3$-manifold and $\rho : \pi_1(M) \to \operatorname{SL}_2(\mathbb C)$ a representation. (Feel free to replace $\rho$ with a flat $\mathfrak{sl}_2$ connection with holonomy $\rho$.) ...
Calvin McPhail-Snyder's user avatar
2 votes
0 answers
83 views

On the "integrality condition" of the bilinear form in the Chern-Simons action

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\mathcal{M}$ be a principal $G$-bundle over a smooth orientable manifold $\mathcal{M}$. Furthermore, let $\langle\cdot,\cdot\...
G. Blaickner's user avatar
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What does it mean for correlation functions to be dominated by the vacuum block for a 2D CFT?

In a 2D CFT, correlation functions dominated by the vacuum block have no conical defects. You can calculate the OPE and determine the correlation function using the D-S equations and cancel out UV ...
Burak Guner's user avatar
9 votes
2 answers
430 views

Formula for the anomalies of spin Chern-Simons theories?

$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons ...
Arun Debray's user avatar
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1 answer
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Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
207 views

Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
Calvin McPhail-Snyder's user avatar
5 votes
0 answers
174 views

References for computing $n$-point correlations in Chern-Simons theory

I am interested in learning how to compute $n$-point correlation functions in Chern-Simons theory, thought of as a TQFT (similar to Witten's work linking that theory to knot theory). I am mostly ...
Malkoun's user avatar
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9 votes
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304 views

What is the current state of research in Chern-Simons theory?

I'm a PhD student in mathematical physics looking for some orientation. As asked in the title, I would like to know the current state of research in Chern-Simons theory. More specifically, what are ...
user158970's user avatar
3 votes
1 answer
412 views

Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?

Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals: abelian Chern-Simons theory on non-spin manifolds --- $$ \int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA )) ...
annie marie cœur's user avatar
6 votes
0 answers
222 views

Equivariant Venice Lemma

In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as Theorem: For ...
Eric Schlarmann's user avatar
5 votes
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228 views

Does Dijkgraaf-Witten theory have a time-reversal symmetry?

By having a time-reversal symmetry I mean that there is a local anti-unitary symmetry (representing the non-trivial element of $Z_2$) of the state-sum construction (or, if you want, of the associated ...
Andi Bauer's user avatar
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11 votes
1 answer
628 views

Importance of the principal bundle in Chern-Simons theory

This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $A$ on a topological 3-...
Andi Bauer's user avatar
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5 votes
1 answer
636 views

Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary

I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant. For a compact 3-manifold, $M$, without ...
Mtheorist's user avatar
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Possibility of defining Chern character form in terms of odd Chern Character form?

Given a complex vector bundle $E\to X$ with a connection $\nabla^E$ and an automorphism $U$ of $E\to X$, one can define an odd Chern character form $\textrm{ch}(\nabla, U)$ in terms of Chern character ...
Ho Man-Ho's user avatar
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4 votes
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3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$. Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named ...
wonderich's user avatar
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2 votes
1 answer
388 views

The exterior derivative of a certain differential form on the space of connections of a surface

Let $Y$ be a closed oriented $2$-dimensional manifold, $G$ a Lie group and $Q \to Y$ a principal $G$-bundle with a given section $q.$ Denote by $\mathcal{A}_Q$ the space of connections on $Q,$ and by $...
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What is variation of the Chern-Simons functional, and why can it be calculated as follows?

Let $G$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $X,$ we let $\...
user avatar
5 votes
1 answer
240 views

The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions. Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...
Valac's user avatar
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4 votes
0 answers
229 views

Chern-Simons and framing dependence$.$

I posted this question to physics.SE last week (cf. here), but it got not attention. I hope it is not too trivial to post it here. According to ref.1, the correlation functions of a Chern-Simons ...
AccidentalFourierTransform's user avatar
13 votes
0 answers
1k views

Eta-Invariant and Atiyah-Patodi-Singer Index Theorem

In Quantum Field Theory and Jones Polynomial (equation 2.16), Witten used a formula relating the APS eta-invariant to the Chern-Simons action. Witten claimed that it is derived from the Atiyah-Patodi-...
Valac's user avatar
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6 votes
0 answers
207 views

Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory

In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
Valac's user avatar
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0 answers
403 views

Chern-Simons theory with non-compact gauge groups G

This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
wonderich's user avatar
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10 votes
1 answer
997 views

The Precise Meaning of the Moduli Space of Flat Connections?

Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists. For 3d Chern-...
wonderich's user avatar
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19 votes
1 answer
1k views

How is Chern-Simons theory related to Floer homology?

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
Henry's user avatar
  • 1,410
8 votes
2 answers
368 views

How do I calculate the modular fusion category from a given Lie algebra and level in Chern-Simons theory?

In Chern-Simons theory, one has modular fusion categories that are labelled by a Lie algebra and a "level", e.g. $SU(2)_2$ ("$SU(2)$ level $2$"). Physically this modular fusion category describes the ...
Andi Bauer's user avatar
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2 votes
0 answers
73 views

Is there an analog of a Chern-Simons formula for the pfaffian $Pf(F)$ of a $SO(2n)$ curvature $F$?

..something similar to $tr(A \wedge dA + 2/3 * A \wedge A \wedge A)$ for $n = 2$ ?
user50311's user avatar
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1 vote
1 answer
247 views

Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?

In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...
Valac's user avatar
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8 votes
0 answers
517 views

What is $SL(2,\mathbb{R})$-Chern-SImons Theory?

I found in physics that Chern-Simons theory is closely related with three dimensional gravity. From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for $$\...
Valac's user avatar
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9 votes
0 answers
329 views

Twisted Chern-Simons, and Twisted Wess-Zumino Term

I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten. Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
Valac's user avatar
  • 615
21 votes
2 answers
865 views

Define the 3d Chern-Simons TQFT on a discrete simplicial complex

Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
wonderich's user avatar
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4 votes
0 answers
148 views

tertiary characteristic class: integration of the Chern-Simons form

Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...
Tobias Diez's user avatar
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6 votes
0 answers
429 views

Integrality of the Chern-Simons form and normalization of the action

I'm somewhat confused about normalization of the Chern-Simons action (for arbitrary compact gauge group). If we have a trivial principal bundle we write $$S(A)=\frac{k}{8\pi^2}\int_M\text{Tr}\left(A\...
Rik Voorhaar's user avatar
11 votes
0 answers
373 views

Equivalence classes of Wilson lines in $SU(2)_k$ Chern-Simons theory

One basic aspect of the 3D TQFT/2D CFT correspondence that I'd like to understand better is the following. It is often said that the ground states of Chern-Simons theory on a (spatial) torus are in ...
user137's user avatar
  • 181
7 votes
2 answers
634 views

Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$

Let $\Gamma$ be a finite subgroup of SU(2) and consider the quotient of $S^3$ by $\Gamma$ via its left action. Pick a simply connected compact Lie group $G$ and take a flat connection on this quotient....
Yuji Tachikawa's user avatar
3 votes
1 answer
407 views

4-dimensional TQFT with/without requiring spin structure

My questions here are focused on $D$-dimensional topological quantum field theories (TQFTs) which are unitary and which have finite dimensional Hilbert space on a closed spatial manifold $M^{d-1}$. ...
miss-tery's user avatar
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8 votes
1 answer
595 views

Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
ಠ_ಠ's user avatar
  • 5,875
4 votes
1 answer
2k views

Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...
ಠ_ಠ's user avatar
  • 5,875
7 votes
0 answers
257 views

Choice of framing in Gravitational Chern Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
Yingfei Gu's user avatar
10 votes
3 answers
2k views

$A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form $$A \wedge dA + \frac{2}{3}A \wedge A \...
N. Shimode's user avatar
3 votes
1 answer
334 views

Chern-Simons invariants of 2-bridge knots

2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below: Looking at some ...
ThiKu's user avatar
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5 votes
3 answers
1k views

2-bridge knots in the Rolfsen's table

2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a_1,a_2,\ldots,a_k\right]$. Has somebody worked out a ...
ThiKu's user avatar
  • 10.2k
6 votes
1 answer
557 views

Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories. How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...
ShuklaS's user avatar
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8 votes
1 answer
401 views

Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ ...
Peter Samuelson's user avatar
21 votes
3 answers
2k views

What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-...
Qiaochu Yuan's user avatar
24 votes
3 answers
3k views

Interpreting the CS/WZW correspondence

It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary ...
Jamie Vicary's user avatar
  • 2,433
22 votes
1 answer
1k views

Classical and Quantum Chern-Simons Theory

Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway. Let $\Sigma$ be a two-manifold and $M$ a ...
Minhyong Kim's user avatar
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