# Questions tagged [chern-simons-theory]

The chern-simons-theory tag has no usage guidance.

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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 ...

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### 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^...

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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 ...

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### 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_{\...

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### 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 $\rho$.)
...

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### 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\...

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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 ...

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### 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 ...

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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 (...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ))
...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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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 ...

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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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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-...

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### 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
$$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$
...

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### 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 ...

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### 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 $n = 2$ ?

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### 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 ...

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### 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
$$\...

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### 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 $P\rightarrow M$ ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

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### 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....

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### 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 $M^{d-1}$. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### $A \wedge A \wedge A$ 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 \...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

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### 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 $3$-...

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### 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 ...

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### 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 ...