Questions tagged [chern-simons-theory]

The tag has no usage guidance.

57 questions
Filter by
Sorted by
Tagged with
123 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 ...
152 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 ...
296 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-...
198 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 ...
48 views

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 ...
102 views

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 ...
190 views

148 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 ...
118 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 ...
289 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-...
130 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 ...
217 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 ...
303 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-...
680 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 $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
135 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 ...
63 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 $n = 2$ ?
143 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 ...
271 views

191 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 ...
535 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....
242 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 $M^{d-1}$. ...
338 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 ...
856 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 ...
162 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 ...
1k views

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 $3$-...
2k 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 ...
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 ...
1k views

The trace of a wedge product of matrices

I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds. I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form ...
1k views

5k views

The Chern-Simons/Wess-Zumino-Witten correspondence

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter ...
750 views

Analog of “Spin” Chern-Simons Theory

3-dimensional Chern-Simons theories, with compact gauge group $G$, are determined by $H^4(BG)$. Looking at $U(1)$, with generator $c_1^2\in H^4(BU(1))=\mathbb{Z}$ for 1st Chern class $c_1$, there are ...
300 views

Set of physical states of FQHE on closed Riemann surface = ?

Disclaimer. One might argue that my question is off topic as it is clearly a question about physics... But I'd like a mathematically phrased answer, and I expect that only a mathematician can offer an ...
510 views

Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups? This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain why ...
612 views

Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...
639 views

SL(2,C) Chern-Simons theory in genus 1

In http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104202513, Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ ...
Cheeger and Simons here defined a family of characteristic classes for principal $G$-bundles ($G$ a Lie group). I'm interested in the special case of \$\hat c_2:H_3(SL(2,\mathbb C);\mathbb Z)\to\...