# Chern-Simons theory Given a Lie group $G$, one can define a Chern-Simons theory with action$S=\frac{k}{4 \pi} \int_{\mathcal{M}} \mathrm{d}^3 x\, \varepsilon^{\mu \nu \rho}\left[A_\mu^a \partial_\nu A_\rho^a+\frac{2}{3} f_{a b c} A_\mu^a A_\nu^b A_\rho^c\right],$where $A$ takes values in the Lie algebra of $G$, $k$ is called the level, and $\mathcal{M}$ is a three-manifold. This defines a [[0607 Topological QFT|TQFT]] because this action is dependent of the the metric. As a consequence, the partition function is a topological invariant of $\mathcal{M}$. ## Observables In a TQFT, all physical observables of the theory must be independent of the metric, so local quantities do not qualify. In Chern-Simons, Wilson line operators constitute all the physical observables of the theory. They are defined via$W_R(C)=\operatorname{Tr}_R P \exp\left({\mathrm{i} \oint_C A_\mu \mathrm{d} x^\mu}\right),$where $R$ is a representation of $G$ and $C$ is a directed loop in $\mathcal{M}$. ## Large-$k$ (weak-coupling limit) - at large $k$, the connection is flat ($F^a_{ij}=0$) - the large $k$ limit of the central charge $c$ is $d=\operatorname{dim}G$, dimension of the gauge group ## Comments - plain CS theory is a string theory but of topological strings - CS theory is most commonly studied in 2+1 dimensions but has natural generalisations to all odd dimensions ## Partition functions on $\Sigma\times S^1$ There are few explicit results for the partition function of Chern-Simons theory on general 3-manifolds. However, for manifolds with topology $\Sigma\times S^1$, the answer is well-known. On these manifolds, the partition function simply computes the trace in the Hilbert space, so$Z=\dim(\mathcal{H}).$For example, take $\Sigma=S^2$ for the group $SU(N)$ at level $k$, then $Z=\dim(\mathcal{H})=1$ because there are no flat connections on $S^2$. For $SU(2)$ on a general surface with genus $g\ge1$, the answer is$\dim(\mathcal{H})=\left(\frac{k+2}{2}\right)^{g-1} \sum_{j=0}^k\left(\sin \frac{(j+1) \pi}{k+2}\right)^{2(g-1)}.$More generally, one can compute the partition function on $\Sigma_{g,n}$ with $n$ punctures. The result for $SU(2)_k$ is given by$Z_{g,n}=\left\langle\prod_{i=1}^n \chi_{l_i}(\phi)\right\rangle_g=\sum_{j=0}^k\left(S_{j 0}\right)^{2-2 g-s} \prod_{i=1}^n S_{j l_i},$where the modular $S$-matrix is given by$S_{i j}=\left(\frac{2}{k+2}\right)^{1 / 2} \sin \frac{(i+1)(j+1) \pi}{k+2}.$ ## Refs - learning materials - [[1999#Dunne (Lectures)]]: more of a condensed matter perspective - David Tong Quantum Hall lectures - Moore at TASI 2019: [Notes](https://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf), [Lec 1](https://youtu.be/PSA8QXp7UrM?feature=shared), [Lec 2](https://youtu.be/iE0DJLNkkvU?feature=shared), [Lec 3](https://youtu.be/J2w5u5q0F0k?feature=shared), [Lec 4](https://youtu.be/zJI6TamZx9M?feature=shared) - [nLab on CS](https://ncatlab.org/nlab/show/Chern-Simons+theory) - [nLab on AdS3-CFT2 and CS-WZW correspondence](https://ncatlab.org/nlab/show/AdS3-CFT2+and+CS-WZW+correspondence) - original paper on CS as TQFT - [[1989#Witten]] - invariants - [[1989#Witten]] - [[1991#Reshetikhin, Turaev]] - string constructions - [[1992#Witten (Jul)]]: open string construction - [[1998#Gopakumar, Vafa]]: closed string construction for the large-$N$ limit and relation to the open string - partition functions on special manifolds - [[1993#Blau, Thompson]]: on $\Sigma\times S^1$ - [[2005#Beasley, Witten]]: Seifert manifolds; using non-Abelian localisation - [[2006#Blau, Thompson]]: circles bundles over $\Sigma$; using Abelianisation ## Related topics - [[0073 AdS3-CFT2]] - [[0090 CS-WZW correspondence]]