---
date: 2021-10-25 00:06
modification date: Monday 25th October 2021 00:06:02
title: Chern Simons and TFTs MSRI Fall 2021
aliases: [Chern Simons and TFTs MSRI Fall 2021]
---

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#todo #projects/active  
Refs:
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# Chern Simons and TFTs MSRI Fall 2021

Website link: <https://www.msri.org/workshops/1026>

## Notes

- Chern-Simons invariant: obstruction to immersing a [3-manifold](Unsorted/Three-manifolds%20MOC.md) in $\RR^4$ conformally.
- APplications to integrability?
- Some talks:
	- [Chern-Simons Theory](Chern-Simons%20Theory) and Fracton
	- How SUSY & Topology Led From Chern-Simons Theory To Solving A Forty Year Old Mathematical Puzzle
	- Chern-Simons, [differential K-theory](differential%20K-theory) and operator theory
	- Astrophysical Observational Signatures of Dynamical Chern-Simons Gravity
	- [categorification](categorification.md)
- [Chern-Simons theory](Chern-Simons%20theory) is a 3-dimensional [TQFT](TQFT.md).
- Slogan: action is proportional to integral of a 3-form.

### Setup

- Take $G$ a Lie group and $\lieg$, can consider $G_\ad$ invariant polynomials on $\lieg$, I'll write this as $k[\lieg]^{G_\ad}$.
	- Todo: solidify what $G_\ad$ is.
	- Need $f(\Ad_g x) = f(x)$ for invariants.
- Take flat principal $G\dash$bundles $P$ on a 3-manifold $M$.
- There is a Chern-Wel morphism $k[\lieg]^{G_\ad} \to H^*(M; \RR)$ of $\CC\dash$algebras.
	- Interesting fact: for $G$ compact or semisimple, $H^*(\B G; \CC) \cong \CC[\lieg]^{G_\Ad}$.
- A type of [gauge theory](gauge%20theory.md)
- [flat connection](flat%20connection.md) : needed for curvature to vanish, corresponds to solving equations of motion.
- Curvature is given as $F = da + A\wedgeprod A$ where $A$ is a connection one-form: 
	- An $E\dash$valued form is a differential operator on $\Globsec{E\tensor \Omega^* M}$ which is map of graded modules, so $D(v\tensor a) = Dv\tensor a + 1^{\abs v}v\tensor da$.
	- Determined by a matrix of 1-forms
- Take a Lie algebra-valued 1 form $A$, then $\Tr(A)$ is a 1-form and the Chern-Simons form is $\Tr(dA \wedgeprod A + c A\wedgepower{3}$.