# Chern Simons and TFTs MSRI Fall 2021

Tags: #todo #projects/active\ Refs: ?

# Chern Simons and TFTs MSRI Fall 2021

• Chern-Simons invariant: obstruction to immersing a 3-manifold in $${\mathbf{R}}^4$$ conformally.
• APplications to integrability?
• Some talks:
• Chern-Simons Theory and Fracton
• How SUSY & Topology Led From Chern-Simons Theory To Solving A Forty Year Old Mathematical Puzzle
• Chern-Simons, differential K-theory and operator theory
• Astrophysical Observational Signatures of Dynamical Chern-Simons Gravity
• categorification
• Chern-Simons theory is a 3-dimensional TQFT.
• Slogan: action is proportional to integral of a 3-form.

### Setup

• Take $$G$$ a Lie group and $${\mathfrak{g}}$$, can consider $G_ { \operatorname{ad}}$ invariant polynomials on $${\mathfrak{g}}$$, I’ll write this as $$k[{\mathfrak{g}}]^{G_ { \operatorname{ad}} }$$.
• Todo: solidify what $G_ { \operatorname{ad}}$ is.
• Need $$f({ \operatorname{Ad} }_g x) = f(x)$$ for invariants.
• Take flat principal $$G{\hbox{-}}$$bundles $$P$$ on a 3-manifold $$M$$.
• There is a Chern-Wel morphism $$k[{\mathfrak{g}}]^{G_ { \operatorname{ad}} } \to H^*(M; {\mathbf{R}})$$ of $${\mathbf{C}}{\hbox{-}}$$algebras.
• Interesting fact: for $$G$$ compact or semisimple, $$H^*({\mathbf{B}}G; {\mathbf{C}}) \cong {\mathbf{C}}[{\mathfrak{g}}]^{G_{ \operatorname{Ad} }}$$.
• A type of gauge theory
• flat connection : needed for curvature to vanish, corresponds to solving equations of motion.
• Curvature is given as $$F = da + A\vee A$$ where $$A$$ is a connection one-form:
• An $$E{\hbox{-}}$$valued form is a differential operator on $${{\Gamma}\qty{E\otimes\Omega^* M} }$$ which is map of graded modules, so $$D(v\otimes a) = Dv\otimes a + 1^{{\left\lvert {v} \right\rvert}}v\otimes da$$.
• Determined by a matrix of 1-forms
• Take a Lie algebra-valued 1 form $$A$$, then $$\operatorname{Tr}(A)$$ is a 1-form and the Chern-Simons form is $$\operatorname{Tr}(dA \vee A + c A{ {}^{ \scriptscriptstyle\wedge^{3} } }$$.