Chern Simons and TFTs MSRI Fall 2021

Notes

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.

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 $\operatorname{Tr}(dA \vee A + c A{ {}^{ \scriptscriptstyle\wedge^{3} } }$.