Chern Simons and TFTs MSRI Fall 2021

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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 in \({\mathbb{R}}^4\) 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 \(k[{\mathfrak{g}}]^{G_{ \operatorname{ad}}} \to H^*(M; {\mathbb{R}})\) of \({\mathbb{C}}{\hbox{-}}\)algebras.
    • Interesting fact: for \(G\) compact or semisimple, \(H^*({\mathbf{B}}G; {\mathbb{C}}) \cong {\mathbb{C}}[{\mathfrak{g}}]^{G_{ \operatorname{Ad} }}\).
  • 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 \(\operatorname{Tr}(dA \vee A + c A{ {}^{ \scriptscriptstyle\wedge^{3} } }\).
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