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 R4 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 g, can consider Gad invariant polynomials on g, I’ll write this as k[g]Gad.
    • Todo: solidify what Gad is.
    • Need f(Adgx)=f(x) for invariants.
  • Take flat principal G-bundles P on a 3-manifold M.
  • There is a Chern-Wel morphism k[g]GadH(M;R) of C-algebras.
    • Interesting fact: for G compact or semisimple, H(BG;C)C[g]GAd.
  • A type of gauge theory
  • flat connection : needed for curvature to vanish, corresponds to solving equations of motion.
  • Curvature is given as F=da+AA where A is a connection one-form:
    • An E-valued form is a differential operator on Γ(EΩM) which is map of graded modules, so D(va)=Dva+1|v|vda.
    • 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(dAA+cA3.
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