Tags: #todo #projects/active\ Refs: ?
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?
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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
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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.
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There is a Chern-Wel morphism k[g]Gad→H∗(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.
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Curvature is given as F=da+A∨A where A is a connection one-form:
- An E-valued form is a differential operator on Γ(E⊗Ω∗M) which is map of graded modules, so D(v⊗a)=Dv⊗a+1|v|v⊗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∨A+cA∧3.