# Simons, Origins of Chern-Simons Theory 

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Motivations: a combinatorial formula for the signature of a 4-manifold.
Triangulate and integrate the first Pontryagin class, yields a term that doesn't vanish coming from the 3-manifold surrounding a vertex.
Setup: $M^3$ closed oriented Riemannian, then $\Frame(M)\to M \in \Prin\Bun_{\SO_3(\RR)}$.
Write $\theta_{ij}$ for the connection form and $\Omega_{ij}$ its curvature form.
Get a 3-form $Q$ which integrates to zero along fibers (since curvature terms are horizontal).
3-manifolds are parallelizable, yielding sections of the frame bundle.
Any two sections $\chi, \chi'$ differ by an integer, so get a well-defined $\Phi(M) \da \int_\chi Q \mod \ZZ$
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Theorems: $\Phi(M)$ is a conformal invariant, and $\Phi(M) = 0$ is necessary for $M$ to admit a conformal immersion into $\RR^4$.
Application: $\Phi(\RP^3) = 1/2$, so although immersible and isometrically embeddable into $\RR^4$, not conformally.

Interpret $\Phi$ as a map from the space of conformal structures to $\RR/\ZZ$, whose critical points are locally conformally flat structures.
Well-known fact: a locally conformally flat simply-connected 3-manifold is diffeomorphic to $S^3$.
Could lead to a shorter proof of 3d smooth Poincaré.
Chern got this invariant to work in all dimensions, led to Annals paper.
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