Chern-Simons theory and cohomological invariants of representation varieties

TL;DR

This paper establishes a local rigidity theorem for representations of discrete groups into reductive symmetric spaces, showing that certain geometric invariants like volume remain constant under deformation, by connecting classical invariant theory with modern geometric analysis.

Contribution

It provides a new proof of local rigidity for geometric structures modeled on reductive homogeneous spaces using cohomological invariants and classical invariant theory.

Findings

Volume of manifolds modeled on G/H is invariant under deformation.

Invariant forms on G/H are generated by Chern-Weil and Chern-Simons forms.

The approach reinterprets classical results to prove rigidity theorems.

Abstract

We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space $G/H$ is constant under deformation of the $G/H$-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter $3$-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of $G$-invariant forms on $G/H$ is generated by ``Chern-Weil forms'' and ``Chern-Simons forms''.