Quantization of Hodge structures of representation varieties

Given a complex algebraic group $G$ and a compact manifold $M$, the set of representations $\rho: \pi_1(M) \to G$ has a natural algebraic structure, the so-called representation variety. In this talk, we will show how the mixed Hodge structures of these varieties can be encoded in a lax monoidal TQFT via a general quantization procedure by means of Saito’s mixed Hodge modules theory.

This strategy recovers the stratification technique developed by Logares, Muñoz and Newstead and offers a new framework in which mirror symmetry conjectures for $E$-polynomials can be addressed.

Joint work with M. Logares and V. Muñoz.