Motivic invariants of moduli stacks of Higgs bundles and bundles with connections: results and speculations

TL;DR

This paper reviews techniques for computing motivic classes of moduli stacks of Higgs bundles and connections, discusses future research directions, and explores conjectural links with the P=W conjecture and geometric representation theory.

Contribution

It introduces new methods for motivic class computations and proposes generalizations of the P=W conjecture inspired by recent work in complex symplectic geometry.

Findings

Motivic classes of moduli stacks are computed for specific cases.

Proposed generalizations of the P=W conjecture.

Connections to the Riemann--Hilbert correspondence and geometric Satake.

Abstract

We review some results and techniques from our papers devoted to the computation of motivic classes of stacks of parabolic Higgs budles and bundles with connections on a curve. In the last section we present some directions for future work, as well as some speculations. The latter include a generalization of the P=W conjecture inspired by the work of Maxim Kontsevich and the third author on the Riemann--Hilbert correspondence for complex symplectic manifolds as well as our running project on the motivic classes of the moduli stacks of nilpotent pairs on the formal disk and geometric Satake correspondence for double affine Grassmannians.