Motivic Donaldson-Thomas Invariants of Parabolic Higgs Bundles and Parabolic Connections on a Curve

Abstract

Let X be a smooth projective curve over a field of characteristic zero and let D be a non-empty set of rational points of X. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on (X, D) and motivic classes of moduli stacks of semistable parabolic Higgs bundles on (X, D). As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

Intersection cohomology of character varieties for punctured Riemann surfaces

“…Similarly to this recursive approach, González-Prieto [GP18] developped a topological quantum field theory associated to character varieties. Fedorov-Soibelman-Soibelman [FSS20] computed the motivic class of the moduli stack of semistable parabolic Higgs bundles.…”

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

“…Namely, in [HL19], Hoskins and Lehaleur established what they called a "motivic non-abelian Hodge correspondence" by proving an equality of the Voevodsky motives of M dR (r, d) and M K X (r, d); their result indeed holds for any characteristic zero field, not just C. In fact, by considering d = 0 and the stacky version of these moduli spaces, a similar result was proved before in [FSS18], for motivic classes in the Grothendieck ring of stacks, but through a completely different technique, namely point counting. This was recently generalized to the parabolic setting in [FSS20].…”