Motivic classes of moduli of Higgs bundles and moduli of bundles with connections

Abstract

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero.We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to −∞.

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

“…Let k be a field of characteristic zero and X be a smooth geometrically connected projective curve over k (geometric connectedness means that X remains connected after the base change to an algebraic closure of k). In [12] we calculated the motivic classes of moduli stacks of semistable Higgs bundles on X. These motivic classes are closely related to Donaldson-Thomas invariants, see [22,23].…”

“…These motivic classes are closely related to Donaldson-Thomas invariants, see [22,23]. In [12] we also calculated the motivic classes of moduli stacks of vector bundles with connections on X by relating them to the motivic classes of stacks of semistable Higgs bundles.…”

“…Our formulas for motivic classes of the moduli stacks above are all given in terms of certain motivic classes B γ called motivic Donaldson-Thomas invariants (see Section 1.5 for this terminology), which we are going to define. First of all, we recall that in [12,Section 2] we defined (following earlier works [11,Section 1], [19], and [22]) the ring of motivic classes of Artin stacks denoted Mot(k). We also defined its dimensional completion Mot(k).…”

“…For a curve X and a partition λ we defined the series J mot λ (z), H mot λ (z) ∈ Mot(k) [[z]] in [12,Section 1.3.2]. The definitions (especially of H mot λ (z)) are somewhat long, so we will not recall them here inviting the reader to look into [12]. We only note that J mot λ (z) and H mot λ (z) are defined in terms of the motivic zeta-function of X (cf.…”

BPS States, Torus Links and Wild Character Varieties

“…Moreover, the motives of the moduli stacks of irregular Higgs bundles, as well as irregular connections over arbitrary fields have been recently computed in [29]. An alternative approach based on wallcrossing for moduli spaces of linear chains on curves was developed in [32], and used in [31] to compute the Hirzebruch genus of moduli spaces of P GL(r, C) Higgs bundles.…”