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

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.

💡 Research Summary

The paper surveys and expands the authors’ recent work on computing motivic classes of moduli stacks of parabolic Higgs bundles and of bundles with irregular ε‑connections on a smooth projective curve over a field of characteristic zero. After recalling the construction of the Grothendieck ring of varieties K₀(Varₖ), its completion Mot(k) equipped with a λ‑ring structure, and the definition of motivic classes for Artin stacks via global quotients, the authors present two guiding examples: the motivic class of the stack of vector bundles (recovering the Kapranov zeta function) and the motivic Göttsche formula for Hilbert schemes of surfaces.

The main objects of study are ε‑connections (ε=0 gives Higgs fields, ε≠0 gives singular connections) together with level‑D parabolic structures and prescribed normal forms ζ at a finite set of marked points D. These data define a monoid Γ_D of discrete invariants (rank, parabolic jumps, degree) and a decomposition of the stack Conn(ε,X,D,ζ) into finite‑type substacks Conn_{γ,d}(ε,X,D,ζ).

The computational framework relies on motivic Hall algebras H_C associated to a 3‑dimensional Calabi–Yau category C (for instance the derived category of coherent sheaves on the curve) and on the motivic quantum torus R_C(V). Using Bridgeland stability, Harder–Narasimhan filtrations and the wall‑crossing factorisation A_Hall^V = ∏_{ℓ⊂V} A_ℓ, the authors apply the integration map H_C → R_C to obtain explicit formulas for the motivic Donaldson–Thomas series. In the Higgs case the Euler form is trivial, so the quantum torus becomes commutative and the DT‑series is expressed as a plethystic exponential \