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

Roman Fedorov, Alexander Soibelman, Yan Soibelman

TL;DR

The paper develops a comprehensive framework to compute motivic invariants (motivic volumes) of moduli stacks of Higgs bundles and bundles with connections on a curve, including parabolic and irregular structures. It leverages motivic Hall algebras and the motivic quantum torus to formulate Donaldson–Thomas-type generating series and factorization formulas, and derives explicit motivic formulas for irregular semistable parabolic connections under genericity conditions. The authors connect global moduli problems to local models via nilpotent endomorphisms, twisted cotangent bundle geometry, and Mellit-style techniques, producing concrete expressions that extend known results (e.g., Behrend–Dhillon, Göttsche) to irregular/parabolic settings and providing a platform for generalized non-abelian Hodge theory. The work outlines promising directions, including generalizations to arbitrary groups, cohomological approaches, motivic Deligne–Simpson correspondences, and geometric Satake/Baer–Satake perspectives, with potential impacts on understanding motivic DT-invariants in broader symplectic and hyperk"ahler contexts.

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.

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

TL;DR

The paper develops a comprehensive framework to compute motivic invariants (motivic volumes) of moduli stacks of Higgs bundles and bundles with connections on a curve, including parabolic and irregular structures. It leverages motivic Hall algebras and the motivic quantum torus to formulate Donaldson–Thomas-type generating series and factorization formulas, and derives explicit motivic formulas for irregular semistable parabolic connections under genericity conditions. The authors connect global moduli problems to local models via nilpotent endomorphisms, twisted cotangent bundle geometry, and Mellit-style techniques, producing concrete expressions that extend known results (e.g., Behrend–Dhillon, Göttsche) to irregular/parabolic settings and providing a platform for generalized non-abelian Hodge theory. The work outlines promising directions, including generalizations to arbitrary groups, cohomological approaches, motivic Deligne–Simpson correspondences, and geometric Satake/Baer–Satake perspectives, with potential impacts on understanding motivic DT-invariants in broader symplectic and hyperk"ahler contexts.

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.
Paper Structure (27 sections, 6 theorems, 30 equations)

This paper contains 27 sections, 6 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Motivic classes of moduli stacks
  4. Ring of motivic classes
  5. Motivic classes of stacks
  6. Two examples of motivic classes of stacks
  7. Motivic classes of moduli stacks of bundles on a curve (after Behrend and Dhillon)
  8. Motivic Göttsche formula
  9. For which stacks are we interested in computing motivic classes?
  10. Higgs Bundles and -connections
  11. Parabolic structures
  12. Stack of parabolic irregular -connections with prescribed normal forms
  13. Motivic Hall algebras and quantum tori
  14. A version for 2-dimensional Calabi--Yau categories
  15. Summary of results
  16. ...and 12 more sections

Key Result

Theorem 2.3.1

We have in $Mot(\mathbf{k})$ the following formula for the motivic class of the stack of vector bundles of rank $r$ and degree $d$: where $\zeta_X(z)$ is Kapranov's motivic zeta function defined as while $\mathop{\mathrm{Jac}}\nolimits(X)$ is the Jacobian of $X$. Here $Sym^nX$ is the $n$-th symmetric power of $X$.

Theorems & Definitions (10)

  • Theorem 2.3.1: BehrendDhillonFedorovSoibelmans
  • Definition 3.1.1
  • Definition 3.2.1
  • Theorem 4.0.1
  • Theorem 5.1.1: MellitNoPuncturesFedorovSoibelmansIrregular
  • Theorem 5.1.2
  • Lemma 5.2.1: FedorovSoibelmansIrregular
  • Theorem 5.2.2: FedorovSoibelmansIrregular
  • Definition 7.4.1: see KontsevichSoibelmanTwistor
  • Conjecture 7.4.2