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Verlinde formulae on complex surfaces: K-theoretic invariants

L. Göttsche, M. Kool, R. A. Williams

TL;DR

This work proposes Verlinde-type formulas for moduli spaces of sheaves and Higgs sheaves on surfaces with pg>0, linking K-theoretic Donaldson and K-theoretic Vafa-Witten invariants. The authors develop a unified framework based on Mochizuki's formula and universal series to express invariants as generating functions controlled by Seiberg-Witten data and nested Hilbert schemes, separating instanton and monopole contributions. They prove universality of the contributing series, perform toric localization to compute them to finite orders, and verify the conjectures in numerous examples, including K3 and various surfaces. The results further extend to higher rank, yield blow-up and disconnected-canonical-divisor formulas, and illuminate the Vafa-Witten theory with μ-classes, thereby connecting Donaldson theory, Vafa-Witten theory, and S-duality in a K-theoretic setting.

Abstract

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between $K$-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and $K$-theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).

Verlinde formulae on complex surfaces: K-theoretic invariants

TL;DR

This work proposes Verlinde-type formulas for moduli spaces of sheaves and Higgs sheaves on surfaces with pg>0, linking K-theoretic Donaldson and K-theoretic Vafa-Witten invariants. The authors develop a unified framework based on Mochizuki's formula and universal series to express invariants as generating functions controlled by Seiberg-Witten data and nested Hilbert schemes, separating instanton and monopole contributions. They prove universality of the contributing series, perform toric localization to compute them to finite orders, and verify the conjectures in numerous examples, including K3 and various surfaces. The results further extend to higher rank, yield blow-up and disconnected-canonical-divisor formulas, and illuminate the Vafa-Witten theory with μ-classes, thereby connecting Donaldson theory, Vafa-Witten theory, and S-duality in a K-theoretic setting.

Abstract

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between -theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and -theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).

Paper Structure

This paper contains 22 sections, 16 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Verlinde formula for moduli of sheaves
  3. Verlinde formula for moduli of Higgs sheaves
  4. K3 surfaces
  5. Instanton contribution and Donaldson invariants
  6. Donaldson invariants
  7. Mochizuki's formula
  8. Universal series
  9. Reduction to toric surfaces
  10. Verifications
  11. Monopole contribution and nested Hilbert schemes
  12. Gholampour-Thomas's formula
  13. Virtual normal bundle and $\mu(c_1(L))$-insertion
  14. Universal series
  15. Reduction to toric surfaces
  16. ...and 7 more sections

Key Result

Theorem 1.4

There exist universal series with the following property. Let $S$ be a smooth projective surface with $p_g(S)>0$, $b_1(S)=0$, and $L \in \operatorname{Pic}(S)$. Let $H,c_1,c_2$ be chosen such that there are no rank 2 strictly Gieseker $H$-semistable Higgs sheaves on $S$ with Chern classes $c_1,c_2$. Let $N:=N_S^H(2,c_1,c_2)$ an where $\ell_a := x^{a K_S - K_S^2 - 3 \chi(\mathcal{O}_S)} (y^{\frac{

Theorems & Definitions (39)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Theorem 2.2: Mochizuki
  • Remark 2.3
  • Remark 2.4
  • ...and 29 more