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Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory

Y. Jiang, M. Kool

TL;DR

The paper addresses a mathematical formulation of S-duality between $SU(r)$ and $SU(r)/\mathbb{Z}_r$ Vafa-Witten partition functions on smooth projective surfaces by employing Yoshioka's moduli of twisted sheaves and Chern characters twisted by a rational $B$-field. It develops a rigorous moduli framework with twisted Higgs pairs, establishes symmetric obstruction theories, and defines $\mathsf Z_{w}(q)$ for all Brauer classes, proving independence from choices in the prime rank setting. For $K3$ surfaces, it derives explicit closed forms and demonstrates that the $S$-duality relation holds as a modular transformation, incorporating flux-sum identities on the $K3$ lattice. The results bridge physical S-duality with algebro-geometric constructions, providing a robust mathematical foundation for twisted Vafa-Witten invariants and their modular properties. The work highlights how Brauer-twisted moduli and B-field twists yield modular generating functions tied to Hilbert schemes and discriminant modular forms, with potential implications for broader geometric and physical contexts.

Abstract

The $\mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $μ_r$-gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong. $S$-duality, a concept from physics, predicts that the $\mathrm{SU}(r)$ and $\mathrm{SU}(r) / \mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$.

Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory

TL;DR

The paper addresses a mathematical formulation of S-duality between and Vafa-Witten partition functions on smooth projective surfaces by employing Yoshioka's moduli of twisted sheaves and Chern characters twisted by a rational -field. It develops a rigorous moduli framework with twisted Higgs pairs, establishes symmetric obstruction theories, and defines for all Brauer classes, proving independence from choices in the prime rank setting. For surfaces, it derives explicit closed forms and demonstrates that the -duality relation holds as a modular transformation, incorporating flux-sum identities on the lattice. The results bridge physical S-duality with algebro-geometric constructions, providing a robust mathematical foundation for twisted Vafa-Witten invariants and their modular properties. The work highlights how Brauer-twisted moduli and B-field twists yield modular generating functions tied to Hilbert schemes and discriminant modular forms, with potential implications for broader geometric and physical contexts.

Abstract

The Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface , was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on -gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational -field, we give a new mathematical definition of the Vafa-Witten partition function when is prime. Our definition uses the period-index theorem of de Jong. -duality, a concept from physics, predicts that the and partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all surfaces and prime numbers .

Paper Structure

This paper contains 10 sections, 11 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Moduli space
  3. Categories of twisted sheaves
  4. Twisted Chern character
  5. Moduli and obstruction theory
  6. Virtual invariants
  7. $\mathrm{SU}(r)/\mathbb Z_r$ partition function
  8. Joyce-Song pairs
  9. Partition functions for $K3$ surfaces
  10. $S$-duality for $K3$ surfaces

Key Result

Theorem 1.4

For any $K3$ surface $S$, prime number $r$, genericThe precise genericity we (and TT2) require is described in Section sec:pfmainthm. polarizations, and $c_1 \in H^2(S,\mathbb Z)$ algebraic, we have

Theorems & Definitions (31)

  • Conjecture 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 21 more