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A proof of all ranks S-duality conjecture for K3 surfaces

Yunfeng Jiang, Hsian-Hua Tseng

Abstract

Using the multiple cover formula of Y. Toda for counting invariants of semistable twisted sheaves over twisted local K3 surfaces we calculate the $\SU(r)/\zz_r$-Vafa-Witten invariants for K3 surfaces for any rank $r$ for the Langlands dual group $\SU(r)/\zz_r$ of the gauge group $\SU(r)$. We generalize and prove the S-duality conjecture of Vafa-Witten for K3 surfaces in any rank $r$ based on the result of Tanaka-Thomas for the $\SU(r)$-Vafa-Witten invariants.

A proof of all ranks S-duality conjecture for K3 surfaces

Abstract

Using the multiple cover formula of Y. Toda for counting invariants of semistable twisted sheaves over twisted local K3 surfaces we calculate the -Vafa-Witten invariants for K3 surfaces for any rank for the Langlands dual group of the gauge group . We generalize and prove the S-duality conjecture of Vafa-Witten for K3 surfaces in any rank based on the result of Tanaka-Thomas for the -Vafa-Witten invariants.

Paper Structure

This paper contains 17 sections, 11 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. S-duality conjecture using twisted Vafa-Witten invariants
  3. Twisted multiple cover formula
  4. All ranks S-duality for K3 surfaces
  5. Outline
  6. Convention
  7. Proof of the S-duality conjecture Theorem \ref{['thm_higher_rank_K3_intro']}
  8. Twisted Vafa-Witten invariants counting semistable Higgs pairs
  9. Joyce-Song twisted stable pairs
  10. Generalized twisted $\mathop{\mathrm{SU}}\nolimits(\mathop{\rm rk})$-Vafa-Witten $\mathop{\mathrm{vw}}\nolimits$-invariants
  11. Generalized twisted $\mathop{\mathrm{SU}}\nolimits(\mathop{\rm rk})$-Vafa-Witten invariants $\mathop{\mathrm{VW}}\nolimits^{\hbox{\tiny tw}}$
  12. All ranks $\mathop{\mathrm{SU}}\nolimits(r)/{\mathbb Z}_r$-Vafa-Witten invariants for K3 surfaces
  13. Trivial $\mu_r$-gerbes
  14. Non trivial essentially trivial $\mu_r$-gerbes
  15. Non essentially trivial $\mu_r$-gerbes
  16. ...and 2 more sections

Key Result

Theorem 1.4

(Jiang-Tseng_multiple) We have

Theorems & Definitions (29)

  • Conjecture 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 19 more