Counting twisted sheaves and S-duality

Abstract

We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surfaces using the moduli spaces of µ r -gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are naturally used to study the period-index theorem for the corresponding µ r -gerbe in the Brauer group of the surfaces. Deformation and obstruction theory of the twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define virtual fundamental classes on the moduli spaces and define the twisted Vafa-Witten invariants using virtual localization and the Behrend function on the moduli spaces. As applications for the Langlands dual group SU(r)/Z r of SU(r), we define the SU(r)/Z r -Vafa-Witten invariants using the twisted invariants for étale gerbes, and prove the S-duality conjecture of Vafa-Witten for the projective plane in rank two and for K3 surfaces in prime ranks. We also conjecture for other surfaces.CONTENTS 2.1. Algebraic stacks and Deligne-Mumford stacks 2.2. Gerbes and essentially trivial gerbes 2.3. Brauer group and optimal gerbes 3. Moduli space of twisted sheaves and twisted Higgs sheaves on surfaces 3.1. Étale gerbe twisted sheaves 3.2. Twisted stability and the moduli stack 3.3. Moduli of X-twisted sheaves for optimal gerbes 3.4. Twisted Higgs sheaves and the moduli stack 4. Perfect obstruction theory, virtual classes and Vafa-Witten invariants 4.1. Perfect obstruction theory for moduli of twisted stable sheaves 4.2. Deformation and obstruction theory for twisted Higgs sheaves 4.3. SU(rk)-twisted Vafa-Witten invariants 4.4. The G m -fixed loci 4.5. Invariants defined by the Behrend function 5. Joyce-Song twisted stable pairs 5.1. Background to count semistable objects 5.2. Hall algebras and the integration map 5.3. Application to semistable S-twisted Higgs sheaves 5.4. Generalized twisted SU(rk)-Vafa-Witten invariants VW tw 5.5. The SU(r)/Z r -Vafa-Witten invariants and proof in rank two for P 2 6. S-duality conjecture for K3 surfaces 1 2 YUNFENG JIANG 6.1. K3 surfaces-Picard numbers, Mukai Lattice and Brauer group 43 6.2. Tanaka-Thomas's Vafa-Witten invariants for K3 surfaces 44 6.3. Twisted Vafa-Witten invariants-essentially trivial gerbes 44 6.4. Twisted Vafa-Witten invariants-optimal gerbes 46 6.5. Proof of S-duality formula in rank two 49 6.6. Vafa-Witten's S-duality conjecture for K3 surfaces 51 6.7. Discussion on higher rank case 54 56 References 56

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

“…The group H 2 (S, µ r ) classifies µ r -gerbes. In [Jia1], the first-named author developed Vafa-Witten theory of µ r -gerbes (see also [Jia2,Jia3,JK,JTs]). In this paper, which is inspired by [Jia1], we take a different approach:…”

“…• We define the SU(r)/Z r partition function by appropriately summing over "Chern character twisted by rational B-field", a notion introduced by D. Huybrechts and P. Stellari [HSt]. We sum differently over the Chern data compared to [Jia1] and in a way which works for arbitrary surfaces. We give a mathematical definition of Z w (q) for any w ∈ H 2 (S, µ r ) and then define Z SU(r)/Zr c 1 (q) by ( 4).…”

On multiple cover formula for local K3 gerbes

“…is the moduli stack of S-twisted sheaves with Mukai vector v. From Yoshioka [51], M tw ω,S (v) is deformation equivalent to the Hilbert scheme Hilb 1´χ (v,v) 2 (S). This is essential to the calculation of Vafa-Witten invariants in [21].…”

“…is by definition the torsion part of the cohomology H 2 (S, O S ) tor . De Jong's theorem [9] implies that the Brauer group Br(S) = Br 1 (S), and Br(S) is the group of isomorphism classes of Azumaya algebras A on S, see [21,Definition 2.10]. Here an Azumaya algebra A on S is an associative (noncommutative) O S -algebra A which is locally isomorphic to a matrix algebra M r (O S ) for some r ą 0.…”

On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

“…One motivation for our study on the Bogomolov-Gieseker inequality for slope semistable torsion free coherent sheaves is the Vafa-Witten theory for projective surfaces and surface Deligne-Mumford stacks in [26], [7], where the Bogomolov-Gieseker inequality for the modified semistable sheaf E will make the moduli space of Gieseker stable sheaves on a root stack surface X empty for c 2 pEq ă 0. The Vafa-Witten theory for surface Deligne-Mumford stacks has applications to prove the S-duality conjecture in [29] which is a functional duality for the generating functions counting SUprq and L SUprq " SUprq{Z r -instantons, see [9], [8], [10]. On the other hand, the Bogomolov-Gieseker inequality for slope semistable torsion free coherent sheaves on a surface Deligne-Mumford stack X is interesting in its own since it will prove some restriction theorem of slope semistable sheaves on X to a large degree divisor inside X .…”