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

Abstract

The 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 SU(r )/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 SU(r ) and SU(r )/Z r partition functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all K 3 surfaces and prime numbers r .Communicated by Jean-Yves Welschinger.

Exact formulae and Turán inequalities for Vafa–Witten invariants of surfaces

“…Thus, with the tools of modular forms at our disposal, we are in a position to obtain precise information on the Vafa–Witten invariants in question. One of the most natural families of surfaces, which we consider throughout, are the surfaces, and there have been several recent results for the generating function of Vafa–Witten invariants in this case [15, 22, 23].…”

“…In [15], Jiang and Kool determined the generating function for SU(p)/Z p Vafa-Witten invariants for prime p. To describe the result, fix a Chern class c 1 ∈ H 2 (S, Z), let µ p be the cyclic group of order p and let [15]). For a K3 surface S, prime p, generic polarization and c 1 ∈ H 2 (S, Z) algebraic, we have the partition function…”