Vafa-Witten invariants from modular anomaly

Abstract

Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the holomorphic functions themselves, thereby providing the Betti numbers of instanton moduli spaces on such surfaces. As a result, we derive a closed formula for the generating functions and their completions for all $N$. Besides, our construction reveals in a simple way instances of fiber-base duality, which can be used to derive new non-trivial identities for generalized Appell functions. It also suggests the existence of new invariants, whose meaning however remains obscure.

26+25 pages, 1 figure; a clarification added; numerous improvements of the presentation, version accepted for publication in Commun.Num.Theor.Phys; references updated

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Mock Theta Functions
• A Farey Tail for Attractor Black Holes
• Fiber-Base Duality and Global Symmetry Enhancement
• Vafa-Witten invariants from exceptional collections
• Invariants of moduli spaces of stable sheaves on ruled surfaces

Cited by in corpus (6)

• Vafa-Witten invariants from exceptional collections
• On the BPS spectrum of 5d SU(2) super-Yang-Mills
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Scaling Black Holes and Modularity
• Sheaves on surfaces and virtual invariants
• Refinement and modularity of immortal dyons