Verlinde formulae on complex surfaces: K-theoretic invariants

Abstract

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between $K$-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and $K$-theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).

Published version. 37 pages

References in corpus (4)

• Equivariant K-theory and refined Vafa-Witten invariants
• Monopole contributions to refined Vafa-Witten invariants
• The Verlinde formula for Higgs bundles
• The equivariant Verlinde formula on the moduli of Higgs bundles

Cited by in corpus (5)

• Virtual Segre and Verlinde numbers of projective surfaces
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• $\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
• Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds