Abstract
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 Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).
Highlights
Let be a smooth projective curve of genus ≥ 2 over C
We study Verlinde type formulae on the moduli space of rank 2 Gieseker stable (Higgs) sheaves on, where is a smooth projective surface satisfying ( ) > 0 and 1( ) = 0
Using Mochizuki’s formula [35, Theorem 1.4.6], the latter can be written in terms of integrals on Hilbert schemes of points
Summary
Let be a smooth projective curve of genus ≥ 2 over C. The Verlinde formula (for rank 2 and trivial determinant), originating from conformal field theory [44], is the following expression: dim 0 ( , L⊗ ) =. This formula has been proved by several people [39, 5, 41, 32, 8, 38, 7, 46] (for rank 2) and [11, 3] (for general rank). We study Verlinde type formulae on the moduli space of rank 2 Gieseker stable (Higgs) sheaves on , where is a smooth projective surface satisfying ( ) > 0 and 1( ) = 0
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