  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).
