Towards refined curve counting on the Enriques surface I: K-theoretic refinements

TL;DR

This paper proposes explicit formulas for K-theoretic refined invariants of the Enriques surface, connecting them to Pandharipande-Thomas invariants, with evidence supporting the conjectures and implications for K3 surfaces.

Contribution

It introduces new conjectural formulas linking K-theoretic refined Vafa-Witten and Pandharipande-Thomas invariants for Enriques surfaces.

Findings

Evidence provided for several cases of the conjecture

Conjectural formulas relate different refined invariants

Comments on the K3 surface case by Thomas

Abstract

We conjecture an explicit formula for the $K$-theoretically refined Vafa-Witten invariants of the Enriques surface. By a wall-crossing argument the conjecture is equivalent to a new conjectural formula for the K-theoretically refined Pandharipande-Thomas invariants of the local Enriques surface. Evidence for the conjecture is given in several cases. We also comment on the case of K3 surfaces previously studied by Thomas.