Equivariant K3 invariants

Abstract

In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformai field theory. We consider the invariants calculated by Yau-Zaslow (capturing the Euler characters of the moduli spaces of D2-branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz-Klemm-Vafa (KKV), and Katz-Klemm-Pandharipande (KKP). We observe that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been used to establish mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway's group. This observation leads us to conjectural descriptions of equivariant versions of the KKV and KKP invariants. A K3 sigma model is specified by a choice of 4-plane in the K3 D-brane charge lattice. Symmetries of K3 sigma models are naturally identified with 4-plane preserving subgroups of the Conway group, according to the work of Gaberdiel-Hohenegger-Volpato, and one may consider corresponding equivariant refined K3 Gopakumar-Vafa invariants. The same symmetries naturally arise in the auxiliary CFT state space, affording a suggestive alternative view of the same computation. We comment on a lift of this story to the generating function of elliptic genera of symmetric products of K3 surfaces, and the connection to work of Oberdieck-Pandharipande on curve counts for the product of a K3 surface with an elliptic curve.