Equivariant K3 Invariants

TL;DR

This paper explores the connection between K3 surface curve invariants and superconformal field theories, revealing new symmetries and equivariant invariants linked to moonshine phenomena and the Conway group.

Contribution

It establishes a novel link between enumerative geometry invariants of K3 surfaces and the chiral ring of an auxiliary superconformal field theory, incorporating equivariant and refined invariants.

Findings

Invariants match Ramond ground states of an auxiliary CFT.

Symmetries correspond to Conway group subgroups.

Connections to mock modular moonshine are identified.

Abstract

In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal 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 show that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been observed to give rise to mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway's group. We also study equivariant versions of these 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.