  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.

High Energy Physics - Theory;Algebraic Geometry;Representation Theory