Derived Equivalences of K3 Surfaces and Twined Elliptic Genera

TL;DR

This paper constructs a bridge between derived equivalences of K3 surfaces, moonshine phenomena, and elliptic genera using vertex operator algebras, revealing new symmetries and connections to moonshine theories.

Contribution

It introduces a novel method to attach weak Jacobi forms to symplectic derived equivalences of K3 surfaces, linking algebraic geometry, string theory, and moonshine.

Findings

Reproduces K3 elliptic genus twining genera in examples

Connects derived equivalences to Conway's group actions

Generalizes to recover Jacobi forms from umbral moonshine

Abstract

We use the unique canonically-twisted module over a certain distinguished super vertex operator algebra---the moonshine module for Conway's group---to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective complex K3 surface that fixes a stability condition in the distinguished space identified by Bridgeland. According to work of Huybrechts, following Gaberdiel--Hohenegger--Volpato, any such derived equivalence determines a conjugacy class in Conway's group, the automorphism group of the Leech lattice. Conway's group acts naturally on the module we consider. In physics the data of a projective complex K3 surface together with a suitable stability condition determines a supersymmetric non-linear sigma model, and supersymmetry preserving automorphisms of such an object may be used to define twinings of the K3 elliptic genus. Our construction recovers the K3 sigma model twining genera precisely in all available examples. In particular, the identity symmetry recovers the usual K3 elliptic genus, and this signals a connection to Mathieu moonshine. A generalization of our construction recovers a number of the Jacobi forms arising in umbral moonshine. We demonstrate a concrete connection to supersymmetric non-linear K3 sigma models by establishing an isomorphism between the twisted module we consider and the vector space underlying a particular sigma model attached to a certain distinguished K3 surface.