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 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.
Highlights
The main result of this paper is a construction which attaches weak Jacobi forms to suitably defined autoequivalences of the bounded derived category of coherent sheaves on a complex projective K3 surface.The origins of our method extend back to the monstrous moonshine phenomenon, initiated by the observations of McKay and Thompson [101,102], Conway–Norton [29], and Queen [95]
Our results have physical significance. They suggest that a certain distinguished super vertex operator algebra is a universal object for supersymmetric non-linear K3 sigma models
This represents a new role for vertex algebra in physics: rather than serving as the “chiral half” of a particular, holomorphically factorizable super conformal field theory, the super vertex operator algebra in question is, simultaneously related to a diverse family of super conformal field theories
Summary
The main result of this paper is a construction which attaches weak Jacobi forms to suitably defined autoequivalences of the bounded derived category of coherent sheaves on a complex projective K3 surface. The analysis of [68] suggests that we can expect to obtain a Jacobi form with level, called a twined elliptic genus, from any supersymmetry-preserving automorphism of a supersymmetric non-linear sigma model. It is generally a difficult matter to compute twined K3 elliptic genera, for the Hilbert spaces attached to supersymmetric non-linear K3 sigma models are only known in a few special cases It has been shown recently by Gaberdiel– Hohenegger–Volpato [68] (cf [75]) that any group of supersymmetry-preserving automorphisms of such a model can be embedded in the Conway group Co0 (Co0 here can be replaced by Co1, but it seems to be more natural to regard Co0 as the operative group). We refer the reader to [3,12,39] for introductory expositions of the deep connection between these notions
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