The Moonshine Module for Conway's Group

We exhibit an action of Conway's group---the automorphism group of the Leech lattice---on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel--Lepowsky--Meurman moonshine module for Conway's group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically-twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically-twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus zero groups otherwise.Comment: 54 pages including 11 pages of tables; minor revisions in v2, submitte

Derived Equivalences of K3 Surfaces and Twined Elliptic Genera

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.Comment: 62 pages including 7 pages of tables; updated references and minor editing in v.2; to appear in Research in the Mathematical Science

Modular Forms on the Double Half-Plane

We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author.Comment: 17 pages: Minor corrections in text (due to a helpful referee), updated affiliations. Accepted for publication in the International Journal for Number Theory (IJNT