Conference

The elliptic genus of K3 and its decomposition into characters of the N=4 superconformal algebra of associated conformal field theories can be viewed as the outset of Mathieu Moonshine. Thus, extended supersymmetry induces additional properties of the elliptic genus, which so far lack a satisfactory geometric interpretation. We investigate the implications of this decomposition on geometric structures that underlie the elliptic genus.

We consider dual pairs of four dimensional heterotic/type IIA CHL models with 16 space-time supersymmetries. We provide strong evidence for the existence of an S-duality acting on the heterotic axion-dilaton by a Fricke involution S --> -1/NS, where N is the order of the orbifold symmetry. While most models are self-dual, in some cases S-duality relates the CHL model to a compactification of type IIA on an orbifold of T^6. We provide a simple criterion to determine whether a model is self-dual or not. Finally, we argue that in self-dual CHL models the lattices of electric and magnetic charges must be N-modular and verify this prediction.

Umbral moonshine attaches mock modular forms and meromorphic Jacobi forms to automorphisms of the Niemeier lattices. It is now known that this association can be recovered from specific, graded modules for the Niemeier lattice automorphism groups. We will describe recent progress in a program to realize these modules explicitly.

We observe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in Zagier's work on traces of singular moduli and Borcherds products. We conjecture the existence of an infinite dimensional graded module for the Thompson group and use the observed relationship to propose a McKay-Thompson series for each conjugacy class of the Thompson group and then construct weakly holomorphic weight one-half forms at higher level that coincide with the proposed McKay-Thompson series. We also observe a discriminant property in this conjectured moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.

This talk will summarize some recent results in bimetric theory, including the existence of the square-root matrix, possible connection to partial masslessness and conformal gravity, the structure of constraints and finally, the cosmological implications of the theory.