Talk

Mathematically the black hole entropy of a hyper Kahler manifold M  as defined by Vafa is related to a special property of the elliptic genus of M. Namely the elliptic genus of M is not just a Jacobi form it but admits a decomposition into the characters of the N=4 super conformal algebra. These are of two types, the massive and massless. The former are essentially theta functions and the later are the Mock theta functions. The collection of the multiplicities of the massless characters defines the entropy. In a series of papers T.

I will construct a tower of strongly reflective modular forms of type $nA_1$. Moreover I give other examples of modular forms (including modular of singular weight) related to the Jacobi theta-series. This is a joint work (in progress) with V. Gritsenko.

The hypermultiplet moduli space in type II string theory compactified on a Calabi-Yau 3-folds provides a framework for a far-reaching generalization of classical and homological mirror symmetry, as well as a convenient packaging of BPS black hole degeneracies consistent with wall-crossing. In addition to the usual action of the monodromy group and discrete Peccei-Quinn symmetries, it should also be invariant under S-duality, which mixes the usual D-brane instantons (or objects in the derived/Fukaya category) with a new type of instantons (NS5-branes, or Kaluza-Klein monopoles).

We construct a family of supersymmetric generalized Kac-Moody superalgebras using automorphic products and show that one of these Lie superalgebras describes a superstring moving on a 10-dimensional torus.

The study of automorphic forms of singular or critical weight can often be reduced to the study of Jacobi forms of singular or critical weight. These can be interpreted as invariants of Weil representations associated to finite quadratic modules. The latter are the key to the understanding of those representations of $SL(2,\mathbb{Z})$ whose kernel is a congruence subgroup. Finally, these representations are intimately connected to the arithmetic theory of integral quadratic forms. This talk provides an overview of these ideas and their interplay.

Physicists Bershadsky-Cecotti-Ooguri-Vafa introduced a certain combination of analytic torsions as a counter part in B-model of elliptic Gromov-Witten invariants of Calabi-Yau threefolds. For Borcea-Voisin threefolds without mirrors, we give an expression of  the BCOV torsion as a nice Borcherds product on the Kaehler moduli of a Del Pezzo surface.

In this talk, I will explain what these Lie algebras, which generalize the semi-simple finite dimensional ones, are and why they were originally studied.  The more interesting ones can be constructed from lattice vertex algebras and hence I will give an idea about this construction.  As we will see, the essential information about the structure of these Lie algebras is contained in a formula known as the denominator formula.  In the cases of interest today, this gives an infinite product expansion of a function on a hyperbolic space transforming nicely under the action of its

There is a direct connection between the Seiberg duality for four dimensional N=1 supersymmetric field theories and the theory of elliptic hypergeometric integrals formulated by the author around 10 years ago. Roemelsberger conjectured in 2007 that superconformal (topological) indices for dual field theories coincide. Dolan and Osborn in 2008 confirmed this for a number of simplest dualities by showing that the indices coincide with the particular elliptic hypergeometric integrals.

The problem of counting quantum degeneracies of certain black holes in string theory has led to interesting new connections with topics in number theory including Siegel modular forms, mock modular forms, and generalized Borcherds superalgebras.