Abstracts for Automorphic forms, Kac-Moody Lie algebras and Strings

Hyperbolic Weyl groups appear as symmetries of many gravitational systems when these systems are studied in extreme limits near space-like singularities. After reviewing the origin of this appearance of arithmetic structures in gravity, the hyperbolic reflection groups will be reinterpreted as modular groups of type similar to $PSL(2,Z)$ but over other integer structures in algebras of higher dimension than the real numbers. This can be used to reformulate the fundamental equation of quantum gravity in this limit in terms of automorphic forms.

We prove that the existence of a strongly reflective modular form of a large  weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some very special cases, it is equal to zero. We construct three new strongly reflective modular forms of singular weight with $10$, $8$ and $6$ variables which produce three towers (8+3+4) of strongly reflective modular forms with the simplest possible divisor.

Given an indefinite lattice with signature $(1,n-1)$, two kinds of non-holomorphic theta functions with nice modular transformation properties can be defined. The first one is the Siegel-Narain theta function which has modular weight $(1,n-1)/2$. The second one is the indefinite theta function defined by Zwegers, which has weight $(0,n)/2$. This talk will discuss a theta function for a lattice with signature $(2,2n-2)$, which combines the properties of the two previously mentioned theta functions.

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).