Talk

Families of Dirac type operators constructed from the supersymmetric Wess-Zumino-Witten model are a useful tool in Fredholm operator realization of twisted K-theory classes on compact Lie groups. They transform in a covariant manner with respect to the action of a central extension of a loop group, the level of the representation giving directly the Dixmier-Douady class of the twisting gerbe. I want to describe a deformation of this system in the language of quantum affine algebras. The loop group covariance property is replaced by a "infinitesimal" Hopf algebra covariance with respect to a quantum enveloping algebra $U_q(\hat g)$ and the Dixmier-Douady class is defined purely algebraically from the action of a central group like element in the Hopf algebra. This is a ongoing project with Antti Harju.
 

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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. Eguchi calculated the multiplicities of the massless characters in the Hilbert scheme of points on K3 surface. He has stated a problem of calculating these multiplicities for an arbitrary hyper Kahler manifold. In this talk we propose a way of solving this problem. We claim that the set of manifolds of the Hilbert schemes of n points on K3 surface is a set of multiplicative denerators of the cobordism ring of the symplectic manifolds, hence the Eguchi calculation defines the entropy of any hyper Kahler manifold. To prove the claim we need to show that the Milnor number of the hilbert scheme of points on K3 is not zero. We also discuss a possible connection of the elliptic genus of K3 and the Joyce invariants.

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). For rigid Calabi-Yau three-folds, the group degenerated by these generators can be identified as a Picard modular subgroup of SU(2,1). For non-rigid ones, it probably includes SL(3,Z) or even larger arithmetic groups. I will use these symmetries to obtain the contributions of NS5-branes at the semi-classical level.

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 automorphism group -- i.e. an automorphic form on a Grassmannian -- with the property that the exponents of the product factors are coefficients of a vector valued modular form.  I will end by mentioning some of the main open questions in this area.

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. Seiberg duality appears to be equivalent to discrete Weyl group symmetries in the parameter space of the latter integrals. In a joint work with G. Vartanov [arXiv:0910.5944] we have systematically analyzed all known dualities and suggested many new ones using known relations for integrlas. In this talk I will briefly outline the structure of indices and integrals and discuss known automorphic properties of these objects.