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

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The Dark Side of Number Theory (Modular Forms and Quantum Black Holes)

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Speaker: 
A. Dabholkar
Date: 
Mon, 2010-05-10 10:30 - 12:00
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

Hyperbolic Weyl groups and gravity

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Speaker: 
A. Kleinschmidt
Date: 
Mon, 2010-05-10 14:00 - 14:50
Location: 
MPIM Lecture Hall
Parent event: 
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.

Superconformal indices, matrix integrals, and duality

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Speaker: 
V. Spiridonov
Date: 
Mon, 2010-05-10 15:00 - 15:50
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

New reflective modular forms and modular varieties of Calabi-Yau type

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Speaker: 
V. Gritsenko
Affiliation: 
Lille
Date: 
Mon, 2010-05-10 16:30 - 17:20
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

An introduction to Borcherds-Kac-Moody Lie algebras, vertex algebras, and related automorphic forms

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Speaker: 
U. Rey
Date: 
Tue, 2010-05-11 10:00 - 10:50
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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

Multi-centered $N=2$ black holes and Mock-Siegel-Narain theta-series

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Speaker: 
J. Manschot
Date: 
Tue, 2010-05-11 11:15 - 12:05
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

Analytic torsion of certain Calabi-Yau threefolds

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Speaker: 
K.-I. Yoshikawa
Date: 
Tue, 2010-05-11 14:00 - 14:50
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

Duality in hypermultiplet moduli spaces

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Speaker: 
B. Pioline
Date: 
Tue, 2010-05-11 15:00 - 15:50
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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).

Representations of $SL(2,\mathbb{Z})$ and automorphic forms of singular and critical weight

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Speaker: 
N.-P. Skoruppa
Date: 
Tue, 2010-05-11 16:30 - 17:20
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

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.

Eisenstein series and scattering amplitudes

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Speaker: 
P. Vanhove
Date: 
Wed, 2010-05-12 10:00 - 10:50
Location: 
MPIM Lecture Hall
Parent event: 
Automorphic forms, Kac-Moody Lie algebras and Strings

Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, $En(Z)$, of simply-laced Lie groups in the $En$ series ($n\le 8$). In  articular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein’s theory,  with coefficients that are automorphic functions.

 

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