# Modular Forms and String Duality (06w5041)

Arriving in Banff, Alberta Saturday, June 3 and departing Thursday June 8, 2006

## Organizers

Helena Verrill (Louisiana State University)

## Objectives

Physical duality symmetries relate special limits of the various consistent string theories (Types I, II, Heterotic string and their cousins, including F-theory) one to another. By comparing the mathematical descriptions of these theories, one reveals often quite deep and unexpected mathematical conjectures. The best known string duality to mathematicians, Type IIA/IIB duality also called {bf mirror symmetry}, has inspired many new developments in algebraic and arithmetic geometry, number theory, toric geometry, Riemann surface theory, and infinite dimensional Lie algebras. Other string dualities such as Heterotic/Type II duality and F-Theory/Heterotic string duality have also, more recently, led to series of mathematical conjectures, many involving elliptic curves, K3 surfaces, and modular forms. Modular forms and quasi-modular forms play a central role in mirror symmetry, in particular, as generating functions counting the number of curves on Calabi--Yau manifolds and describing Gromov--Witten invariants. This has led to a realization that time is ripe to assess the role of number theory, in particular, that of modular forms, in mirror symmetry and string dualities in general.

One of the principal goals of this workshop is to look at modular and quasi-modular forms, congruence zeta-functions, Galois representations, and $L$-series for dual families of Calabi--Yau varieties with the aim of interpreting duality symmetries in terms of arithmetic invariants associated to the varieties in question. Over the last decades, a great deal of work has been done on these problems. In particular it appears that we need to modify the classical theories of Galois representations (in particular, the question of modularity) and modular forms, among others, for families of Calabi--Yau varieties in order to accommodate quantum corrections''.

As dictated by the research interests of the participating members, the research activities will be focused on the following themes:

(A) Arithmetic of Calabi--Yau varieties defined over number fields:
Arithmetic of elliptic curves, K3 surfaces, Calabi--Yau threefolds, and higher dimensional Calabi--Yau varieties defined over number fields in connection with string dualities. These will include the following topics and problems: Interpretation of string duality phenomena of Calabi--Yau varieties in terms of zeta-functions and $L$-series of the varieties in question, the modularity conjectures for Calabi--Yau varieties, the conjectures of Birch and Swinnerton-Dyer for elliptic curves and Abelian varietes, the conjectures of Beilinson-Bloch on special values of $L$-series and algebraic cycles, and intermediate Jacobians of Calabi--Yau threefolds. Calabi-Yau varieties of CM (complex multiplication) type and their possible connections to rational conformal field theories and, in the elliptic fibered case, behavior under F-Theory/Heterotic string duality.

(B) Mirror symmetry for families of Calabi--Yau varieties:
Characterization of mirror maps in connection with the mirror moonshine phenomenon and, via Fourier-Laplace transform, the classification of Q-Fano threefolds. In particular, differential equations associated to modular and quasi-modular forms related to GKZ-hypergeometric systems and, more generally, to Picard-Fuchs differential systems will be investigated.

(C) Modular and quasi-modular forms in string duality:
Modular forms and quasi-modular forms have appeared frequently in mirror symmetry contexts, e.g., in the generating functions counting the number of simply ramified covers of elliptic curves with marked points, in Gromov--Witten invariants, and also as mirror maps. The appearance of modular and quasi-modular forms in string dualities, e.g., in the Harvey-Moore conjectures of Heterotic-Type II duality, will be investigated. Understanding why modular and quasi-modular forms play central roles in string dualities is one of our goals.