# Modular Forms in String Theory (16w5009)

## Organizers

(Louisiana State University)

(University of Alberta)

(Hokkaido University of Education at Hakodate, Japan)

(Queen's University)

## Description

The Banff International Research Station will host the "Modular Forms in String Theory" workshop from September 25th to September 30th, 2016.

In the last two and half decades, the world has seen explosive
interactions between Number Theory, Arithmetic and Algebraic Geometry,
and Theoretical Physics (in particular, String Theory). To name a few,
the classical modular forms, quasi-modular forms, Jacobi forms, and more
recently Mock and quantum modular forms have appeared in many areas of
Physics such as mirror symmetry, topological quantum field theory, Gromov-Witten
invariants, conformal field theory, Calabi-Yau manifolds and in block hole
entropy. On the other hand, modularity questions of Calabi-Yau varieties
or families of Calabi-Yau varieties in connection with Langlands Program
are getting considerable attention and feedbacks from physics; zeta-functions
and L-series of Calabi-Yau varieties enter scenes
at various places in physics. Concretely, the generating functions of
Gromov-Witten invariants counting some geometric/physical quantities
are described in terms of modular, quasimodular or Jacobi forms. Via
renormalization, it is observed that Feynman integrals are related to multiple
zeta-values, and purportedly to motives. Calculations of the energy and
charge degeneracies of black holes lead surprisingly to Jacobi forms and
Siegel modular forms.

Mathematicians (number theorists) and physicists (string theorists) had
been very few interactions between the two sets of researchers till the
workshop series on Number Theory and Physics at the Crossroads'' started
about 10 years ago with Noriko Yui being one of the chief architects.
Due to these efforts, we witnessed very active and intensive
interactions of both camps to cross boundaries and establish relatively
comfortable rapport that led to exciting new ideas and collaborations.
Some of the outcomes have been documented by a number of proceeding
volumes. Recently, there have been strong desires among mathematicians and
physicists for the next workshop due to the recent rapid developments on
arithmetic and geometric modularity theorems, as well as new theories of modular
forms including mock modular forms, vector-valued modular forms
and quantum modular forms appearing in string theory. In addition, we will use this
workshop to celebrate the work of Professor Noriko Yui and recognize her
contributions in number theory with relation to physics.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT)..