Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

University Of California-San Diego, La Jolla CA

Investigators

Abstract

This proposal will help support the workshop on "Modular forms around string theory", scheduled for September 16-20 2013 at the Fields Institute in Toronto, Canada. The workshop is part of the thematic program on Calabi-Yau varieties at the Fields Institute. The NSF support will complement the existing funding and will be used to cover some of the travel expenses of the participants based in the United States. The Fields Institute is a major international research center with extensive experience in organizing similar events. The workshop in question focuses on the appearance of modular forms in many different contexts: in the physics of Calabi-Yau manifolds, such as generating functions for certain physical quantities; in the arithmetic of Calabi-Yau manifolds, as part of the general Langlands philosophy in the study of Galois representations of Calabi-Yau varieties defined over number fields; in the structure of geometric invariants on Calabi-Yau varieties, such as Donaldson-Thomas invariants. All of these represent extremely active topics of research, and the workshop aims to synthesize the different points of view brought by each of these perspectives. Calabi-Yau manifolds play a vital role at the crossroads of mathematics and physics. While they first arose as important objects in geometry, Calabi-Yau manifolds really came to prominence after they appeared naturally in string theory. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time. To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small "curled up" geometric object. These objects are Calabi-Yau manifolds. Their introduction into physics led to a great deal of new mathematics. A crucial property of Calabi-Yau manifolds is that they both have very delicate flatness properties (satisfying so-called Ricci flatness) and they can be defined using systems of polynomials equations. As a result, they can be viewed as both algebro-geometric and arithmetic objects, the latter if the polynomial equations have coefficients in a number field. Intimate connections have been found between the physical, algebro-geometric and artihmetic features of Calabi-Yau manifolds. The goal of this workshop is to explore these relationships, making new connections between researchers with these three different perspectives. This award is co-funded by the Algebra and Number Theory and the Geometric Analysis programs.