This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Generalized geometries form a class of almost complex manifolds with reduced structure groups, which have become of central importance to the study of realistic string theory models. These are natural generalizations of Calabi-Yau manifolds and are of mathematical interest in their own right. In one class important for string theory, the canonical structure one seeks on a generalized Calabi-Yau manifold is governed by a ``warp factor equation'' that couples a balanced Hermitian metric to an anti self-dual connection of a vector bundle. When the manifold is Kahler Calabi-Yau and the vector bundle is the tangent bundle, this system reduces to the Calabi conjecture for Ricci-flat metrics. The mathematical understanding of generalized geometries is still in its nascent stage. The purpose of this proposal is to develop this field further, into a full-fledged extension of Kahler Calabi-Yau geometries. We will focus on the following tightly interconnected problems: constructing new solutions to string theory in this class; characterizing the deformations and specifically the moduli of these spaces; understanding "worldsheet instantons" and their enumerative geometry in these manifolds; and an understanding of "generalized calibrations," the analog of calibrated submanifolds of special holonomy manifolds.
The proposed project is to study the mathematics of a new class of geometric objects called "generalized geometries", and the appearance of this class in string theory. Mathematically, these structures provide interesting and natural extensions of a well-known class of geometric constructions in Calabi-Yau geometry. Physically, these extensions are known to be required to capture essential features of particle physics and cosmology, and will push string theorists closer to the goal of making contact with observations. The project is a multi-institutional and interdisciplinary effort, involving mathematicians and physicists at Brandeis, Harvard, and Texas A&M.
