FRG Collaborative Research: Generalized Geometry, String Theory and Deformations

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This project is a multi-institutional (Harvard, Brandeis and Texas A&M), interdisciplinary (mathematics and physics) effort to study the mathematical theory of generalized geometries and its applications to string theory. 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 fully nonlinear complex Monge Ampere type system 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. Generalized geometries arise as the internal component of spacetime of string theory models that are much closer to our real world than previous constructions. The mathematics that governs such a geometry is a natural extension of deep problems in geometric analysis, algebraic geometry, and deformation theory. The physics that is behind the new geometry provides inspiration and novel approaches to posing and solving the mathematical problems. Their solutions will in turn lead to greater understanding of fundamental problems in physics, making this a truly interdisciplinary collaboration.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 deformations and specifically the moduli of these spaces; computing the dimension of the space's light fields; and an understanding of "generalized calibrations" - the analog of calibrated submanifolds of special holonomy manifolds.