FRG: Collaborative Research: generalized geometries in string theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。广义几何形成了一类具有约化结构群的几乎复流形，这类流形对现实弦理论模型的研究至关重要。这些都是自然的推广卡-丘流形和数学兴趣在自己的权利。在弦理论中有一类重要的性质，人们在广义卡-丘流形上寻找的正则结构是由一个“翘曲因子方程”决定的，这个方程把一个平衡的厄米特度规与一个向量丛的反自对偶联络耦合起来。当流形是Kahler Calabi-Yau，向量丛是切丛时，这个系统归结为Ricci平坦度量的Calabi猜想。广义几何的数学理解仍处于萌芽阶段。 本建议的目的是进一步发展这一领域，成为一个成熟的扩展卡勒卡-丘几何。我们将专注于以下紧密相关的问题：在这门课上构造弦理论的新解;描述这些空间的变形，特别是模;理解“世界面瞬子”及其在这些流形中的枚举几何;以及对“广义校准，“特殊完整流形的校准子流形的类似物。拟议的项目是研究一类新的几何对象的数学，称为“广义几何”，以及这门课在弦理论中的出现在数学上，这些结构提供了有趣的和自然的扩展一类著名的几何结构在卡-丘几何。从物理上讲，这些扩展被认为是捕捉粒子物理学和宇宙学的基本特征所必需的，并将推动弦理论家更接近与观测联系的目标。该项目是一个多机构和跨学科的努力，涉及数学家和物理学家在布兰迪斯，哈佛和得克萨斯州A M。

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.