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。广义几何是一类具有简化结构群的几乎复杂流形，对现实弦理论模型的研究具有重要意义。这些是Calabi-Yau流形的自然推广，它们本身就具有数学意义。在一个对弦理论很重要的类中，我们在广义Calabi-Yau流形上寻找的正则结构是由一个“扭曲因子方程”控制的，它将一个平衡的厄米度规耦合到一个向量束的反自对偶连接上。当流形为Kahler Calabi- yau且向量束为切束时，该系统简化为ricci平面度量的Calabi猜想。广义几何的数学理解仍处于萌芽阶段。该方案的目的是进一步发展这一领域，使其成为Kahler Calabi-Yau几何图形的全面扩展。我们将专注于以下紧密相连的问题：在本课程中构建弦理论的新解；表征这些空间的变形，特别是模；理解这些流形中的“世界表实例”及其枚举几何；以及对“广义标定”的理解，即特殊完整流形的标定子流形的类比。这个计划是研究一类新的几何物体的数学，称为“广义几何”，以及这类物体在弦理论中的出现。在数学上，这些结构提供了Calabi-Yau几何中一类著名的几何结构的有趣和自然的扩展。从物理上讲，这些扩展被认为是捕获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相联系的目标。该项目是一个多机构和跨学科的努力，涉及布兰代斯大学、哈佛大学和德克萨斯农工大学的数学家和物理学家。

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.