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)提供资金的。广义几何构成了一类具有简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具有重要意义。这些都是Calabi-Yau流形的自然推广，它们本身就具有数学意义。在弦理论中重要的一类中，人们在广义Calabi-Yau流形上寻找的正则结构是由一个将平衡的厄米度规耦合到向量丛的反自对偶连接的‘’翘曲因子方程‘’所支配的。当流形是Kahler Calabi-Yau，向量丛是切丛时，这个系统退化为Ricci平坦度量的Calabi猜想。对广义几何的数学理解仍处于萌芽阶段。这项提议的目的是进一步开发这一油田，成为Kahler Calabi-Yau几何图形的全面延伸。我们将集中在下列紧密相关的问题上：构造这类弦理论的新解；刻画这些空间的变形，特别是这些空间的模；理解“世界表瞬子”及其在这些流形中的计数几何；以及理解“广义校准”，即类似于特殊完整流形的已校准子流形。建议的项目是研究一类新的几何对象的数学，称为“广义几何”，以及这类几何对象在弦理论中的出现。在数学上，这些结构提供了Calabi-Yau几何学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捉粒子物理学和宇宙学的基本特征所必需的，并将把弦理论家推向与观测接触的目标。该项目是一项多机构和跨学科的努力，涉及布兰迪斯大学、哈佛大学和德克萨斯A&A；M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.