FRG Collaborative Research: Generalized Geometry, String Theory and Deformations

FRG 合作研究：广义几何、弦理论和变形

• 批准号: 1159412
• 负责人: Shing-Tung Yau
• 金额: $ 30.99万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2017-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1159412&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Generalized; Geometry

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.

该项目是一个多机构（哈佛大学、布兰代斯大学和德克萨斯农工大学）、跨学科（数学和物理学）共同努力的项目，旨在研究广义几何的数学理论及其在弦理论中的应用。广义几何形成了一类具有简化结构群的几乎复杂的流形，这对于现实弦理论模型的研究至关重要。这些是 Calabi-Yau 流形的自然推广，并且本身就具有数学意义。在对弦理论很重要的一类中，人们在广义卡拉比-丘流形上寻求的规范结构是由完全非线性复数蒙日安培型系统控制的，该系统将平衡厄米度量与矢量丛的反自对偶连接耦合起来。当流形是 Kahler Calabi-Yau 并且向量丛是切丛时，该系统简化为 Ricci 平坦度量的 Calabi 猜想。广义几何是作为弦理论模型时空的内部组成部分而出现的，它比以前的构造更接近我们的现实世界。支配这种几何的数学是几何分析、代数几何和变形理论中深层问题的自然延伸。新几何背后的物理学为提出和解决数学问题提供了灵感和新颖的方法。他们的解决方案反过来将导致对物理学基本问题的更好理解，使之成为真正的跨学科合作。对广义几何的数学理解仍处于初级阶段。 该提案的目的是进一步发展这一领域，成为 Kahler Calabi-Yau 几何学的全面扩展。我们将重点关注以下紧密相连的问题：在本课程中构建弦理论的新解决方案；表征变形，特别是这些空间的模量；计算空间光场的尺寸； 以及对“广义校准”的理解 - 特殊完整流形的校准子流形的模拟。