FRG: Collaborative Research: Generalized Geometry, String Theory and Deformations

FRG：协作研究：广义几何、弦理论和变形

• 批准号: 1159404
• 负责人: Melanie Becker
• 金额: $ 19.4万
• 依托单位: Texas A&M 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 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1159404&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.

该项目是多机构（哈佛，布兰代斯和德克萨斯A＆M），跨学科（数学和物理）的努力，用于研究广义几何形状的数学理论及其在弦理论中的应用。广义的几何形状形成一类几乎复杂的歧管，结构组减少，这对于研究现实弦理论模型的研究至关重要。这些是卡拉比远流形的自然概括，并且本身具有数学兴趣。在一个对字符串理论重要的类中，人们寻求在广义的calabi-yau歧管上寻求的规范结构，受一个完全非线性的复杂蒙吉安培类型系统的控制，该系统将平衡的Hermitian指标与矢量束的反偶联连接结合在一起。当歧管是Kahler Calabi-Yau，而矢量束为切线束时，该系统将减少到Calabi猜想的RICCI-FLAT指标。广义的几何形状是弦理论模型的时空内部组成部分，与以前的结构相比，它更接近我们的现实世界。控制这种几何形状的数学是几何分析，代数几何和变形理论中深处问题的自然扩展。新的几何形状背后的物理学为摆姿势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的基本问题有更多的了解，使其成为真正的跨学科合作。对广义几何形状的数学理解仍处于其新生阶段。 该提议的目的是进一步发展该领域，成为Kahler Calabi-Yau几何形状的成熟扩展。我们将重点关注以下紧密互连的问题：在此类中构建新的解决方案；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 - 特殊自动歧管的校准亚策略的类似物。