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

这个项目是一个多机构(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)的努力，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完全非线性的复Monge Ampere型系统所支配的，该系统将平衡的厄米度规耦合到向量丛的反自对偶连接。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。广义几何作为弦理论模型的时空的内部组成部分出现，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、代数几何和形变理论中深层问题的自然延伸。新几何学背后的物理学为提出和解决数学问题提供了灵感和新的方法。他们的解决方案将反过来导致对物理学基本问题的更好理解，使这成为一种真正的跨学科合作。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。我们将集中讨论以下紧密相关的问题：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数；计算空间光场的维度；以及理解“广义校准”--特殊完整流形的已校准子流形的模拟。

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.