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

这个项目是一个多机构（哈佛，布兰迪斯和得克萨斯州A M），跨学科（数学和物理）的努力，研究广义几何的数学理论及其应用弦理论。广义几何形成了一类具有约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至关重要。这些都是自然的推广卡-丘流形和数学兴趣在自己的权利。在弦理论中有一类重要的性质，在广义卡-丘流形上寻找的正则结构是由一个完全非线性的复蒙日-安培型系统所支配的，这个系统将一个平衡的厄米特度量与一个向量丛的反自对偶联络耦合起来。当流形是Kahler Calabi-Yau，向量丛是切丛时，这个系统归结为Ricci平坦度量的Calabi猜想。广义几何作为弦理论模型时空的内部分量而出现，这些模型比以前的构造更接近我们的真实的世界。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。他们的解决方案将反过来导致更好地理解物理学中的基本问题，使之成为一个真正的跨学科合作。广义几何的数学理解仍处于萌芽阶段。 本建议的目的是进一步发展这一领域，成为一个成熟的扩展卡勒卡-丘几何。我们将专注于以下紧密相连的问题：构建新的解决方案，弦理论在这类;表征变形，特别是这些空间的模量;计算空间的光场的尺寸;和理解“广义校准”-校准特殊holonomy流形的子流形的模拟。

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.