Geometry of Strings and Gravity

弦理论和引力背景下的基本物理问题通常可以纯粹地表述为几何问题，弦理论的真空空间，特别是它们上的超对称(BPS)物体的光谱。在真空模数空间的某些“墙”上，BPS天体可以衰变成质量较小的BPS天体。这样的翻墙现象有着极其丰富而神秘的数学结构。建议从不同的角度来研究穿墙现象，包括具有高超对称性的真空，应用镜像对称和弦对偶等工具。了解BPS天体的光谱对于计算黑洞熵和拓扑弦理论是很重要的。PI建议发现物理描述这些弦真空所必需的新的几何不变量，并探索最近定义的满足所有已知一致性条件的准局部质量的含义。这些项目解决了令人兴奋的问题，这些问题对理论物理、微分和代数几何都有重要的影响。该提议的核心是培训博士后研究员和研究生，以开展前沿跨学科研究，并有效地与物理学和数学界进行互动。

Problems of fundamental physics in the context of string theory and gravity can often be rephrased purely as problems in geometry, the space of vacua of string theory and in particular the spectrum of supersymmetric (BPS) objects on them. On certain "walls" in the moduli space of vacua, a BPS object can decay to less massive BPS objects. Such a wallcrossing phenomenon has an extremely rich and mysterious mathematical structure. It is proposed to study wall-crossings from various points of views including vacua with high number of supersymmetries, applying tools of mirror symmetry and string dualities. Understanding the spectrum of BPS objects is important for calculating black hole entropy and topological string theory. The PI proposes to uncover new geometrical invariants which are necessary for physically characterizing these string vacua, and explore the implications of a recently defined quasi-local mass which satisfies all known consistency conditions. The projects address exciting questions that have important implications for theoretical physics, and for differential and algebraic geometry. Central to the proposal is the training of postdoctoral fellows and graduate students to conduct cutting edge cross-disciplinary research and to be effective in interacting with both the physics and mathematics communities.