Geometry of Strings and Gravity

弦理论和重力背景下的基础物理问题通常可以被纯粹地重新表述为几何问题，弦理论的真空空间，特别是超对称（BPS）物体的光谱。在真空模空间的某些“壁”上，BPS物体可以衰变成质量更小的BPS物体。这种隔墙现象具有极其丰富而神秘的数学结构。本文提出了利用镜像对称和弦对偶性等工具，从具有大量超对称的真空等不同角度研究壁交问题。了解BPS天体的光谱对于计算黑洞熵和拓扑弦理论具有重要意义。PI建议揭示新的几何不变量，这对于物理表征这些弦真空是必要的，并探索最近定义的准局部质量的含义，它满足所有已知的一致性条件。这些项目涉及对理论物理、微分几何和代数几何具有重要意义的令人兴奋的问题。该提案的核心是培养博士后和研究生进行前沿的跨学科研究，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.