The Asymptotic Geometry of Moduli Spaces

这个项目是基于几何学和物理学之间的界面的最新发展。在数学中，有许多关于中心对象（“希钦模空间”和“K3空间”）的错综复杂的解释，这些解释来自于研究低能量子系统的理论物理学家。PI使用了来自几何分析的新工具，这些工具非常适合于验证关于希钦模空间和K3空间的几何的极其微妙的几何关系。这种策略通常会扩展目前通过几何分析可以做的事情的极限，同时导致对这些物理结构的新见解。由于希钦模空间在几何学中占据了杰出的地位，在许多数学领域的十字路口，包括代数几何，微分几何，表示论和几何分析，这项工作具有跨学科的影响。这个项目是集中在渐近几何希钦模空间和家庭的K3曲面。一个主要目标是验证物理学家盖奥托、摩尔和尼兹克对希钦模空间上的超凯勒度规的美丽的几何描述。第二个目标是验证物理学家Kachru，Tripathy和Zimet对椭圆纤维K3表面上的超凯勒度量的类似描述。Hitchin模空间和椭圆纤维K3曲面都是代数完全可积系统，且都具有超Kaehler度量。两者都是在半维的基础上纤维化的，并且通用纤维是阿贝尔变种。有些纤维是奇异的，在这些奇异纤维附近，这些结构是最有趣和最困难的。建设性的分析技术和几何微局部分析的方法似乎是最合适的方式来验证从物理学的假设，因为它们非常适合分析奇异微分算子，自然出现。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project is based on recent developments at the interface between geometry and physics. There are a number of intricate conjectures about central objects in mathematics (“Hitchin moduli spaces” and “K3 spaces”) coming from theoretical physicists studying particular quantum systems at low energies. The PI uses new tools coming out of geometric analysis which are well-suited for verifying the extremely delicate conjectures about the geometry of the Hitchin moduli space and K3 spaces. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these physics conjectures. Because Hitchin moduli spaces occupy a distinguished position in geometry at the crossroads of a number of mathematical fields including algebraic geometry, differential geometry, representation theory, and geometric analysis, this work has cross-disciplinary impact.This project is centered on the asymptotic geometry of Hitchin moduli spaces and families of K3 surfaces. A main goal is to verify the beautiful conjectural description of the hyperkaehler metric on the Hitchin moduli space by the physicists Gaiotto, Moore, and Neitzke. A second goal is to verify a similar conjectural description of hyperkaehler metrics on elliptically-fibered K3 surfaces by the physicists Kachru, Tripathy, and Zimet. The Hitchin moduli spaces and elliptically-fibered K3 surfaces are both algebraic completely integrable systems admitting a hyperkaehler metric. Both are fibered over a half-dimensional base and the generic fibers are abelian varieties. Some fibers are singular, and these conjectures are most interesting and most difficult near these singular fibers. Constructive analytic techniques and the methods of geometric microlocal analysis seem to be the most appropriate way to verify conjectures from physics because they are well-suited to analyzing the singular differential operators that naturally appear.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.