The Asymptotic Geometry of Moduli Spaces

这个项目是基于最近的发展，在几何和物理之间的接口。有许多关于数学中心对象的复杂猜想（“希钦模空间”和“K3空间”）来自理论物理学家在低能下研究特定量子系统。PI使用了来自几何分析的新工具，这些工具非常适合验证关于希钦模空间和K3空间的几何形状的极其微妙的猜想。这种策略通常会扩展目前通过几何分析所能做的事情的极限，同时也会对这些物理猜想产生新的见解。由于Hitchin模空间在几何中占据了一个特殊的位置，在许多数学领域的十字路口，包括代数几何、微分几何、表示理论和几何分析，这项工作具有跨学科的影响。这个项目的中心是Hitchin模空间和K3曲面族的渐近几何。一个主要目标是验证物理学家Gaiotto、Moore和Neitzke在Hitchin模空间上对超凯勒度规的美丽推测描述。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推测描述。Hitchin模空间和椭圆纤维K3曲面都是允许一个超凯勒度量的代数完全可积系统。两种纤维都是在半维基础上制成的，通用纤维是阿贝尔品种。有些纤维是单一的，这些猜想在这些单一纤维附近是最有趣和最困难的。构造分析技术和几何微局部分析方法似乎是验证物理学猜想的最合适的方法，因为它们非常适合分析自然出现的奇异微分算子。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.