Algebraic Geometry Inspired by Theoretical Physics

本项目将研究代数几何中的几种情况，在这些情况下，由于与镜像对称有关的原因，由李群定义的空间预计具有枚举或上同调不变量，这些不变量与群被朗兰兹对偶取代的不变量相关或相等。目的是计算这些不变量并验证镜像对称性的预测。主要研究的例子有：(1)结构群为约化李群的黎曼曲面上平面连接空间的弦Hodge多项式；(2)磁单极子Nahm方程解的模空间的弦Hodge多项式；(3)环群的量子上同调。代数几何是数学中最经典的部分之一，研究多项式方程的解。但它的目的是证实最近关于镜像对称的一些假设，这是一个最近从理论物理学中出现的令人兴奋的想法。镜像对称提出，关于宇宙基本结构的某些弦理论可以用两种看似不同但等效的数学方式来表述。本项目将探索支持最近关于如何根据另一个公式构建其中一个公式的建议之一的证据。

This project will study several situations in algebraic geometry where, for reasons connected with mirror symmetry, spaces defined in terms of a Lie group are expected to have enumerative or cohomological invariants which are related, or equal, to those where the group is replaced by its Langlands dual. The aim is to compute these invariants and verify the predictions of mirror symmetry. Among the principal examples to be studied are: (1) the stringy Hodge polynomials of the spaces of flat connections on a Riemann surface with structure group a reductive Lie group; (2) the stringy Hodge polynomials of the moduli spaces of solutions to Nahm's equations of magnetic monopoles; (3) the quantum cohomology of a loop group.This is research in algebraic geometry, one of the most classical parts of mathematics, concerned with finding solutions of polynomial equations. But it is aimed at corroborating some very recent hypotheses about mirror symmetry, an exciting idea which has lately emerged from theoretical physics. Mirror symmetry proposes that certain string theories about the fundamental structure of the universe can be formulated mathematically in two seemingly different but equivalent ways. This project will explore the evidence supporting one of the recent suggestions for how to construct one of these formulations in terms of the other.