Asymptotic Localization, Moser-Calogero Systems and Super Symmetric Yang-Mills Theory

摘要奖：DMS-9971834主要研究者：Kirill L.考虑经典粒子与Moser-Calogero势相互作用的配分函数（热力学极限）的计算问题。 我们计算配分函数的方法可以看作是辛几何中著名的Duistermaat-Heckman局部化定理的一个版本。这个定理的标准要求是哈密顿量产生一个圆作用量。许多物理上有趣的系统，例如有限粒子Moser-Calogeros系统不满足这个条件。 我们研究了当辛流形维数变大时的局部化和凸性定理。目前Moser-Calogero系统是研究的一个重要课题。结果表明，某些超对称杨-米尔斯（SUSY YM）理论可以与Moser-Calogero系统的谱曲线空间相一致。 我们发现Moser-Calogero粒子配分函数的计算问题与SUSY YM理论问题之间存在着新的、深刻的联系。物理学可以用两种不同的语言表述，但问题的数学性质是相同的。这些方面的进展为谱曲线几何、辛几何、特别是凸性定理、Toeplitz行列式理论和谱理论提供了新的见解。

AbstractAward: DMS-9971834Principal Investigator: Kirill L. VaninskyWe consider the problem of computation of partition function (inthermodynamic limit) of classical particles interacting withMoser-Calogero potentials. Our approach to the computation ofthe partition function can be viewed as a version of thewell-known in symplectic geometry Duistermaat-Heckmanlocalization theorem. The standard requirement in such theorem isthat the Hamiltonian produces a circle action. Many physicallyinteresting systems, for example, finite-particle Moser-Calogerosystems do not satisfy this condition. We study our versions oflocalization and convexity theorems when the dimension ofsymplectic manifold become large.At the present time the Moser-Calogero systems are subject ofintensive investigations. It turned out that some Super-SymmetricYang--Mills (SUSY YM) theories can be identified with the spaceof spectral curves of the Moser-Calogero systems. We found anovel and deep connection between problem of computation ofpartition function of Moser-Calogero particles and problems ofSUSY YM theory. The physics can be formulated in two differentlanguages, but mathematical nature of the problem is thesame. Progress in these directions offers new insights into thegeometry of spectral curves, symplectic geometry and, inparticular, convexity theorems, theory of Toeplitz determinantsand spectral theory.