Principal Investigator: Kirill L. Vaninsky
We consider the problem of computation of partition function (in thermodynamic limit) of classical particles interacting with Moser-Calogero potentials. Our approach to the computation of the partition function can be viewed as a version of the well-known in symplectic geometry Duistermaat-Heckman localization theorem. The standard requirement in such theorem is that the Hamiltonian produces a circle action. Many physically interesting systems, for example, finite-particle Moser-Calogero systems do not satisfy this condition. We study our versions of localization and convexity theorems when the dimension of symplectic manifold become large.
At the present time the Moser-Calogero systems are subject of intensive investigations. It turned out that some Super-Symmetric Yang--Mills (SUSY YM) theories can be identified with the space of spectral curves of the Moser-Calogero systems. We found a novel and deep connection between problem of computation of partition function of Moser-Calogero particles and problems of SUSY YM theory. The physics can be formulated in two different languages, but mathematical nature of the problem is the same. Progress in these directions offers new insights into the geometry of spectral curves, symplectic geometry and, in particular, convexity theorems, theory of Toeplitz determinants and spectral theory.
