Toda lattices and Toric varieties for real semisimple Lie algebras

Ohio State University, The, Columbus OH

Investigators

Abstract

DMS-0071523 Luis Casian This project concerns the topology (integral homology, cohomology and cell decompositions) of certain real toric varieties that arise when isospectral manifolds of a (signed) Toda lattice are compactified. The Toda lattice can be solved explicitly as an integrable hamiltonian system, but the geometrical feature of the solutions has not been clarified. In Lie-theoretic terms, these toric varieties consist of closures of generic orbits of a split Cartan Subgroup acting on a real flag manifold of a semisimple Lie algebra. An interesting problem is then to describe, in detail, their structure, which has some similarities with the structure of real flag manifolds. The topology of these varieties is well-known in the complex case; however the real case poses new difficulties which have not been tackled before. Extensions of this main problem are also considered which include some Kac-Moody versions of the original problem, the full Kostant-Toda lattice and, in general, the structure of real flag manifolds. The study of these toric varieties is physically motivated by the appearance of the indefinite (signed) Toda lattices in the context of symmetry reduction of the Wess-Zumino-Novikow-Witten (WZNW) model which is one of the most important model equation for conformal field theory. The toric varieties under Study can then be seen to give a concrete description of (an expected) regularization of the integral manifolds of these indefinite Toda lattices, where infinities (i.e. blow up points) of the solutions of these Toda systems glue everything into a smooth compact manifold. Also the study of isospectral manifolds of the Toda lattices is useful to understand the geometry of matrix eigenvalue algorithm based on QR or LU factorization. The present project will clarify a global aspect of the integrable systems