Some Toric Manifolds and a Path Integral

Details Some Toric Manifolds and a Path Integral Some Toric Manifolds and a Path Integral

2003-07-09

arxiv.org/abs/math/0307116v1

Progress in Math. 231, The Orbit Method in Geometry and Physics. In Honor of A.A.Kirillov., Birkhauser, 2003, 395-419

Mathematics

Symplectic Geometry

Preprint version of the published paper. LaTeX. 20 pages

Scientific paper

The toric manifolds in question were invented by Bott and studied by Grossberg and Karshon under the name "Bott towers". Interest in them comes from their relation to characters of semisimple Lie groups and geometric quantization. We offer another construction of them, introduce a family symplectic structures as curvatures of line bundles, work out canonical coordinates (action-angle variables), derive positivity criteria for the curvature, and determine the range of the moment map. A character of the torus group is introduced as an integral in the spirit of Kirillov's character formula and shown to equal the sum over the integral points in the moment polytope by methods from convex analysis and equivariant cohomology. The result is compared with an explicit calculation of a path integral, based on a definition devised for the occasion.

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