University of Toronto's Symplectic Geometry Seminar

February 18, 2008, 2:10pm
Bahen 6183

Dana Fine

University of Massachusetts, Dartmouth

Supersymmetic Quantum Mechanics and the Gauss-Bonnet-Chern Theorem


Alvarez-Gaume' and Witten derive the Gauss-Bonnet-Chern theorem heuristically from a supersymmetric path integral expression for the supertrace of the heat kernel of the Laplacian on forms. Blau notes this follows, again heuristically, from interpreting the path integral via the Mathai-Quillen formalism.

This talk describes how to make these arguments rigorous: Start with the path integral expression for the partition function of supersymmetric quantum mechanics on a Riemannian manifold. Restrict to piecewise geodesic paths with $n$ pieces. The resulting sequence of integrals on finite-dimensional manifolds converges to the heat kernel of the Laplacian on forms. For closed paths, the corresponding sequence converges to the supertrace of the heat kernel. Each integral in the sequence is the integral of a Mathai-Quillen Thom form on the tangent bundle to the finite-dimensional manifold. After interchanging the large-n limit of the sequence with the small-''time'' limit of each term, this localizes to the integral over the original manifold of the Pfaffian of the curvature of the Levi-Cevita connection.

This construction is sufficiently general to permit immediate generalizations to other index theorems.