Appears in collection : Integrability, Anomalies and Quantum Field Theory

The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals.Similar methods can be developed to give a geometric formula for semi simple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg's trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators. In joint work with Shu SHEN, we have also shown how to extend these formulas to orbital integrals involving general elements of the center of the enveloping algebra.