Index Theorems and Loop Space Geometry

Abstract

We investigate the evaluation of the Dirac index using symplectic geometry in the loop space of the corresponding supersymmetric quantum mechanical model. In particular, we find that if we impose a simple first class constraint, we can evaluate the Callias index of an odd dimensional Dirac operator directly from the quantum mechanical model which yields the Atiyah-Singer index of an even dimensional Dirac operator in one more dimension. The effective action obtained by BRST quantization of this constrained system can be interpreted in terms of loop space symplectic geometry, and the corresponding path integral for the index can be evaluated exactly using the recently developed localization techniques.

15 pages, report CERN-TH-6471 and HU-TFT-92-15

Cited by in corpus (5)

• Two Dimensional Gauge Theories Revisited
• Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories
• Equivariance, BRST and Superspace
• Phase Space Isometries and Equivariant Localization of Path Integrals in Two Dimensions
• Derivation of Index Theorems by Localization of Path Integrals