  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.

High Energy Physics - Theory