Index Theorems and Loop Space Geometry

TL;DR

This paper connects index theorems with loop space geometry by evaluating Dirac indices through symplectic geometry and localization techniques in supersymmetric quantum mechanics, revealing new computational methods for topological invariants.

Contribution

It introduces a novel approach to compute Dirac indices using symplectic geometry in loop space, linking Callias and Atiyah-Singer indices via quantum mechanical models.

Findings

Exact evaluation of the index via localization techniques.

Relation between Callias and Atiyah-Singer indices.

Interpretation of the effective action in terms of 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.