Dirac Operator on Complex Manifolds and Supersymmetric Quantum Mechanics

TL;DR

This paper introduces a supersymmetric quantum mechanics model on complex manifolds that connects the Dirac operator with the twisted Dolbeault complex, providing new physical proofs of key mathematical theorems.

Contribution

It presents a novel N=2 SQM model linking Dirac operators and complex geometry, offering simplified proofs of fundamental mathematical results.

Findings

Equivalence of twisted Dirac and Dolbeault complexes

New physical proofs of Atiyah-Singer theorem

Extension of the Dirac operator concept to generic complex manifolds

Abstract

We explore a new simple N=2 SQM model describing the motion over complex manifolds in external gauge fields. The nilpotent supercharge Q of the model can be interpreted as a (twisted) exterior holomorphic derivative, such that the model realizes the twisted Dolbeault complex. The sum Q + \bar Q can be interpreted as the Dirac operator: the standard Dirac operator if the manifold is K\"ahler and a certain "truncated" Dirac operator for a generic complex manifold. Focusing on the K\"ahler case, we give new simple physical proofs of the two mathematical facts: (i) the equivalence of the twisted Dirac and twisted Dolbeault complexes and (ii) the Atiyah-Singer theorem.