Dirac Operator on Complex Manifolds and Supersymmetric Quantum Mechanics

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.