A path integral derivation of $\chi_y$-genus

TL;DR

This paper provides a direct derivation of the Hirzebruch $hi_y$-genus formula for complex manifolds using supersymmetric quantum mechanics, offering a new perspective on classical index theorems.

Contribution

It introduces a novel derivation method for the $hi_y$-genus formula based on supersymmetric quantum mechanics, bypassing traditional algebraic geometry techniques.

Findings

Derivation of the $hi_y$-genus formula from supersymmetric quantum mechanics

Connection between index theorems and quantum mechanics methods

Unified perspective on classical topological invariants

Abstract

The formula for the Hirzebruch $\chi_y$-genus of complex manifolds is a consequence of the Hirzebruch-Riemann-Roch formula. The classical index formulae for Todd genus, Euler number, and Signature correspond to the case when the complex variable $y=$ 0, -1, and 1 respectively. Here we give a {\it direct} derivation of this nice formula based on supersymmetric quantum mechanics.