A Comment on the chi_y Genus and Supersymmetric Quantum Mechanics

TL;DR

This paper introduces a modification to supersymmetric quantum mechanics with a fermion mass term, deriving the chi_y genus formula and connecting it to moduli space dynamics in supersymmetric theories.

Contribution

It presents a novel modification of supersymmetric quantum mechanics that naturally leads to the chi_y genus formula, linking mathematical topology with physical moduli space dynamics.

Findings

Derived the integral formula for the chi_y genus using modified supersymmetric quantum mechanics.

Connected the modification to moduli space dynamics of monopoles and instantons in supersymmetric theories.

Showed the natural emergence of the modification in theories with partially broken supersymmetry.

Abstract

In this note we show that a simple modification of supersymmetric quantum mechanics involving a mass term for half the fermions naturally leads to a derivation of the integral formula for the chi_y genus, a quantity that interpolates between the Euler characteristic and arithmetic genus. We note that this modification naturally arises in the moduli space dynamics of monopoles or instantons in theories with 16 supercharges partially broken to 8 supercharges by mass terms.