Oscillator Calculus on Coadjoint Orbits and Index Theorems

TL;DR

This paper explores the relationship between quantum spin chain systems with supersymmetry and geometric index theorems on coadjoint orbits, providing new insights into their spectra and topological invariants.

Contribution

It establishes that certain supersymmetric quantum systems are truncations of sigma models on coadjoint orbits and computes their indices, linking spectral problems to geometric index calculations.

Findings

Computed Witten indices matching Dolbeault and de Rham indices.

Revealed equivalence between spectra of Laplace operators and spin chain Hamiltonians.

Connected quantum spectral problems with geometric index theorems.

Abstract

We consider quantum mechanical systems of spin chain type, with finite-dimensional Hilbert spaces and $\mathcal{N}=2$ or $\mathcal{N}=4$ supersymmetry, described in $\mathcal{N}=2$ superspace in terms of nonlinear chiral multiplets. We prove that they are natural truncations of 1D sigma models, whose target spaces are $\mathsf{SU}(n)$ (co)adjoint orbits. As a first application, we compute the Witten indices of these finite-dimensional models showing that they reproduce the Dolbeault and de Rham indices of the target space. The problem of finding the exact spectra of generalized Laplace operators on such orbits is shown to be equivalent to the diagonalization of spin chain Hamiltonians.