Supersymmetric quantum mechanics living on topologically nontrivial Riemann surfaces

TL;DR

This paper develops a new non-Hermitian supersymmetric quantum mechanics framework on complex Riemann surfaces, exploring topologically nontrivial curves and antilinear mappings between partner operators.

Contribution

It introduces a novel non-Hermitian representation of supersymmetric quantum mechanics on complex curves with nontrivial topology, emphasizing the non-uniqueness of partner mappings.

Findings

Construction of supersymmetric quantum mechanics on Riemann surfaces.

Use of antilinear maps between partner operators.

Analysis of non-uniqueness in topologically nontrivial mappings.

Abstract

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators $H^{(\pm)}$ is chosen antilinear. Secondly, both these components of a super-Hamiltonian ${\cal H}$ are defined along certain topologically nontrivial complex curves $r^{(\pm)}(x)$ which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map ${\cal T}$ between "tobogganic" partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is emphasized.