Supersymmetric quantum mechanics living on topologically non-trivial Riemann surfaces

Supersymmetric quantum mechanics is constructed in a new non-Hermitian representation. Firstly, the map between the partner operators H ^(±) is chosen antilinear. Secondly, both these components of a super-Hamiltonian $ mathcal{H} $ mathcal{H} are defined along certain topologically non-trivial complex curves r ^(±)(x) which spread over several Riemann sheets of the wave function. The non-uniqueness of our choice of the map $ mathcal{T} $ mathcal{T} between ‘tobogganic’ partner curves r ⁽⁺⁾(x) and r ⁽⁻⁾(x) is emphasized.