Meromorphic N=2 Wess–Zumino supersymmetric quantum mechanics

Abstract

The ordinary (holomorphic) N=2 Wess–Zumino model in supersymmetric quantum mechanics is extended to the case where the superpotential V(z) is a meromorphic function on C■{∞}. The extended model is analyzed in a mathematically rigorous way. Self‐adjoint extensions and the essential self‐adjointness of the supercharges are discussed. The supersymmetric Hamiltonian defined by one of the self‐adjoint extensions of the supercharges has no fermionic zero‐energy states (‘‘vanishing theorem’’). It is proven that if V(z) has only one pole at z=0 in C, then the supercharges are essentially self‐adjoint on C^∞₀(R²■{0};C⁴). The special case where V(z)=λz^−p(p∈N,λ∈C■{0}) is analyzed in detail to prove the following facts: (i) the number of the bosonic zero‐energy ground state(s) is equal to p−1; (ii) the supercharges are not Fredholm.