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 CU{ CO}. 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 @, then the supercharges are essentially self-adjoint on Cc (IX*\ {O};c") . The special case where V(z) = ;lz-P (#+I&Q=\{O]) is analyzed in detail to prove the following facts: (i) the number of the bosonic zero-energy ground state(s) is equal top -1; (ii) the supercharges are not Fredholm.

Spectral analysis of a Dirac operator with a meromorphic potential

“…We remark that the definition of H + (V ) (respectively H − (V )) here is different form that in [3], although it turns out that they coincide (see Theorems 2.5(ii) and 2.6(ii) in Section 2). We also note that the potential |∂V | 2 (respectively H I (V )) has a pole at z = 0 with order 2(deg V 1 + 1) 4 (respectively deg V 1 + 2).…”

“…Hence, from a purely mathematical point of view, it may be more suitable to define Q ± (V ) and Q(V ) with ∂V replaced by a meromorphic function W . But, in this respect, we follow the notations used in early works [1][2][3]6].…”

“…The basic methods we use to analyze the operator Q(V ) is to consider its square Q(V ) 2 as was done in [2,3,6] (see also [10,Chapter 5]). This leads us to introduce the following operators:…”

“…The nonnegativity ofH − is obvious. Thus, we need only to show the self-adjointness of H − with D(H − ) = D(∆) ∩ D(|∂V | 2 ) (this was not proven in [3]). By (2.3) and (2.6), we see that (2.10) holds for all Ψ ∈ C ∞ 0 (R 2 \ {0}; C 2 ).…”

Berry phase and supersymmetric topological index

“…2, i.e., holomorphic supersymmetric quantum mechanics which was investigated in many articles. [13][14][15] It was shown that the Hamiltonian of the supersymmetric harmonic oscillator has a pure point spectrum and there is the only single zero-mode in the fermionic sub-space. So the Witten index for such an operator is equal to Ϫ1.…”