Berry phase and supersymmetric topological index

Abstract

The sequences of supersymmetry for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian are revised. The condition (so-called supersymmetric adiabatic evolution condition) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek–Zee (degenerated case) phases of superpartners are taking place is pointed out. The analog of the Witten index (supersymmetric Berry index) is determined. The final expression for the new index has compact form of indBH=sDet U≡Det Uτ, where U is the cyclic evolution operator generated by supersymmetric Hamiltonian H and τ is a supersymmetric involution. As the examples of the suggested concept of the supersymmetric adiabatic evolution the holomorphic quantum mechanics on a complex plane and meromorphic quantum mechanics on Riemann surfaces are considered. The supersymmetric Berry indices for the models are calculated.