Perturbative corrections to effective zero-mode hamiltonian in supersymmetric QED

Abstract

We consider the supersymmetric QED in the finite volume. In this problem the dynamic variables can be divided into two classes: “slow variables” (the zero-momentum Fourier harmonic of the electromagnetic field A i (0) and its supersymmetric counterpart) and “fast variables” (all others). The effective hamiltonian involving only slow variables can be constructed in the framework of the Born-Oppenheimer approach. In the lowest nontrivial order it describes free motion in the space of A i (0). In this paper corrections in the Born-Oppenheimer parameter to the effective hamiltonian are found. They do not vanish and the resultant H ̂ eff describes a nontrivial N = 2 supersymmetric extension of the σ-model on a 3-dimensional conformally flat manifold. The possibility of such an extension has escaped attention so far. We calculated also the corrections to the effective hamiltonian of the chiral (0 + 1)-dimensional supersymmetric QED. Corrections of this kind can in principle alter predictions for the Witten index made on the basis of the lowest order H ̂ eff in some models.