Extended supersymmetry and its applications in quantum mechanical models associated with self-dual gauge fields

Abstract

We study certain new models of supersymmetric quantum mechanics. The explicit form of the corresponding superfield and component actions, as well as of the quantum Hamiltonians and supercharges is given. It is shown that the Hamiltonian H=D*D, where D is flat four-dimensional Dirac operator in an external self-dual gauge background, Abelian or non-Abelian, is supersymmetric with N=4 supersymmetry. A generalization of this Hamiltonian to the motion on a curved conformally flat four-dimensional manifold exists. For an Abelian self-dual background, the corresponding Lagrangian can be derived from certain harmonic superspace expressions. If the Hamiltonian involves a non-Abelian self-dual gauge field, one can construct the Lagrangian formulation of it by introducing auxiliary bosonic variables with Wess-Zumino type action. For a special class of such Lagrangians when the gauge group is SU(2) and the gauge field is expressed in the `t Hooft ansatz form, it is possible to give a superfield description using the harmonic superspace formalism. As a new explicit example, the N=4 mechanics with Yang monopole in R^5 (= instanton on S^4) is considered. Independently, a similar system with N=4 supersymmetry in three dimensions also admits the superfield description. Although the three-dimensional system involves different superfields, its component Lagrangian and Hamiltonian appear to be the three-dimensional reduction of the mentioned four-dimensional system. The off-shell N=4 supersymmetry requires the gauge field to be a static form of the 't Hooft ansatz for the four-dimensional self-dual SU(2) gauge fields, that is a particular solution of Bogomolny equations for BPS monopoles.