Supersymmetric inhomogeneous field theories in 1+1 dimensions

We study supersymmetric inhomogeneous field theories in 1+1 dimensions which have explicit coordinate dependence. Although translation symmetry is broken, part of supersymmetries can be maintained. In this paper, we consider the simplest inhomogeneous theories with one real scalar field, which possess an unbroken supersymmetry. The energy is bounded from below by the topological charge which is not necessarily nonnegative definite. The bound is saturated if the first-order Bogomolny equation is satisfied. Non-constant static supersymmetric solutions above the vacuum involve in general a zero mode although the system lacks translation invariance. We consider two inhomogeneous theories obtained by deforming supersymmetric sine-Gordon theory and phi(6) theory. They are deformed either by overall inhomogeneous resealing of the superpotential or by inhomogeneous deformation of the vacuum expectation value. We construct explicitly the most general supersymmetric solutions and obtain the BPS energy spectrum for arbitrary position-dependent deformations. Nature of the solutions and their energies depend only on the boundary values of the inhomogeneous functions. The vacuum of minimum energy is not necessarily a constant configuration. In some cases, we find a one-parameter family of degenerate solutions which include a non-vacuum constant solution as a special case.

Automated summary

This paper studies supersymmetric inhomogeneous field theories in 1+1 dimensions with explicit coordinate dependence. Despite the breaking of translation symmetry, some supersymmetries can still be preserved. The simplest inhomogeneous theories with one real scalar field, possessing unbroken supersymmetry, are considered. The energy is bounded from below by the topological charge, which is not necessarily nonnegative definite. The bound is saturated when the first-order Bogomolny equation is satisfied. Non-constant static supersymmetric solutions, although lacking translation invariance, generally involve a zero mode. Two inhomogeneous theories obtained by deforming supersymmetric sine-Gordon theory and phi(6) theory are investigated. The most general supersymmetric solutions and the BPS energy spectrum for arbitrary position-dependent deformations are explicitly constructed.