Supersymmetric Inhomogeneous Field Theories in 1+1 Dimensions

TL;DR

This paper explores supersymmetric inhomogeneous field theories in 1+1 dimensions, analyzing how explicit coordinate dependence affects supersymmetry, solutions, and energy bounds, with explicit constructions for deformed sine-Gordon and $$ theories.

Contribution

It introduces new inhomogeneous supersymmetric models with explicit coordinate dependence and constructs their general solutions and energy spectra, extending understanding of supersymmetry in non-uniform backgrounds.

Findings

Supersymmetry can be preserved despite explicit coordinate dependence.

Energy bounds are determined by topological charge, with solutions depending on boundary conditions.

Existence of degenerate solutions including non-vacuum states.

Abstract

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 rescaling 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.