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 ϕ6 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.
Highlights
In this paper, we will address these issues in supersymmetric theories where the translation symmetry is explicitly broken
We study supersymmetric inhomogeneous field theories in 1+1 dimensions which have explicit coordinate dependence
The energy is independent of the integration constant and we can have a one-parameter family of supersymmetric solutions above the vacuum, i.e., the solution possesses a zero mode even though the system has no translation invariance
Summary
We would like to note that the value of the energy in inhomogeneous case has no absolute meaning just as in usual nonsupersymmetric theories This is because one can always add a field-independent term W1(x) to the superpotential which has no physical effect,. In theories with nontrivial boundary values of the field, the energy spectrum can change by adding this term, since solutions belonging to different topological sectors would give different energy shifts. This kind of ambiguity, exists even in usual homogeneous theories. In the present case, it is evident that the central charge T is shifted by the same amount in (2.15), cancelling the energy shift
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