BPS saturated solitons in N=2 two-dimensional theories on R×S (domain walls in theories with compactified dimensions)

We discuss topologically stable solitons in two-dimensional theories with extended supersymmetry assuming that the spatial coordinate is compact. This problem arises in consideration of domain walls in popular theories with compactified extra dimensions. Contrary to naive expectations, it is shown that solitons on a cylinder can be BPS saturated. In the case of one chiral superfield, a complete theory of the BPS saturated solitons is worked out. We describe the classical solutions of the BPS equations. Depending on the choice of the Kähler metric, the number of such solutions can be arbitrarily large. Although the property of BPS saturation is preserved order by order in perturbation theory, nonperturbative effects eliminate the majority of the classical BPS states upon passing to the quantum level. The number of quantum BPS states is found. It is shown that the N=2 field theory includes an auxiliary N=1 quantum mechanics, Witten's index of which counts the number of BPS particles.