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We discuss topologically stable solitons in two-dimensional theories with theextended supersymmetry assuming that the spatial coordinate is compact. Thisproblem arises in the consideration of the domain walls in the popular theorieswith compactified extra dimensions. Contrary to naive expectations, it is shownthat the solitons on the cylinder can be BPS saturated. In the case of onechiral superfield, a complete theory of the BPS saturated solitons is workedout. We describe the classical solutions of the BPS equations. Depending on thechoice of the Kahler metric, the number of such solutions can be arbitrarilylarge. Although the property of the BPS saturation is preserved order by orderin perturbation theory, nonperturbative effects eliminate the majority of theclassical BPS states upon passing to the quantum level. The number of thequantum BPS states is found. It is shown that the N=2 field theory includes anauxiliary N=1 quantum mechanics, Witten's index of which counts the number ofthe BPS particles.Comment: LaTeX, 23 pages, 7 Postscript figure

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

BPS saturated solitons in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mn /></mml:math>two-dimensional theories on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>R</mml:mi><mml:mo>×</mml:mo><mml:mi>S</mml:mi></mml:math>(domain walls in theories with compactified dimensions)

BPS saturated solitons inN=2two-dimensional theories onR×S(domain walls in theories with compactified dimensions)