N=2 sigma model with a twisted mass and superpotential: Central charges and solitons

We consider supersymmetric sigma models on the Kähler target spaces, with a twisted mass. The Kahler spaces are assumed to have holomorphic Killing vectors. The introduction of a superpotential of a special type is known to be consistent with N= 2 superalgebra (Alvarez-Gaumé and Freedman). We show that the algebra acquires central charges in the anticommutators {Q _L,Q_L} and {Q_R,Q_R}. These central charges have no parallels, and they can exist only in two dimensions. The central extension of the N= 2 superalgebra we find paves the way to a novel phenomenon-spontaneous breaking of a part of the supersymmetry. In the general case, 1/2 of the supersymmetry is spontaneously broken (the vacuum energy density is positive), while the remaining 1/2 is realized linearly. In the model at hand, the standard fermion number is not defined, so that the Witten index as well as the Cecotti-Fendley-Intriligator-Vafa index are useless. We show how to construct an index for counting short multiplets in internal algebraic terms which is well defined in spite of the absence of the standard fermion number. Finally, we outline the derivation of the quantum anomaly in the anticommutator {Q̄_L,Q_R}.