  We consider supersymmetric sigma models on the Kahler target spaces, with
twisted mass. The Kahler spaces are assumed to have holomorphic Killing
vectors. Introduction of a superpotential of a special type is known to be
consistent with N=2 superalgebra (Alvarez-Gaume 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 found paves the
way to a novel phenomenon -- spontaneous breaking of a part of supersymmetry.
In the general case 1/2 of 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 derivation of the quantum anomaly
in {\bar Q_L, Q_R}.
