Cecotti-Fendley-Intriligator-Vafa index in a box

The Cecotti-Fendley-Intriligator-Vafa (CFIV) index in two-dimensional N=(2,2) models is revisited. We address the problem of "elementary" (nontopological) excitations over a kink solution, in the one-kink sector (using the Wess-Zumino or Landau-Ginzburg models with two vacua as examples). In other words, we limit ourselves to the large-β limit. The excitation spectrum over the BPS-saturated at the classical level kink is discretized through a large box with judiciously chosen boundary conditions. The boundary conditions are designed in such a way that half of supersymmetry is preserved as well as the BPS kink itself, and relevant zero modes. The excitation spectrum acquires a mass gap. All (discretized) excited states enter in four-dimensional multiplets (two bosonic states + two fermionic). Their contribution to indCFIV vanishes level-by-level. The ground state contribution produces |indCFIV|=1.