Cecotti-Fendley-Intriligator-Vafa Index in a Box

The Cecotti-Fendley-Intriligator-Vafa (CFIV) index in two-dimensional ${\mathcal N} =(2,2)$ 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-$\beta$ 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 $\mathrm{ind}_{\rm CFIV}$ vanishes level by level. The ground state contribution produces $|\mathrm{ind}_{\rm CFIV}|=1$.