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 ind CFIV vanishes level-by-level. The ground state contribution produces |ind CFIV|=1.
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We are grateful to S. Cecotti, K. Intriligator, C. Vafa, M. Voloshin and A. Vainshtein for useful discussions. The work of M.S. is supported in part by DOE grant DE-FG02-94ER-40823 at the University of Minnesota. The work of A.M. is supported in part by Swiss National Science Foundation, FASI RF 14.740.11.0347 (2009–2013) and RFBR 10-02-00509 .