Cecotti-Fendley-Intriligator-Vafa Index in a Box

TL;DR

This paper revisits the CFIV index in 2D N=(2,2) models, analyzing non-topological excitations over kinks using boundary conditions that preserve supersymmetry, revealing a discretized spectrum with a mass gap and a ground state index of 1.

Contribution

It introduces a method to discretize the excitation spectrum over BPS kinks in large box limits while preserving supersymmetry, clarifying the index contributions.

Findings

Excited states form multiplets with zero index contribution.

The spectrum has a mass gap due to boundary conditions.

Ground state contribution yields an index magnitude of 1.

Abstract

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 $|\mathrm{ind}_{\rm CFIV}|=1$.