Counting fermionic zero modes on M5 with fluxes

TL;DR

This work extends the counting of fermionic zero modes on M5 branes to include background flux, reformulating the problem as a flux-dependent linear system on divisor cohomology. It shows that the index χ_D(F) can differ from the fluxless holomorphic Euler characteristic, enabling instanton-generated superpotentials in cases previously forbidden. The authors derive a general expression χ_D(F) = χ_D − (h^{0,2} − n) with n determined by a flux-induced linear constraint, and illustrate the mechanism with a K3 × K3 example, where flux choice can preserve or break supersymmetry and alter the zero-mode count accordingly. These results broaden the conditions under which non-perturbative effects contribute to moduli stabilization and have potential implications for string cosmology and the landscape of flux vacua.

Abstract

We study the Dirac equation on an M5 brane wrapped on a divisor in a Calabi--Yau fourfold in the presence of background flux. We reduce the computation of the normal bundle U(1) anomaly to counting the solutions of a finite--dimensional linear system on cohomology. This system depends on the choice of flux. In an example, we find that the presence of flux changes the anomaly and allows instanton corrections to the superpotential which would otherwise be absent.