A Note on 4D Heterotic String Vacua, FI-terms and the Swampland

TL;DR

The paper addresses FI-term cancellation in 4D $N=1$ heterotic vacua with anomalous $U(1)$ factors by proposing a lattice-based conjecture: the positive hull $\Lambda_0$ of charges of massless chiral fields forms a sublattice of the full charge lattice $\Lambda$. If true, there exist D-flat directions generated by holomorphic operators that cancel the FI-term, yielding nearby SUSY vacua and linking to swampland considerations. The authors provide evidence from Abelian $(0,2)$ orbifold compactifications, showing that appropriate massless states (including twisted-sector states) can generate the required sublattice, sometimes relying on multi-field vevs. They further argue for a connection to the Weak Gravity Conjecture, suggesting that the lattice structure may underlie BH instabilities and FI-term cancellations as a quantum-gravity consistency check, with caveats in non-heterotic or non-perturbative duals. Overall, the work proposes a concrete mechanism tying massless spectra to FI-term cancellation and WGC considerations, with implications for the landscape/Swampland of heterotic vacua.

Abstract

We present a conjecture for the massless sector of perturbative 4D $N=1$ heterotic $(0,2)$ string vacua, including $U(1)^n$ gauge symmetries,one of them possibly anomalous (like in standard heterotic compactifications). Mathematically it states that the positive hull generated by the charges of the massless chiral multiplets spans a sublattice of the full charge lattice. We have tested this conjecture in many heterotic $N=1$ compactifications in 4D. Our motivation for this conjecture is that it allows to understand a very old puzzle in $(0,2)$ $N=1$ heterotic compactification with an anomalous $U(1)$. The conjecture guarantees that there is always a D-flat direction cancelling the FI-term and restoring $N=1$ SUSY in a nearby vacuum. This is something that has being verified in the past in a large number of cases, but whose origin has remained obscure for decades. We argue that the existence of this lattice of massless states guarantees the instability of heterotic non-BPS extremal blackholes, as required by Weak Gravity Conjecture arguments. Thus the pervasive existence of these nearby FI-cancelling vacua would be connected with WGC arguments.