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

TL;DR

This paper proposes a conjecture about the charge lattice structure of massless sectors in 4D heterotic string vacua, linking it to anomaly cancellation, supersymmetry restoration, and the Weak Gravity Conjecture.

Contribution

It introduces a new conjecture on the charge lattice in heterotic vacua and tests it across various compactifications, providing insights into FI-term cancellation and black hole stability.

Findings

Conjecture holds in many heterotic compactifications.

Lattice structure ensures existence of FI-term cancelling vacua.

Supports connection between string vacua and Weak Gravity Conjecture.

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.