A note on 4D heterotic string vacua, FI-terms and the swampland

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 a lattice generated by 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. Fil: Aldazabal, Gerardo Marcelo. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Ibáñez, Luis E.. Universidad Autónoma de Madrid; España