Symmetry Points of $\mathcal{N}=1$ Modular Geometry

TL;DR

This paper investigates the critical points of scalar potentials in 4d $ abla=1$ supergravity with modular symmetry, linking vacuum types to superpotential weights and analyzing asymptotic behavior consistent with Swampland principles.

Contribution

It demonstrates that enhanced discrete symmetry points are always critical points if no new massless fields appear, and relates vacuum nature to superpotential weights in modular invariant theories.

Findings

Enhanced symmetry points are critical points of the scalar potential.

Vacuum type (dS, AdS, Minkowski) is determined by superpotential weight.

Scalar potential asymptotics are constrained by Swampland principles.

Abstract

We consider 4d $\mathcal{N}=1$ supergravity theories with modular symmetry, where the modulus $\tau$ is the upper half-plane modulo $SL(2,\mathbf{Z})$ action. We focus on enhanced discrete gauge symmetry points $\tau=i, \exp(2\pi i/3)$, and argue that, if there are no new additional massless fields at these points, they will always be critical points of the scalar potential. Moreover, we show that whether these correspond to dS, AdS, or Minkowski vacua can be generically determined simply by the weight of the superpotential under modular transformations. We also analyze the asymptotics of the scalar potential and find that compatibility with the Swampland principles implies that, if nonvanishing, the scalar potential decays either exponentially or double-exponentially, and that the asymptotic slope is bounded. The slope is governed by the superpotential weight as well as by real-analytic modular contributions to the K\"ahler potential.