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

TL;DR

This work analyzes 4d ${\cal N}=1$ supergravity with modular symmetry acting on a single modulus ${\tau}$ in ${\mathfrak{H}}/SL(2,\mathbb{Z})$. By requiring modular invariance of the generating functional ${G=K+\log|W|^2}$, the authors show that the symmetry points ${\tau}=i$ and ${\tau}=e^{2\pi i/3}$ are always critical points of the scalar potential (assuming no extra massless fields), and that the vacuum type there is determined by the modular weight of the superpotential ${W}$ (with or without a multiplier system). They develop a framework in which the Kähler potential can include real-analytic modular contributions, and use Swampland constraints to bound the asymptotic behavior of the potential, finding either exponential or double-exponential decay with a slope bounded by the cusp and weight data. The paper provides a symmetry-based classification of possible dS, AdS, and Minkowski vacua at the elliptic points, and derives how real-analytic corrections and multiplier systems shape the vacuum structure and asymptotics. This approach offers a principled route to constraining vacua in modular-invariant EFTs and points toward extensions to multi-modulus settings and explicit string theory realizations.

Abstract

We consider 4d $\mathcal{N}=1$ supergravity theories with modular symmetry, where the modulus $τ$ is the upper half-plane modulo $SL(2,\mathbf{Z})$ action. We focus on enhanced discrete gauge symmetry points $τ=i, \exp(2πi/3)$, 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 Kähler potential.