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<p>We consider 4d <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N}=1 \)</EquationSource> </InlineEquation> supergravity theories with modular symmetry, where the modulus <i>τ</i> is the upper half-plane modulo SL(2<i>,</i> <b>Z</b>) action. We focus on enhanced discrete gauge symmetry points <i>τ</i> = <i>i,</i> exp(2<i>πi/</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 Kähler potential.</p>

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Symmetry points of \( \mathcal{N}=1 \) modular geometry

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月19日

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  • Amineh Mohseni,
  • Cumrun Vafa

摘要

We consider 4d N = 1 \( \mathcal{N}=1 \) supergravity theories with modular symmetry, where the modulus τ is the upper half-plane modulo SL(2, 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.