Modular Symmetries and the Swampland Conjectures

TL;DR

This work uses modular (duality) symmetries of moduli spaces to probe swampland constraints beyond perturbation theory in 4d ${\rm N}=1$ string vacua. It shows that modular-invariant non-perturbative potentials diverge at infinite distance due to towers of light states, aligning with the swampland distance conjecture and dynamically preventing access to global symmetries; these effects are illuminated via dual descriptions such as Type I'/Horava-Witten, where towers correspond to D0-branes or KK modes. The authors find that, across a range of single- and two-modulus modular-invariant potentials, there are dS maxima and saddles but no dS minima, and the refined dS conjecture is satisfied. This establishes a concrete bridge between modular invariance and quantum-gravity consistency, suggesting that duality structure may generically enforce swampland constraints in low-supersymmetry settings.

Abstract

Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d $\mathcal{N}=1$ compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the Kähler and complex structure moduli, in particular $SL(2,Z)$ T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I' or Horava-Witten theory, in which the $SL(2,Z)$ acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. The non-perturbative examples explored point to potentials not leading to weak coupling at infinite distance, but rather diverging in the asymptotic corners of moduli space, dynamically forbidding the access to points with global symmetries. We perform a study of general modular invariant potentials and find that there are dS maxima and saddle points but no dS minima, and that all examples explored obey the refined dS conjecture.

Figures (3)

• Figure 1: Fundamental domain for $PSL(2,{\bf Z})$. The self-dual points are located at $T=i,\rho$. Other extrema are found on the border of the modular domain.
• Figure 3: Left: Scalar potential for $W=j^{1/3}/\eta^6$ (i.e. $n=1$, $m=0$, ${\cal P} =1$): Right: A zoom around its dS maximum.
• Figure 4: $\frac{\left|\nabla V\right|}{V}$ for $W=j^{1/3}/\eta^6$. The ratio is always bigger than one and grows linearly with $\text{Im} \ T$.