Rolling with modular symmetry: quintessence and de Sitter in heterotic orbifolds

Abstract

Modular invariance is a fundamental symmetry in string compactifications, constraining both the structure of the effective theory and the dynamics of moduli and matter fields. It has also gained renewed importance in the context of swampland conjectures and, independently, flavour physics. We investigate a modular-invariant scalar potential arising from heterotic orbifolds, where the flavour structure and moduli dynamics are jointly shaped by the underlying geometry. Focusing on a string-inspired, two-moduli truncation, we uncover a rich vacuum structure featuring anti-de Sitter minima and unstable de Sitter saddle points. We identify large regions in moduli space supporting multifield hilltop quintessence consistent with observations. All solutions satisfy refined swampland de Sitter bounds. Our results illustrate how modular symmetry can guide the construction of controlled, string-motivated quintessence scenarios within consistent effective theories.

Figures (11)

• Figure 1: Fundamental domain of $\mathrm{SL}(2,\mathds{Z}_{})$ for the modulus $\tau$.
• Figure 2: The $\mathds{T}^2/\mathds{Z}_{3}$ orbifold. The three inequivalent fixed points under the $\mathds{Z}_{3}$ action are shown as blue dots. The green shaded region denotes the fundamental domain of the orbifold.
• Figure 3: Hierarchical classification scheme applied to critical points of the scalar potential $V$, based on their properties. Initial filtering identifies candidate critical points where $\nabla V \sim 0$. Then, we discriminate the cases where $\mathop{\mathrm{Re}}\nolimits(S)$ diverges. The eigenvalues of the Hessian $\nabla^i\nabla_j V$ are then evaluated to assess the stability of each solution: cases with all positive eigenvalues, one negative eigenvalue, or multiple negative eigenvalues are distinguished. We then evaluate whether the dominant contribution arises from $\mathop{\mathrm{Re}}\nolimits(S)$. Based on the outcomes of these filters, each point is assigned to a physical class: Vacuum, Gauge runaway, Many tachyons, Tachyonic axion, Tachyonic gauge coupling, or Not a critical point.
• Figure 4: Z-score radar chart of the parameter and moduli values for the different classes of solutions. Each polygon compares how the Z-scores of the various parameters and moduli distribute, based on their mean values and $\sigma$ variance to facilitate direct comparison. The innermost (outermost) circle corresponds to the mean value minus (plus) $1\sigma$ for all variables. The table provides these statistical values for each variable in our search. The unphysical classes Not a critical point and Gauge runaway are omitted.
• Figure 5: Distribution of the classes from \ref{['fig:Tree']} in the complex $\tau$-plane. Solutions corresponding to stable vacua are represented by yellow stars, whereas those identified as Tachyonic axion are marked with pink squares. Tachyonic gauge coupling solutions are represented with blue triangles, while different colours distinguish the remaining cases. Solutions classified as Tachyonic axion cluster in specific angular regions with a "penacho"-like structure, while the apparent tendency of the Not a critical point class to occupy complementary sectors is not statistically robust. Stable vacua occur more often near the boundaries of the fundamental domain, though they also appear sporadically throughout the region without a dominant spatial pattern.