Small Cosmological Constants in String Theory

TL;DR

This work constructs explicit supersymmetric $ ext{AdS}_4$ vacua in type IIB string theory on orientifolded Calabi–Yau threefold hypersurfaces, achieving exponentially small $W_0$ through perturbatively flat flux vacua and racetrack nonperturbative effects. It ensures full stabilization of all Kähler moduli via a sufficient set of constant Pfaffian nonperturbative terms on rigid divisors, while controlling $ ext{GV}$-driven worldsheet corrections to the Kähler potential through high-degree invariants. The authors develop computational tools to identify orientifolds, compute GV invariants, and verify convergence of the worldsheet instanton series in high $h^{1,1}$ settings, producing multiple explicit vacua with vacuum energies as tiny as $|V_0| \sim 10^{-144} M_{ ext{pl}}^4$ (and smaller in other examples). The results demonstrate hierarchical scale separation with AdS length far exceeding the KK scale, and they provide a robust framework for exploring small-$W_0$ landscapes, while noting that uplift to de Sitter remains for future work.

Abstract

We construct supersymmetric $\mathrm{AdS}_4$ vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the $α'$ expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude $ < 10^{-123}$ in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol.

Figures (6)

• Figure 1: A triangulation of $\Theta^{(2)}.$
• Figure 2: Convergence of worldsheet instanton sum for $(h^{2,1},h^{1,1})=(5,113)$. Left: We plot the log-magnitude $\mathrm{log}(\xi_n)$, cf. \ref{['eq:xi']}, of the $n$-th term in the instanton series associated with a sample of 1728 potent rays in $\mathcal{M}_\infty(X)$, spanning a 101-dimensional cone. Right: a histogram of the slopes of $\mathrm{log}(\xi_n)$ with respect to $n$ for the set of potent rays. It is apparent that the sum converges.
• Figure 3: Convergence of worldsheet instanton sum for the second vacuum in $(h^{2,1},h^{1,1})=(5,113)$. Left: We plot the log-magnitude $\mathrm{log}(\xi_n)$, cf. \ref{['eq:xi']}, of the $n$-th term in the instanton series associated with a sample of 1728 potent rays in $\mathcal{M}_\infty(X)$, spanning a 101-dimensional cone. Right: a histogram of the slopes of $\mathrm{log}(\xi_n)$ with respect to $n$ for the set of potent rays. It is apparent that the sum converges, but the instanton series decays more slowly towards large degree in comparison to the first flux vacuum in $(h^{2,1},h^{1,1})=(5,113)$, cf. Figure \ref{['fig:convergence_5-113-4627']}.
• Figure 4: Convergence of worldsheet instanton sum for $(h^{2,1},h^{1,1})=(7,51)$. Left: We plot the log-magnitude $\mathrm{log}(\xi_n)$, cf. \ref{['eq:xi']}, of the $n$-th term in the instanton series associated with a sample of 758 potent rays in $\mathcal{M}_\infty(X)$, spanning a 48-dimensional cone. Right: a histogram of the slopes of $\mathrm{log}(\xi_n)$ with respect to $n$ for the set of potent rays. It is apparent that the sum converges.
• Figure 5: Convergence of worldsheet instanton sum for $(h^{2,1},h^{1,1})=(5,81)$. Left: We plot the log-magnitude $\mathrm{log}(\xi_n)$, cf. \ref{['eq:xi']}, of the $n$-th term in the instanton series associated with a sample of 727 potent rays in $\mathcal{M}_\infty(X)$, spanning a 76-dimensional cone. Right: a histogram of the slopes of $\mathrm{log}(\xi_n)$ with respect to $n$ for the set of potent rays. It is apparent that the sum converges.