de Sitter no-go's for Riemann-flat manifolds and a link to semidefinite optimisation

TL;DR

The paper analyzes the existence of de Sitter vacua in string/M-theory flux compactifications on Riemann-flat manifolds with Casimir energy, proving a no-go for $dS$ minima in dimensions $d>3$ and showing that $dS_4$ is excluded across supersymmetric setups, while $dS_3$ minima remain possible in principle with carefully chosen fluxes and duality-breaking mechanisms. The authors recast the moduli-stabilization problem into a semidefinite programming framework, linking it to matrix-programs familiar from the CFT bootstrap, and demonstrate how to leverage existing SDP/numerical bootstrap tools to search for viable $dS_3$ vacua. A concrete toy model based on M-theory on $T^8$ down to $d=3$ illustrates how to parametrize flux/Casimir contributions and explore feasible regions in a reduced moduli space, highlighting the utility of the bootstrap-inspired approach for navigating the string landscape. The work provides a clear classification of constraints across theories with 16 and 32 supercharges and sets a practical program for extending the SDP method to broader compactifications (e.g., Calabi–Yau, nilmanifolds), with potential implications for realistic cosmologies and swampland considerations.

Abstract

We establish a no-go theorem in the context of string and M-theory flux compactifications on Riemann-Flat manifolds with Casimir energy. Specifically, we show that no dS minimum exists in this setup in dimension $d>3$. The case of dS$_3$ minima is not excluded, but their actual fate can only be ascertained via an explicit construction. We also point out that the problem of finding dS minima on RFM's and more general flux compactifications is mathematically equivalent to a semidefinite programming problem, identical to those studied in CFT bootstrap, and hence the search for dS can benefit from the existing vast literature and numerical tools. We illustrate this in a toy model.

Figures (2)

• Figure 1: In this paper, we look for dS minimum solutions where a negative Casimir energy contribution (blue dashed line) and, at least, two flux contributions (red lines) generate a combined potential (black solid line) for the volume modulus $R$ with a local positive metastable minimum. There is no curvature contribution since the compactification manifold is Riemann-flat. Due to Einstein frame rescalings, the value of the minimum can easily become small in Planck units even for moderately large $R$.
• Figure 2: Minimum value of $V_{\text{Cas}}$ that satisfies all constraints of our semidefinite optimisation problem (\ref{['diff']}) for each value of $(\mathcal{C}^{(1)},\mathcal{C}^{(2)})$, and different combinations of fluxes. White regions have no solution to the constraints and are therefore excluded. We also show the points corresponding to the Casimir potential $\mathcal{C}(\phi)$ computed for $\phi\in[0.5,2]$ and two different choices of spin structure on $T^8$, labeled using the notation of ValeixoBento:2025yhz. The color in the allowed region represents the value of the objective function for the optimisation problem, the Casimir coefficient $\mathcal{C}$. Lower values/darker hues correspond to solutions which are under better control.