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De Sitter hunting in a classical landscape

Ulf H. Danielsson, Sheikh S. Haque, Paul Koerber, Gary Shiu, Thomas Van Riet, Timm Wrase

TL;DR

Addresses the problem of realizing de Sitter vacua within classical Type IIA string theory by constructing O6-plane compactifications on $SU(3)$-structure manifolds with smeared sources. The authors develop a dual 4D/10D framework, classify homogeneous group-manifold geometries and their orbifold/orientifold quotients, and systematically search for de Sitter critical points of the 4D potential, using a universal ansatz as a guiding tool. They find numerous candidate de Sitter extrema across several geometries but all exhibit tachyonic instabilities, and flux/charge quantization in the tree-level smeared regime prevents a properly quantized, large-volume, weakly coupled solution in the explicit SU(2)$\times$SU(2) model. The work provides a comprehensive methodological guide, highlights the central obstacles to fully stable de Sitter vacua in this setup, and outlines key avenues for future research, including backreaction, non-geometric fluxes, and broader geometric classes.

Abstract

We elaborate on the construction of de Sitter solutions from IIA orientifolds of SU(3)-structure manifolds that solve the 10-dimensional equations of motion at tree-level in the approximation of smeared sources. First we classify geometries that are orbifolds of a group manifold covering space which, upon the proper inclusion of O6 planes, can be described within the framework of N=1 supergravity in 4D. Then we scan systematically for de Sitter solutions, obtained as critical points of an effective 4D potential. Apart from finding many new solutions we emphasize the challenges in constructing explicit classical de Sitter vacua, which have sofar not been met. These challenges are interesting avenues for further research and include finding solutions that are perturbatively stable, satisfy charge and flux quantization, and have genuine localized (versus smeared) orientifold sources. This paper intends to be self-contained and pedagogical, and thus can serve as a guide to the necessary technical tools required for this line of research. In an appendix we explain how to study flux and charge quantization in the presence of a non-trivial H-field using twisted homology.

De Sitter hunting in a classical landscape

TL;DR

Addresses the problem of realizing de Sitter vacua within classical Type IIA string theory by constructing O6-plane compactifications on -structure manifolds with smeared sources. The authors develop a dual 4D/10D framework, classify homogeneous group-manifold geometries and their orbifold/orientifold quotients, and systematically search for de Sitter critical points of the 4D potential, using a universal ansatz as a guiding tool. They find numerous candidate de Sitter extrema across several geometries but all exhibit tachyonic instabilities, and flux/charge quantization in the tree-level smeared regime prevents a properly quantized, large-volume, weakly coupled solution in the explicit SU(2)SU(2) model. The work provides a comprehensive methodological guide, highlights the central obstacles to fully stable de Sitter vacua in this setup, and outlines key avenues for future research, including backreaction, non-geometric fluxes, and broader geometric classes.

Abstract

We elaborate on the construction of de Sitter solutions from IIA orientifolds of SU(3)-structure manifolds that solve the 10-dimensional equations of motion at tree-level in the approximation of smeared sources. First we classify geometries that are orbifolds of a group manifold covering space which, upon the proper inclusion of O6 planes, can be described within the framework of N=1 supergravity in 4D. Then we scan systematically for de Sitter solutions, obtained as critical points of an effective 4D potential. Apart from finding many new solutions we emphasize the challenges in constructing explicit classical de Sitter vacua, which have sofar not been met. These challenges are interesting avenues for further research and include finding solutions that are perturbatively stable, satisfy charge and flux quantization, and have genuine localized (versus smeared) orientifold sources. This paper intends to be self-contained and pedagogical, and thus can serve as a guide to the necessary technical tools required for this line of research. In an appendix we explain how to study flux and charge quantization in the presence of a non-trivial H-field using twisted homology.

Paper Structure

This paper contains 29 sections, 129 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The 4D and 10D picture
  3. Dimensional reduction and effective 4D theories
  4. The "universal" ansatz in 10D
  5. Classification of geometries
  6. Homogenous $\mathop{\rm SU}(3)$-structures
  7. Group manifold geometry and geometric moduli
  8. All unipotent real six-dimensional Lie algebras
  9. Discrete symmetries, orbifolds and orientifolds
  10. Discrete subgroups of $\mathop{\rm SU}(3)$
  11. The standard $\mathbb{Z}_2\times \mathbb{Z}_2$ orientifold example
  12. The non-standard $\mathbb{Z}_2\times \mathbb{Z}_2$ orientifold example
  13. General abelian and non-abelian orbifolds
  14. The standard orientifold of $\Delta(12)$
  15. The non-standard orientifold of $\Delta(12)$
  16. ...and 14 more sections

Figures (2)

  • Figure 1: The homogeneous $\mathop{\rm SU}(3)$-structure spaces $G/H$. The isotropy group is necessarily contained in $\mathop{\rm SU}(3)$ for a $G$-invariant $\mathop{\rm SU}(3)$-structure.
  • Figure 2: Plots of radius and dilaton for the family of properly quantized dS solutions on $\mathop{\rm SU}(2) \times \mathop{\rm SU}(2)$.