# Open problems on classical de Sitter solutions

## TL;DR

This paper reviews open problems and conjectures regarding classical de Sitter solutions in string theory, highlighting the lack of proofs and potential areas for counterexamples within the parameter space.

## Contribution

It clarifies the current state of the conjectures, develops new analytical tools, and identifies unexplored regions where counterexamples might exist.

## Key findings

- No proven classical de Sitter solutions with parallel sources
- Uncertainty about the stability of intersecting sources solutions
- Remaining parameter space where conjectures could be challenged

## Abstract

Classical 10d string backgrounds with a 4d de Sitter space-time, D-brane and orientifold sources, are commonly believed to satisfy the following: 1. There is no classical de Sitter solution with parallel sources. 2. Classical de Sitter solutions with intersecting sources are unstable. 3. Classical de Sitter solutions cannot have at the same time a large internal volume, a small string coupling, a bounded number of orientifolds and quantized fluxes. These three conjectures are of particular relevance to the swampland program, and if true, they challenge the connection of string theory to cosmology. We restrict here to a standard solution ansatz for which the problem is well-defined, and we still fail to prove analytically any of these conjectures. While developing new tools and obtaining new constraints, we identify remaining corners of parameter space where counter-examples to these conjectures could be found.