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

TL;DR

This paper proves that de Sitter minima cannot exist on Riemann-flat manifolds with Casimir energy in string and M-theory for dimensions greater than three, linking the problem to semidefinite programming and numerical methods.

Contribution

It establishes a no-go theorem for de Sitter minima on Riemann-flat manifolds in higher dimensions and connects the problem to semidefinite optimization techniques.

Findings

No de Sitter minima for d>3 on Riemann-flat manifolds with Casimir energy.

De Sitter minima search is equivalent to a semidefinite programming problem.

Illustration of the approach in a toy model.

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.