Non-supersymmetric heterotic strings on $AdS_{4}\times S^{3}\times S^{3}$

We analyze the stability properties of a family of anti-de Sitter flux compactifications of the tachyon-free non-supersymmetric heterotic string in ten dimensions. In contrast with simpler such solutions, the solutions include two independent unbounded fluxes, leading to richer instability phenomena. In particular, when the two fluxes are sufficiently close in magnitude, the perturbative spectrum develops tachyonic modes, which can be projected out by an orbifold action. When the fluxes are far apart, tachyonic modes are absent, and the geometry displays inverse scale separation, where a factor of the internal manifold becomes parametrically larger than the anti-de Sitter factor. Still, non-perturbative instabilities in the form of brane nucleation are always available decay channels, and tend to drive the two fluxes closer together, eventually triggering the tachyonic instability when present.