Non-integrability on AdS$_3$ supergravity backgrounds

TL;DR

This paper proves the complete non-integrability of certain AdS3 supergravity backgrounds, showing they do not admit Liouvillian solutions, except in trivial T-dual cases, using differential Galois theory on string embeddings.

Contribution

It applies differential Galois theory to demonstrate non-integrability of specific AdS3 supergravity backgrounds related to brane configurations.

Findings

No Liouvillian solutions exist for the Hamiltonian system.

Non-integrability holds except for trivial T-dual backgrounds.

The backgrounds are associated with D8-D6-D4-D2 brane setups.

Abstract

We investigate classical integrability on two recently discovered classes of backgrounds in massive IIA supergravity. These vacua are of the form AdS$_3\times\,$S$^2\times\mathbb{R}\times\,$CY$_2$, they preserve small $\mathcal{N}=(0,4)$ supersymmetry and are associated with D8$-$D6$-$D4$-$D2 Hanany-Witten brane set-ups. We choose an appropriate string embedding and use differential Galois theory on its associated Hamiltonian system, intending to produce the conditions under which Liouvillian solutions may occur. By constraining the parameters of the system according to the consistency of the associate brane set-ups we prove that no such conditions exist, yielding the complete non-integrability of these vacua. That is, up to the trivial cases where the background reduces to the Abelian and non-Abelian T-dual of AdS$_3\times\,$S$^3\times\,$T$^4$.