Generalize the pseudo action to non-trivially fibred backgrounds

Generalize the type IIB generalized pseudo action, including the boundary terms and the associated Stokes-theorem justification, to holographic backgrounds in which the internal space is non-trivially fibred over the external spacetime, such as the asymptotically ${\rm EAdS}_3\times S^3\times M_4$ saddles of the D1-D5-P system.

Background

The generalized type IIB pseudo action is formulated using an internal-degree counting argument that ensures certain boundary nine-forms are globally defined and permits the application of Stokes’ theorem. This argument was established for factorized backgrounds of the form $M_d\times X_{10-d}$ and does not directly cover non-trivial fibrations of the internal space over the external factor.

The paper notes that asymptotically ${\rm EAdS}_3\times S^3\times M_4$ saddles arising from the D1-D5-P system fall into this broader class, because the $S^3$ can be fibred over the external factor through flat connections encoding angular potentials. A complete extension of the pseudo-action construction to such geometries is therefore left unresolved.