Boundary-Value Problem in Type IIB Supergravity and Holography

In precision holography, the Euclidean on-shell action of type IIB supergravity often fails to reproduce the leading large-$N$ free energy of the dual field theory: the type IIB pseudo action, for instance, vanishes identically on the ${\mathrm{E}\mathrm{A}\mathrm{d}\mathrm{S}}_5\times S^5$ background. We address this issue by revisiting the boundary-value problem of the Euclidean type IIB pseudo action from first principles. Classifying the admissible boundary conditions through the variational principle, we construct a generalized pseudo action whose boundary terms implement the choice of ensemble — fixed potentials versus fixed quantized Page charges — in which the holographic comparison is performed. For the fixed five-form-charge ensemble, the resulting boundary term reproduces the recently proposed topological correction to the Pasti–Sorokin–Tonin formulation of type IIB supergravity for holographic backgrounds of interest. We then test the generalized pseudo action on two complementary backgrounds: the ${\mathrm{E}\mathrm{A}\mathrm{d}\mathrm{S}}_3\times S^3\times M_4$ near-horizon geometry of the D1-D5 system and the warped ${\mathrm{E}\mathrm{A}\mathrm{d}\mathrm{S}}_6\times S^2\times \mathrm{\Sigma }$ solutions dual to five-dimensional SCFTs. In both cases the ten-dimensional on-shell action agrees exactly with that of the corresponding lower-dimensional supergravity, and thereby with the leading large-$N$ free energy of the dual SCFT. Our results establish the choice of boundary conditions — and hence of ensemble — as an essential ingredient of precision holography directly in type IIB supergravity, extending recent analyses in eleven-dimensional supergravity.