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 $\mathrm{EAdS}_5\times S^5$ 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 $\mathrm{EAdS}_3\times S^3\times M_4$ near-horizon geometry of the D1-D5 system and the warped $\mathrm{EAdS}_6\times S^2\timesΣ$ 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.