On type IIB supergravity action on $M^5\times X^5$ solutions

While the 10d type IIB supergravity action evaluated on AdS${}_5\times S^5$ solution vanishes, the 5d effective action reconstructed from the equations of motion using $M^5\times S^5$ compactification ansatz is proportional to the AdS${}_5$ volume. The latter is consistent with the conformal anomaly interpretation in AdS/CFT context. We show that this paradox can be resolved if, in the case of $M^5\times X^5$ topology, the 10d action contains an additional 5-form dependent “topological” term $\int F_{5M}\wedge F_{5X}$. The presence of this term is suggested also by gauge-invariance considerations in the PST formulation of type IIB supergravity action. We show that this term contributes to the 10d action evaluated on the D3-brane solution.