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

TL;DR

This paper investigates the discrepancy between the 10d type IIB supergravity action and its 5d effective counterpart on $M^5 imes X^5$ solutions, proposing a topological term to resolve the paradox within the AdS/CFT framework.

Contribution

It introduces a topological 5-form dependent term in the 10d action for $M^5 imes X^5$ topologies, clarifying the action's evaluation and its relation to conformal anomalies.

Findings

The 10d action on AdS$_5\times S^5$ vanishes, but the 5d effective action is proportional to volume.

A topological term involving $F_{5M} \wedge F_{5X}$ resolves the paradox.

The topological term affects the action evaluated on D3-brane solutions.

Abstract

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.