Type IIB supergravity action on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>M</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>5</mml:mn></mml:msup></mml:math> solutions
Stefan A. Kurlyand, A.A. Tseytlin
Abstract
While the ten-dimensional type IIB supergravity action evaluated on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ solution vanishes, the five-dimensional effective action reconstructed from equations of motion using the ${M}^{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ compactification ansatz is proportional to the ${\mathrm{AdS}}_{5}$ volume. The latter is consistent with the conformal anomaly interpretation in $\mathrm{AdS}/\mathrm{CFT}$ context. We show that this paradox can be resolved if, in the case of ${M}^{5}\ifmmode\times\else\texttimes\fi{}{X}^{5}$ topology, the ten-dimensional action contains an additional 5-form-dependent ``topological'' term $\ensuremath{\int}{F}_{5M}\ensuremath{\wedge}{F}_{5X}$. The presence of this term is suggested also by gauge-invariance considerations in the Pasti-Sorokin-Tonin formulation of type IIB supergravity action. We show that this term contributes to the ten-dimensional action evaluated on the D3-brane solution.