Finite action in 5D gauged supergravity and the dilatonic conformal anomaly for dual quantum field theory

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Abstract
Gauged supergravity (SG) with a single scalar (dilaton) and arbitrary scalar potential is considered. Such dilatonic gravity describes special renormalization group flows in extended SG where scalars lie in a one-dimensional submanifold of total space. The surface counterterm and finite action for such gauged SG in three-, four-, and five-dimensional asymptotically AdS space are derived. Using a finite action and a consistent gravitational stress tensor (local surface counterterm prescription) the regularized expressions for free energy, entropy, and mass of a 4D dilatonic AdS black hole are found. The same calculation is done within standard reference background subtraction. The dilaton-dependent conformal anomaly from 3D and 5D gauged SGs is calculated using AdS conformal field theory correspondence. Such an anomaly should correspond to two- and four-dimensional dual quantum field theory which is classically and not exactly conformally invariant, respectively. The candidate c functions from 3D and 5D SGs are suggested. These c functions which have fixed points in an asymptotically AdS region are expressed in terms of a dilatonic potential and they are positively defined and monotonic for a number of potentials.

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