Abstract
Extremal black branes upon compactification in the near horizon throat region are known to give rise to AdS2 dilaton-gravity-matter theories. Away from the throat region, the background has nontrivial profile. We interpret this as holographic renormalization group flow in the 2-dim dilaton-gravity-matter theories arising from dimensional reduction of the higher dimensional theories here. The null energy conditions allow us to formulate a holographic c-function in terms of the 2-dim dilaton for which we argue a c-theorem subject to appropriate boundary conditions which amount to restrictions on the ultraviolet theories containing these extremal branes. At the infrared AdS2 fixed point, the c-function becomes the extremal black brane entropy. We discuss the behaviour of this inherited c-function in various explicit examples, in particular compactified nonconformal branes, and compare it with other discussions of holographic c-functions. We also adapt the holographic renormalization group formulated in terms of radial Hamiltonian flow to 2-dim dilaton-gravity-scalar theories, which while not Wilsonian, gives qualitative insight into the flow equations and β-functions.
Highlights
Finite energy excitations in AdS2 [6, 7]
Away from the throat region, the background has nontrivial profile. We interpret this as holographic renormalization group flow in the 2-dim dilaton-gravity-matter theories arising from dimensional reduction of the higher dimensional theories here
The null energy conditions allow us to formulate a holographic c-function in terms of the 2-dim dilaton for which we argue a c-theorem subject to appropriate boundary conditions which amount to restrictions on the ultraviolet theories containing these extremal branes
Summary
We review our discussion in [35], where we studied 2-dim AdS2 dilaton-gravity matter theories arising from dimensional reduction of extremal charged black branes in Einstein-Maxwell and hyperscaling violating Lifshitz theories in 4-dimensions. Turning on perturbations to the dilaton and the metric, we see that the quadratic part of the bulk action gives coupled linearized equations governing these perturbations, while the leading correction comes from the Gibbons-Hawking term. As in the relativistic case earlier, a T 2-redux followed by a Weyl transformation and replacing the gauge fields F1μν, F2μν in terms of the dilaton and the scalar field Ψ gives an equivalent dilaton-gravity-scalar action with an effective interaction potential U (Φ, Ψ) for This admits an AdS2 background with constant dilaton and scalar field, which is the near horizon AdS2 throat of the 4-dim extremal hvLif black brane. The leading correction to AdS2 comes from the Gibbons-Hawking term linear in the dilaton perturbation, which gives the Schwarzian derivative action as mentioned earlier. We study more general 2-dim dilaton-gravity-matter theories: in such cases, the appearance of the Schwarzian follows from a more general argument [37, 45], related to a conformal anomaly arising after properly renormalizing the on-shell action
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