Abstract
AbstractWe show that a class of Einstein-Maxwell-Dilaton (EMD) theories are re- lated to higher dimensional AdS-Maxwell gravity via a dimensional reduction over com- pact Einstein spaces combined with continuation in the dimension of the compact space to non-integral values (‘generalized dimensional reduction’). This relates (fairly complicated) black hole solutions of EMD theories to simple black hole/brane solutions of AdS-Maxwell gravity and explains their properties. The generalized dimensional reduction is used to infer the holographic dictionary and the hydrodynamic behavior for this class of theories from those of AdS. As a specific example, we analyze the case of a black brane carrying a wave whose universal sector is described by gravity coupled to a Maxwell field and two neutral scalars. At thermal equilibrium and finite chemical potential the two operators dual to the bulk scalar fields acquire expectation values characterizing the breaking of con- formal and generalized conformal invariance. We compute holographically the first order transport coefficients (conductivity, shear and bulk viscosity) for this system.
Highlights
We show that a class of Einstein-Maxwell-Dilaton (EMD) theories are related to higher dimensional AdS-Maxwell gravity via a dimensional reduction over compact Einstein spaces combined with continuation in the dimension of the compact space to non-integral values (‘generalized dimensional reduction’)
The results depend smoothly on σ as long as σ > d/2. This can be seen by inspection of the results but it is intuitively clear: the dimension of the torus, (2σ − d), should be positive. It follows that one can use the dimensional reduction in order to establish a holographic dictionary for this theory when1 σ > d/2 (this translates into a constraint on the slope of the potential, δ2 < 2/(d − 1))
One can check that the results established in [22] by a direct analysis of the field equations etc. are reproduced exactly. This method was applied successfully to probe non-conformal branes [26]: the corresponding holographic dictionary was obtained from the results in [27] in this way. This method can be used in order to set up holography for any theory that is related to a theory for which the holographic dictionary is known via such a ’generalized consistent reduction’
Summary
We will consider how higher-dimensional AdS(-Maxwell) gravity reduces to Einstein-Maxwell-Dilaton (EMD) theories via a (generalized) consistent (non-)diagonal Kaluza-Klein reduction. Run from 0 to d, and denote lower, (d + 1)-dimensional spacetime coordinates, while lowercase latin indices a, b, . Will typically run from d + 1 to 2σ and denote internal coordinates. Refer to the higher-dimensional spacetime coordinates and run from 0 to 2σ. Are higher-dimensional indices and run from 0 to 2σ − 1, while lowercase latin indices i, j, . Are lower-dimensional boundary indices and run from 0 to d− 1. The Maxwell terms with straight latin uppercase originate from higher dimensions, A = AAdxA, with field strength. The reductions of interest are over Einstein manifolds. Recall that a p-dimensional Einstein manifold X(p) satisfies.
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