We study a new contraction of a d + 1 dimensional relativistic conformal algebra where n + 1 directions remain unchanged. For n = 0, 1 the resultant algebras admit infinite dimensional extension containing one and two copies of Virasoro algebra, respectively. For n > 1 the obtained algebra is finite dimensional containing an so(2, n + 1) subalgebra. The gravity dual is provided by taking a Newton-Cartan like limit of the Einstein's equations of AdS space which singles out an AdS n+2 spacetime. The infinite dimensional extension of n = 0, 1 cases may be understood from the fact that the dual gravities contain AdS 2 and AdS 3 factor, respectively. We also explore how the AdS/CFT correspondence works for this case where we see that the main role is playing by AdS n+2 base geometry.

Abstract:We study butterfly effect in D-dimensional gravitational theories containing terms quadratic in Ricci scalar and Ricci tensor. One observes that due to higher order derivatives in the corresponding equations of motion there are two butterfly velocities. The velocities are determined by the dimension of operators whose sources are provided by the metric. The three dimensional TMG model is also studied where we get two butterfly velocities at generic point of the moduli space of parameters. At critical point two velocities coincide.

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We study the collective excitations in a relativistic fluid with an anomalous U (1) current. In 3 + 1 dimensions at zero chemical potential, in addition to ordinary sound modes we find two propagating modes in presence of an external magnetic field. The first one which is a transverse degenerate mode, propagates with a velocity proportional to the coefficient of gravitational anomaly; this is in fact the Chiral Alfvén wave recently found in [1]. Another one is a wave of density perturbation, namely a chiral magnetic wave (CMW). The velocity dependence of CMW on the chiral anomaly coefficient is well known. We compute the dependence of CMW's velocity on the coefficient of gravitational anomaly as well. We also show that the dissipation splits the degeneracy of CAW. At finite chiral charge density we show that in general there may exist five chiral hydrodynamic waves. Of these five waves, one is the CMW while the other four are mixed Modified Sound-Alfvén waves. It turns out that in propagation transverse to the magnetic field no anomaly effect appear while in parallel to the magnetic field we find sound waves become dispersive due to anomaly. arXiv:1509.08878v3 [hep-th]