[en] In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under q-deformed Plancherel measure. The matrix model is a q analog of Gross-Witten-Wadia (GWW) matrix model. In the large N limit the model exhibits a third order phase transition between no-gap and gapped phases, which is a q-deformed version of the GWW phase transition. We show that the no-gap phase of this matrix model captures the asymptotic growth of Young diagrams equipped with q-deformed Plancherel measure. The no-gap solutions also satisfy a differential equation which is the q-analogue of the automodel equation. We further provide a droplet description for these growing Young diagrams. Quantising these droplets we identify the Young diagrams with coherent states in the Hilbert space. We also elaborate the connection between moments of Young diagrams and the infinite number of commuting Hamiltonians obtained from the large N droplets and explicitly compute the moments for asymptotic Young diagrams.

[en] Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large N Young diagrams are also mapped to quantum (large N) fluctuations of the coherent states.

[en] We study phase transition between electrically charged Ricci-flat black holes and AdS soliton spacetime of Horowitz and Myers in five dimensions. Boundary topology for both of them is S¹ x S¹ x R². We consider AdS-Reissner-Nordstroem black hole and R-charged black holes and find that phase transition of these black holes to AdS soliton spacetime depends on the relative size of two boundary circles. We also perform the stability analysis for these black holes. In order to use the AdS/CFT correspondence, we work in the grand canonical ensemble

[en] We examine the Euclidean action approach, as well as that of Wald, to the entropy of black holes in asymptotically AdS spaces. From the point of view of holography these two approaches are somewhat complementary in spirit and it is not obvious why they should give the same answer in the presence of arbitrary higher derivative gravity corrections. For the case of the AdS₅ Schwarzschild black hole, we explicitly study the leading correction to the Bekenstein-Hawking entropy in the presence of a variety of higher derivative corrections studied in the literature, including the Type IIB R⁴ term. We find a nontrivial agreement between the two approaches in every case. Finally, we give a general way of understanding the equivalence of these two approaches

[en] In this paper we generalize the construction of locally boosted black brane space time to higher derivative gravities. We consider the Gauss-Bonnet term (with coefficient α') as a toy example. We find the solution to the α' corrected Einstein equations to first order in the boundary derivative expansion. This allows us to find the α' corrections to the boundary stress tensor in the presence of the Gauss-Bonnet term in the bulk action. We therefore obtain the ratio of shear viscosity to entropy which agrees with other methods of computation in the literature.

[en] We analyze the dynamics of weakly coupled finite temperature U(N) gauge theories on S"3 by studying a class of effective unitary matrix model. Solving Dyson-Schwinger equation at large N, we find that different phases of gauge theories are characterized by gaps in eigenvalue distribution over a unit circle. In particular, we obtain no-gap, one-gap and two-gap solutions at large N for a class of matrix model we are considering. The same effective matrix model can equivalently be written as a sum over representations (or Young diagrams) of unitary group. We show that at large N, Young diagrams corresponding to different phases can be classified in terms of discontinuities in number of boxes in two consecutive rows. More precisely, the representation, where there is no discontinuity, corresponds to no-gap and one-gap solution, where as, a diagram with one discontinuity corresponds to two-gap phase, mentioned above. This observation allows us to write a one to one relation between eigenvalue distribution function and Young tableaux distribution function for each saddle point, in particular for two-gap solution. We find that all the saddle points can be described in terms of free fermions with a phase space distribution for no-gap, one-gap and two-gap phases.

[en] The dynamics of finite temperature U(N) gauge theories on S³ can be described, at weak coupling, by an effective unitary matrix model. Here we present an exact solution to these models, for any value of N, in terms of a sum over representations. Taking the large N limit of this solution provides a new perspective on the deconfinement transition which is supposed to be dual to the Hawking-Page transition. The large N phase transition manifests itself here in a manner similar to the Douglas-Kazakov phase transition in 2d Yang-Mills theory. We carry out a complete analysis of the saddle representation in the simplest case involving only the order parameter TrU. We find that the saddle points corresponding to thermal AdS, the small black hole and the large black hole can all be described in terms of free fermions. They all admit a simple phase space description a la the BPS geometries of Lin, Lunin and Maldacena

[en] The shear viscosity coefficient of strongly coupled boundary gauge theory plasma depends on the horizon value of the effective coupling of transverse graviton moving in a black hole background. The proof for the above statement is based on the canonical form of graviton's action. But in presence of generic higher derivative terms in the bulk Lagrangian the action is no longer canonical. We give a procedure to find an effective action for graviton (to first order in coefficient of higher derivative term) in canonical form in presence of any arbitrary higher derivative terms in the bulk. From that effective action we find the effective coupling constant for transverse graviton which in general depends on the radial coordinate r. We also argue that horizon value of this effective coupling is related to the shear viscosity coefficient of the boundary fluid in higher derivative gravity. We explicitly check this procedure for two specific examples: (1) four derivative action and (2) eight derivative action (Weyl⁴ term). For both cases we show that our results for shear viscosity coefficient (upto first order in coefficient of higher derivative term) completely agree with the existing results in the literature.

[en] We compute constitutive relations for a charged $(2+1)$ dimensional Schrödinger fluid up to first order in derivative expansion, using holographic techniques. Starting with a locally boosted, asymptotically AdS, $4+1$ dimensional charged black brane geometry, we uplift that to ten dimensions and perform $TsT$ transformations to obtain an effective five dimensional local black brane solution with asymptotically Schrödinger isometries. By suitably implementing the holographic techniques, we compute the constitutive relations for the effective fluid living on the boundary of this space-time and extract first order transport coefficients from these relations. Schrödinger fluid can also be obtained by reducing a charged relativistic conformal fluid over light-cone. It turns out that both the approaches result the same system at the end. Fluid obtained by light-cone reduction satisfies a restricted class of thermodynamics. Here, we see that the charged fluid obtained holographically also belongs to the same restricted class.

[en] It is now well understood that the coefficient of shear viscosity of boundary fluid can be obtained from the horizon values of the effective coupling of transverse graviton in bulk spacetime. In this paper we observe that to find the shear-viscosity coefficient it is sufficient to know only the near-horizon geometry of the black hole spacetime. One does not need to know the full analytic solution. We consider several examples including non-trivial matter (dilaton, gauge fields) coupled to gravity in presence of higher derivative terms and calculate shear viscosity for both extremal and non-extremal black holes only studying the near-horizon geometry. In particular, we consider higher derivative corrections to extremal R-charged black holes and compute η/s in presence of three independent charges. We also consider asymptotically Lifshitz spacetime whose dual black hole geometry cannot be found analytically. We study the near horizon behaviour of these black holes and find η/s for its dual plasma at Lifshitz fixed point.