We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is −3/2 log(area), in agreement with that for the Schwarzschild black hole obtained in the four dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the right expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toral boundary of the black hole.

We study the time-independent modes of a massless scalar field in various black hole backgrounds, and show that for these black holes, the time-independent mode is localized at the horizon. A similar analysis is done for time-independent, equilibrium modes of the five-dimensional plane AdS black hole. A self-adjointness analysis of this problem reveals that in addition to the modes corresponding to the usual glueball states, there is a discrete infinity of other equilibrium modes with imaginary mass for the glueball. We suggest these modes may be related to a Savvidy-Nielsen-Olesen-like vacuum instability in QCD.

We discuss certain recent mathematical advances, mainly due to Perelman, in the theory of Ricci flows and their relevance for renormalization group (RG) flows. We consider nonlinear sigma models with closed target manifolds supporting a Riemannian metric, dilaton, and 2-form B-field. By generalizing recent mathematical results to incorporate the B-field and by decoupling the dilaton, we are able to describe the 1-loop β-functions of the metric and B-field as the components of the gradient of a potential functional on the space of coupling constants. We emphasize a special choice of diffeomorphism gauge generated by the lowest eigenfunction of a certain Schrödinger operator whose potential and kinetic terms evolve along the flow. With this choice, the potential functional is the corresponding lowest eigenvalue, and gives the order α ′ correction to the Weyl anomaly at fixed points of (g(t), B(t)). The lowest eigenvalue is monotonic along the flow, and since it reproduces the Weyl anomaly at fixed points, it accords with the c-theorem for flows that remain always in the first-order regime. We compute the Hessian of the lowest eigenvalue functional and use it to discuss the linear stability of points where the 1-loop β-functions vanish, such as flat tori and K3 manifolds.

A novel method, based on superpotentials is proposed for obtaining the quasi-normal modes of anti-deSitter black holes. This is inspired by the case of the three dimensional BTZ black hole, where the quasi-normal modes can be obtained exactly and are proportional to the surface gravity. Using this approach, the quasi-normal modes of the five dimensional Schwarzschild anti-deSitter black hole are computed numerically. The modes again seem to be proportional to the surface gravity for very small and very large black holes. They reflect the well-known instability of small black holes in anti-deSitter space.

Quasinormal modes for scalar field perturbations of a Schwarzschild-de Sitter (SdS) black hole are investigated. An analytical approximation is proposed for the problem. The quasinormal modes are evaluated for this approximate model in the limit when black hole mass is much smaller than the radius of curvature of the spacetime. The model mirrors some striking features observed in numerical studies of time behaviour of scalar perturbations of the SdS black hole. In particular, it shows the presence of two sets of modes relevant at two different time scales, proportional to the surface gravities of the black hole and cosmological horizons respectively. These quasinormal modes are not complete -another feature observed in the numerical studies. Refinements of this model to yield more accurate quantitative agreement with numerical results are discussed. Further investigations of this model are outlined, which would provide a valuable insight into time behaviour of perturbations in the SdS spacetime.

We demonstrate the irreversibility of a wide class of world-sheet renormalization group (RG) flows to first order in α ′ in string theory. Our techniques draw on the mathematics of Ricci flows, adapted to asymptotically flat target manifolds. In the case of somewhere-negative scalar curvature (of the target space), we give a proof by constructing an entropy that increases monotonically along the flow, based on Perelman's Ricci flow entropy. One consequence is the absence of periodic solutions, and we are able to give a second, direct proof of this. If the scalar curvature is everywhere positive, we instead construct a regularized volume to provide an entropy for the flow. Our results are, in a sense, the analogue of Zamolodchikov's c-theorem for world-sheet RG flows on noncompact spacetimes (though our entropy is not the Zamolodchikov C-function).

We consider non-spherically symmetric perturbations of the uncharged black string/flat black brane in the large dimension (D) limit of general relativity. We express the perturbations in a simplified form using variables introduced by Ishibashi and Kodama. We apply the large D limit to the equations, and show that this leads to decoupling of the equations in the near-horizon and asymptotic regions. It also enables use of matched asymptotic expansions to obtain approximate analytical solutions and to analyze stability of the black string/brane. For a large class of non-spherically symmetric perturbations, we prove that there are no instabilities in the large D limit. For the rest, we provide additional matching arguments that indicate that the black string/brane is stable. In the static limit, we show that for all nonspherically symmetric perturbations, there is no instability. This is proof that the Gross-Perry-Yaffe mode for semiclassical black hole perturbations is the unique unstable mode even in the large D limit. This work is also a direct analytical indication that the only instability of the black string is the Gregory-Laflamme instability.

We study quantum gravity on dS3 using the Chern-Simons formulation of three-dimensional gravity. We derive an exact expression for the partition function for quantum gravity on dS3 in a Euclidean path integral approach. We show that the topology of the space relevant for studying de Sitter entropy is a solid torus. The quantum fluctuations of de Sitter space are sectors of configurations of point masses taking a discrete set of values. The partition function gives the correct semi-classical entropy. The sub-leading correction to the entropy is logarithmic in horizon area, with a coefficient −1. We discuss this correction in detail, and show that the sub-leading correction to the entropy from the dS/CFT correspondence agrees with our result. A comparison with the corresponding results for the AdS3 BTZ black hole is also presented.