Entanglement entropy measures the correlation between different subsystems of a quantum system. Recently, Ryu and Takayanagi [1] proposed a holographic version of the entanglement entropy (S=(the area of a minimal co-dimensional 2 surface)/4G ) which was proved by Fursaev [2] using the Ads/CFT correspondence. In this talk, I will first give a brief introduction to the concepts of entanglement entropy in quantum field theory, and the gravity/field theory duality. Then I will sketch Fursaev proof of the holographic formula for entanglement entropy and give a simple covariant recipe proposed by Takayanagi, et al [3] to calculate this quantity. Some examples will be worked out as time permits.
It has been known for at least 25 years now,due to work of I. Frenkel, J. Lepowsky, A. Meurman, B. Fischer, R. Griess, and others, that there exists a (holomorphic sector) conformal field theory (CFT) with central charge c=24 whose partition function is an elliptic modular function J(z)(well known in number theory), and whose states form a representation of a particular finite group G,called the "monster" (since G has an enormous number of elements,approximately 10 to the 54th power). Remarkably, one can relate factors of degrees of irreducible modules of G to factors of the famous Kac determinant of conformal weight h=10.Also J(z) can be used to explain the weird fact of nature that exp(pi.square root(163)) is "very nearly equal to a whole number". We discuss this circle of ideas, and how the asymptotic growth of J(z) can be used to compute Bekenstein-Hawking black hole entropy in AdS_3 gravity. We point out that entropy correction terms correspond, in fact, to sub-leading asymptotic terms.
Komar integral relations in Einstein gravity are useful in deriving certain basic properties of stationary black hole solutions. We will give the basic definition of a Komar integral relation and then ask whether they exist in more general higher derivative gravity theories. We will see that the only purely quadratic gravity theory with a Komar integral relation is the one based on the Gauss-Bonnet Lagrangian. It turns out more generally that Lovelock gravity theories, of which Einstein and Gauss-Bonnet gravity are two examples, can also be shown to have Komar integral relations.
In this talk I will review the Friedmann law of brane models - models where matter is confined to a hypersurface embedded in a space with larger dimension. In particular, I will review the cosmological behavior of compact and non compact codimension-one brane models. I will also discuss, if time allows, the (rather peculiar) properties of codimension-2 models.
Although supersymmetry (SUSY), for most of the cases, is introduced on the level of field theory, SUSY in quantum mechanics is simpler, more elegant and yet encompasses a lot of concepts one encounters in field theory. In this chalk talk, I will introduce the supersymmetric Hamiltonian formalism and its use for solving Schrodinger's equation with complicated potentials. After that I will discuss the concept of SUSY breaking. Witten's index and the general factorization method will also be dicussed if time allows.
Recently there have been interesting reformulations of Einstein's field equations for scalar field cosmologies, both for isotropic and anisotropic models, in terms of generalized types of Ermakov-Milne-Pinney equations. Inspired by this work, we have discovered an alternative Schrodinger formulation of Einstein's equations in a Friedmann-Lemaitre-Robertson-Walker universe. This provides for an alternate method of obtaining exact solutions of the field equations. After presenting this initial work, I will briefly demonstrate analogous Schrodinger models that have subsequently been found for both Bianchi I and Bianchi V cosmologies.
Quantized Hamiltonian operators for constant magnetic fields on the hyperbolic plane can be expressed in terms of well-known Maass operators which occur in the theory of automorphic forms. A trace formula for these operators projected to a Riemann surface of genus at least 2 can therefore be given-in the form of a generalized Selberg trace formula. We discuss this interesting connection between physics and number theory; many other such connections exist. In particular we consider the influence of a constant magnetic field on the structure of certain zeta functions.