14 research outputs found
Quantum Liouville theory and BTZ black hole entropy
In this paper I give an explicit conformal field theory description of
(2+1)-dimensional BTZ black hole entropy. In the boundary Liouville field
theory I investigate the reducible Verma modules in the elliptic sector, which
correspond to certain irreducible representations of the quantum algebra
U_q(sl_2) \odot U_{\hat{q}}(sl_2). I show that there are states that decouple
from these reducible Verma modules in a similar fashion to the decoupling of
null states in minimal models. Because ofthe nonstandard form of the Ward
identity for the two-point correlation functions in quantum Liouville field
theory, these decoupling states have positive-definite norms. The explicit
counting from these states gives the desired Bekenstein-Hawking entropy in the
semi-classical limit when q is a root of unity of odd order.Comment: LaTeX, 33 pages, 4 eps figure
Lambda<0 Quantum Gravity in 2+1 Dimensions II: Black Hole Creation by Point Particles
Using the recently proposed formalism for Lambda<0 quantum gravity in 2+1
dimensions we study the process of black hole production in a collision of two
point particles. The creation probability for a BH with a simplest topology
inside the horizon is given by the Liouville theory 4-point function projected
on an intermediate state. We analyze in detail the semi-classical limit of
small AdS curvatures, in which the probability is dominated by the exponential
of the classical Liouville action. The probability is found to be exponentially
small. We then argue that the total probability of creating a horizon given by
the sum of probabilities of all possible internal topologies is of order unity,
so that there is no exponential suppression of the total production rate.Comment: v1: 30+1 pages, figures, v2: 34+1 pages, agruments straightened ou
Hamiltonian structure and quantization of 2+1 dimensional gravity coupled to particles
It is shown that the reduced particle dynamics of 2+1 dimensional gravity in
the maximally slicing gauge has hamiltonian form. This is proved directly for
the two body problem and for the three body problem by using the Garnier
equations for isomonodromic transformations. For a number of particles greater
than three the existence of the hamiltonian is shown to be a consequence of a
conjecture by Polyakov which connects the auxiliary parameters of the fuchsian
differential equation which solves the SU(1,1) Riemann-Hilbert problem, to the
Liouville action of the conformal factor which describes the space-metric. We
give the exact diffeomorphism which transforms the expression of the spinning
cone geometry in the Deser, Jackiw, 't Hooft gauge to the maximally slicing
gauge. It is explicitly shown that the boundary term in the action, written in
hamiltonian form gives the hamiltonian for the reduced particle dynamics. The
quantum mechanical translation of the two particle hamiltonian gives rise to
the logarithm of the Laplace-Beltrami operator on a cone whose angular deficit
is given by the total energy of the system irrespective of the masses of the
particles thus proving at the quantum level a conjecture by 't Hooft on the two
particle dynamics. The quantum mechanical Green's function for the two body
problem is given.Comment: 34 pages LaTe
Hamiltonian structure of 2+1 dimensional gravity
A summary is given of some results and perspectives of the hamiltonian ADM
approach to 2+1 dimensional gravity. After recalling the classical results for
closed universes in absence of matter we go over the the case in which matter
is present in the form of point spinless particles. Here the maximally slicing
gauge proves most effective by relating 2+1 dimensional gravity to the Riemann-
Hilbert problem. It is possible to solve the gravitational field in terms of
the particle degrees of freedom thus reaching a reduced dynamics which involves
only the particle positions and momenta. Such a dynamics is proven to be
hamiltonian and the hamiltonian is given by the boundary term in the
gravitational action. As an illustration the two body hamiltonian is used to
provide the canonical quantization of the two particle system.Comment: 13 pages,2 figures,latex, Plenary talk at SIGRAV2000 Conferenc
Accessory parameters for Liouville theory on the torus
We give an implicit equation for the accessory parameter on the torus which
is the necessary and sufficient condition to obtain the monodromy of the
conformal factor. It is shown that the perturbative series for the accessory
parameter in the coupling constant converges in a finite disk and give a
rigorous lower bound for the radius of convergence. We work out explicitly the
perturbative result to second order in the coupling for the accessory parameter
and to third order for the one-point function. Modular invariance is discussed
and exploited. At the non perturbative level it is shown that the accessory
parameter is a continuous function of the coupling in the whole physical region
and that it is analytic except at most a finite number of points. We also prove
that the accessory parameter as a function of the modulus of the torus is
continuous and real-analytic except at most for a zero measure set. Three
soluble cases in which the solution can be expressed in terms of hypergeometric
functions are explicitly treated.Comment: 30 pages, LaTex; typos corrected, discussion of eq.(74) improve
Gravitational Backreaction of Matter Inhomogeneities
The non-linearity of Einstein's equations makes it possible for small-scale
matter inhomogeneities to affect the Universe at cosmological distances. We
study the size of such effects using a simple heuristic model that captures the
most important backreaction effect due to nonrelativistc matter, as well as
several exact solutions describing inhomogeneous and anisotropic expanding
universes. We find that the effects are or smaller, where
is the Hubble parameter and the typical size scale of inhomogeneities. For
virialized structures this is of order , where is the
characteristic peculiar velocity.Comment: 16 page
Classical conformal blocks from TBA for the elliptic Calogero-Moser system
The so-called Poghossian identities connecting the toric and spherical
blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for
the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain
expressions for the classical 4-point block on the sphere. The main motivation
for this line of research is the longstanding open problem of uniformization of
the 4-punctured Riemann sphere, where the 4-point classical block plays a
crucial role. It is found that the obtained representation for certain 4-point
classical blocks implies the relation between the accessory parameter of the
Fuchsian uniformization of the 4-punctured sphere and the eCMY functional.
Additionally, a relation between the 4-point classical block and the ,
twisted superpotential is found and further used to re-derive the
instanton sector of the Seiberg-Witten prepotential of the , supersymmetric gauge theory from the classical block.Comment: 25 pages, no figures, latex+JHEP3, published versio