64 research outputs found
Enumeration of chord diagrams on many intervals and their non-orientable analogs
Two types of connected chord diagrams with chord endpoints lying in a
collection of ordered and oriented real segments are considered here: the real
segments may contain additional bivalent vertices in one model but not in the
other. In the former case, we record in a generating function the number of
fatgraph boundary cycles containing a fixed number of bivalent vertices while
in the latter, we instead record the number of boundary cycles of each fixed
length. Second order, non-linear, algebraic partial differential equations are
derived which are satisfied by these generating functions in each case giving
efficient enumerative schemes. Moreover, these generating functions provide
multi-parameter families of solutions to the KP hierarchy. For each model,
there is furthermore a non-orientable analog, and each such model likewise has
its own associated differential equation. The enumerative problems we solve are
interpreted in terms of certain polygon gluings. As specific applications, we
discuss models of several interacting RNA molecules. We also study a matrix
integral which computes numbers of chord diagrams in both orientable and
non-orientable cases in the model with bivalent vertices, and the large-N limit
is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio
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
Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality
We exploit the properties of the three-dimensional hyperbolic space to
discuss a simplicial setting for open/closed string duality based on (random)
Regge triangulations decorated with null twistorial fields. We explicitly show
that the twistorial N-points function, describing Dirichlet correlations over
the moduli space of open N-bordered genus g surfaces, is naturally mapped into
the Witten-Kontsevich intersection theory over the moduli space of N-pointed
closed Riemann surfaces of the same genus. We also discuss various aspects of
the geometrical setting which connects this model to PSL(2,C) Chern-Simons
theory.Comment: 35 pages, references added, slightly revised introductio
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
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
- …