192 research outputs found
On quantum gravity coupled to a \s-model
This contribution is a review of the method of isomonodromic quantization of
dimensionally reduced gravity. Our approach is based on the complete separation
of variables in the isomonodromic sector of the model and the related
``two-time" Hamiltonian structure. This allows an exact quantization in the
spirit of the scheme developed in the framework of integrable systems. Possible
ways to identify a quantum state corresponding to the Kerr black hole are
discussed. In addition, we briefly describe the relation of this model with
Chern Simons theory.Comment: 9 pages, LaTeX style espcrc2, to appear in Proceedings of 29th
International Symposium Ahrenshoop, Buckow, 199
Exact relativistic treatment of stationary counter-rotating dust disks III. Physical Properties
This is the third in a series of papers on the construction of explicit
solutions to the stationary axisymmetric Einstein equations which can be
interpreted as counter-rotating disks of dust. We discuss the physical
properties of a class of solutions to the Einstein equations for disks with
constant angular velocity and constant relative density which was constructed
in the first part. The metric for these spacetimes is given in terms of theta
functions on a Riemann surface of genus 2. It is parameterized by two physical
parameters, the central redshift and the relative density of the two
counter-rotating streams in the disk. We discuss the dependence of the metric
on these parameters using a combination of analytical and numerical methods.
Interesting limiting cases are the Maclaurin disk in the Newtonian limit, the
static limit which gives a solution of the Morgan and Morgan class and the
limit of a disk without counter-rotation. We study the mass and the angular
momentum of the spacetime. At the disk we discuss the energy-momentum tensor,
i.e. the angular velocities of the dust streams and the energy density of the
disk. The solutions have ergospheres in strongly relativistic situations. The
ultrarelativistic limit of the solution in which the central redshift diverges
is discussed in detail: In the case of two counter-rotating dust components in
the disk, the solutions describe a disk with diverging central density but
finite mass. In the case of a disk made up of one component, the exterior of
the disks can be interpreted as the extreme Kerr solution.Comment: 30 pages, 20 figures; to appear in Phys. Rev.
Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications
In this work we find the isomonodromic (Jimbo-Miwa) tau-function
corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss
several applications of this result. First, we get an explicit expression for
the G-function (solution of Getzler's equation) of the Hurwitz Frobenius
manifolds. Second, in terms of this tau-function we compute the genus one
correction to the free energy of hermitian two-matrix model. Third, we find the
Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with
quasi-permutation monodromy matrices. Finally, we get a new expression (analog
of genus one Ray-Singer formula) for the determinant of Laplace operator in the
Poincar\'e metric on Riemann surfaces of an arbitrary genus.Comment: The direct proof of variational formulas on branched coverings is
added. The title is modified due to observed coincidence of isomonodromic
tau-function of Hurwitz Frobenius manifolds with Bergman tau-function on
Hurwitz spaces introduced by the author
A Periodic Analog of the Schwarzschild Solution
We construct a new exact solution of Einstein's equations in vacuo in terms
of Weyl canonical coordinates. This solution may be interpreted as a black hole
in a space-time which is periodic in one direction and which behaves
asymptotically like the Kasner solution with Kasner index equal to ,
where is the period and is the mass of the black hole. Outside the
horizon, the solution is free of singularities and approaches the Schwarzschild
solution as .Comment: 6 pages, preprint DESY-TH 94-03
Tau function and the Prym class
We use the formalism of the Bergman tau functions to study the geometry of
moduli spaces of holomorphic quadratic differentials on complex algebraic
curves. We introduce two natural tau functions and interpret them as
holomorphic sections of certain line bundles on the moduli space. Analyzing the
asymptotic behavior of these tau functions near the boundary of the moduli
space we get two non-trivial relation in the rational Picard group of the
moduli space of quadratic differential.Comment: 20 page
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