134 research outputs found
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
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.
On solutions of the Schlesinger Equations in Terms of -Functions
In this paper we construct explicit solutions and calculate the corresponding
-function to the system of Schlesinger equations describing isomonodromy
deformations of matrix linear ordinary differential equation whose
coefficients are rational functions with poles of the first order; in
particular, in the case when the coefficients have four poles of the first
order and the corresponding Schlesinger system reduces to the sixth Painlev\'e
equation with the parameters , our construction leads to a
new representation of the general solution to this Painlev\'e equation obtained
earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions
Isomonodromic tau function on the space of admissible covers
The isomonodromic tau function of the Fuchsian differential equations
associated to Frobenius structures on Hurwitz spaces can be viewed as a section
of a line bundle on the space of admissible covers. We study the asymptotic
behavior of the tau function near the boundary of this space and compute its
divisor. This yields an explicit formula for the pullback of the Hodge class to
the space of admissible covers in terms of the classes of compactification
divisors.Comment: a few misprints corrected, journal reference adde
Self-dual SU(2) invariant Einstein metrics and modular dependence of theta-functions
We simplify Hitchin's description of SU(2)-invariant self-dual Einstein
metrics, making use of the tau-function of related four-pole Schlesinger
system.Comment: A wrong sign in the formula for W_1 is corrected; we thank Owen
Dearricott who pointed out this mistake in the original version of the pape
On the quantization of isomonodromic deformations on the torus
The quantization of isomonodromic deformation of a meromorphic connection on
the torus is shown to lead directly to the Knizhnik-Zamolodchikov-Bernard
equations in the same way as the problem on the sphere leads to the system of
Knizhnik-Zamolodchikov equations. The Poisson bracket required for a
Hamiltonian formulation of isomonodromic deformations is naturally induced by
the Poisson structure of Chern-Simons theory in a holomorphic gauge fixing.
This turns out to be the origin of the appearance of twisted quantities on the
torus.Comment: 13 pages, LaTex2
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