350 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
Isomonodromic Quantization of Dimensionally Reduced Gravity
We present a detailed account of the isomonodromic quantization of
dimensionally reduced Einstein gravity with two commuting Killing vectors. This
theory constitutes an integrable ``midi-superspace" version of quantum gravity
with infinitely many interacting physical degrees of freedom. The canonical
treatment is based on the complete separation of variables in the isomonodromic
sectors of the model. The Wheeler-DeWitt and diffeomorphism constraints are
thereby reduced to the Knizhnik-Zamolodchikov equations for . The
physical states are shown to live in a well defined Hilbert space and are
manifestly invariant under the full diffeomorphism group. An infinite set of
independent observables \`a la Dirac exists both at the classical and the
quantum level. Using the discrete unitary representations of , we
construct explicit quantum states. However, satisfying the additional
constraints associated with the coset space requires solutions
based on the principal series representations, which are not yet known. We
briefly discuss the possible implications of our results for string theory.Comment: 36 pages, LATE
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
Tau function and moduli of differentials
The tau function on the moduli space of generic holomorphic 1-differentials
on complex algebraic curves is interpreted as a section of a line bundle on the
projectivized Hodge bundle over the moduli space of stable curves. The
asymptotics of the tau function near the boundary of the moduli space of
1-differentials is computed, and an explicit expression for the pullback of the
Hodge class on the projectivized Hodge bundle in terms of the tautological
class and the classes of boundary divisors is derived. This expression is used
to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum
of the Lyapunov exponents associated with the Teichm\"uller flow on the Hodge
bundle.Comment: misprints corrected; journal reference adde
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