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D. Korotkin

Publications and source records attributed to D. Korotkin.

At least 37 records · Page 2Linked to original sources
arXiv2003

Tau-function on Hurwitz spaces

We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ``Wirtinger tau-function''). In higher genus we compute the modulus square of the Wirtinger tau-function.

math-ph↗
arXiv2001

Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$

In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe divisor of zeros of the tau-function (so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to determinant of Laplacian operator on the Riemann surface.

math-ph↗
arXiv2001

Isomonodromic deformations and Hurwitz spaces

A class of Riemann-Hilbert problems corresponding to quasi-permutation monodromy matrices is solved in terms of Szegö kernel on auxiliary Riemann surfaces. The tau-function of Schlesinger system turns out to be closely related to determinant of Cauchy-Riemann operator. The link between theta-divisor and Malgrange's divisor in the theory of Schlesinger equations is established.

math-ph↗
arXiv1999

Schlesinger system, Einstein equations and hyperelliptic curves

We review recent developments in the method of algebro-geometric integration of integrable systems related to deformations of algebraic curves. In particular, we discuss the theta-functional solutions of Schlesinger system, Ernst equation and self-dual SU(2)-invariant Einstein equations.

gr-qc↗
arXiv1999

Introduction to the functions on compact Riemann surfaces and theta-functions

We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the treatment is organized to give a background for further applications to non-linear differential equations.

solv-int↗
arXiv1999

Schlesinger transformations for elliptic isomonodromic deformations

Schlesinger transformations are discrete monodromy preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.

solv-int↗
arXiv1997

Yangian Symmetry in Integrable Quantum Gravity

Dimensional reduction of various gravity and supergravity models leads to effectively two-dimensional field theories described by gravity coupled G/H coset space sigma-models. The transition matrices of the associated linear system provide a complete set of conserved charges. Their Poisson algebra is a semi-classical Yangian double modified by a twist which is a remnant of the underlying coset structure. The classical Geroch group is generated by the Lie-Poisson action of these charges. Canonical quantization of the structure leads to a twisted Yangian double with fixed central extension at a critical level.

hep-th↗
arXiv1997

Canonical quantization of cylindrical gravitational waves with two polarizations

The canonical quantization of the essentially nonlinear midisuperspace model describing cylindrically symmetric gravitational waves with two polarizations is presented. A Fock space type representation is constructed. It is based on a complete set of quantum observables. Physical expectation values may be calculated in arbitrary excitations of the vacuum. Our approach provides a non-linear generalization of the quantization of the collinearly polarized Einstein-Rosen gravitational waves.

gr-qc↗
arXiv1997

Quantization of coset space sigma-models coupled to two-dimensional gravity

The mathematical framework for an exact quantization of the two-dimensional coset space sigma-models coupled to dilaton gravity, that arise from dimensional reduction of gravity and supergravity theories, is presented. Extending previous results the two-time Hamiltonian formulation is obtained, which describes the complete phase space of the model in the isomonodromic sector. The Dirac brackets arising from the coset constraints are calculated. Their quantization allows to relate exact solutions of the corresponding Wheeler-DeWitt equations to solutions of a modified (Coset) Knizhnik-Zamolodchikov system. On the classical level, a set of observables is identified, that is complete for essential sectors of the theory. Quantum counterparts of these observables and their algebraic structure are investigated. Their status in alternative quantization procedures is discussed, employing the link with Hamiltonian Chern-Simons theory.

hep-th↗
arXiv1996

Integrable Classical and Quantum Gravity

In these lectures we report recent work on the exact quantization of dimensionally reduced gravity, i.e. 2d non-linear (G/H)-coset space sigma-models coupled to gravity and a dilaton. Using methods developed in the context of flat space integrable systems, the Wheeler-DeWitt equations for these models can be reduced to a modified version of the Knizhnik-Zamolodchikov equations from conformal field theory, the insertions given by singularities in the spectral parameter plane. This basic result in principle permits the explicit construction of solutions, i.e. physical states of the quantized theory. In this way, we arrive at integrable models of quantum gravity with infinitely many self-interacting propagating degrees of freedom.

hep-th↗
arXiv1996

Poisson Realization and Quantization of the Geroch Group

The conserved nonlocal charges generating the Geroch group with respect to the canonical Poisson structure of the Ernst equation are found. They are shown to build a quadratic Poisson algebra, which suggests to identify the quantum Geroch algebra with Yangian structures.

gr-qc↗
arXiv1996

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 $SL(2,R)$. 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 à la Dirac exists both at the classical and the quantum level. Using the discrete unitary representations of $SL(2,R)$, we construct explicit quantum states. However, satisfying the additional constraints associated with the coset space $SL(2,R)/SO(2)$ 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.

hep-th↗
arXiv1996

On $2D$ 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.

gr-qc↗
arXiv1995

An Integrable Model of Quantum Gravity

We present a new quantization scheme for $2D$ gravity coupled to an $SU(2)$ principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a ``clock field''. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the Knizhnik-Zamolodchiokov equations.

hep-th↗
arXiv1994

"Separation of Variables and Hamiltonian Formulation for the Ernst Equation"

It is shown that the vacuum Einstein equations for an arbitrary stationary axisymmetric space-time can be completely separated by re-formulating the Ernst equation and its associated linear system in terms of a non-autonomous Schlesinger-type dynamical system. The conformal factor of the metric coincides (up to some explicitly computable factor) with the $τ$-function of the Ernst equation in the presence of finitely many regular singularities. We also present a canonical formulation of these results, which is based on a ``two-time" Hamiltonian approach, and which opens new avenues for the quantization of such systems.

hep-th↗
arXiv1994

The Ernst Equation on a Riemann Surface

The Ernst equation is formulated on an arbitrary Riemann surface. Analytically, the problem reduces to finding solutions of the ordinary Ernst equation which are periodic along the symmetry axis. The family of (punctured) Riemann surfaces admitting a non-trivial Ernst field constitutes a ``partially discretized'' subspace of the usual moduli space. The method allows us to construct new exact solutions of Einstein's equations in vacuo with non-trivial topology, such that different ``universes'', each of which may have several black holes on its symmetry axis, are connected through necks bounded by cosmic strings. We show how the extra topological degrees of freedom may lead to an extension of the Geroch group and discuss possible applications to string theory.

gr-qc↗