arXiv subjects

M. Olshanetsky

Publications and source records attributed to M. Olshanetsky.

44 records · Page 3Linked to original sources

arXiv1999

Non-autonomous Hamiltonian systems related to highest Hitchin integrals

We describe non-autonomous Hamiltonian systems coming from the Hitchin integrable systems. The Hitchin integrals of motion depend on the W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals dependent on the complex structure (W_2-structure) of the basic curve and times are coordinate on the Teichmuller space. The corresponding flows are the monodromy preserving equations such as the Schlesinger equations, the Painleve VI equation and their generalizations. The equations corresponding to the highest integrals are monodromy preserving conditions with respect to changing of the W_k-structures (k>2). They are derived by the symplectic reduction from the gauge field theory on the basic curve interacting with W_k-gravity. As by product we obtain the classical Ward identities in this theory.

math-ph↗

arXiv1999

Painlevé type equations and Hitchin systems

In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles in the marked points. We discuss relations of these equations with the Whitham quantization of the Hitchin systems and with the classical limit of the Knizhnik-Zamolodchikov-Bernard equations. The main example is the one-parameter family of Painlevé VI equation and its multicomponent generalization.

math-ph↗

arXiv1998

The $q$-Fourier transform of $q$-distributions

We consider functions on the lattice generated by the integer powers of $q^2$ for $0<q<1$ and construct the $q$-analog of Fourier transform based on the Jackson integral in the space of distributions on the lattice.

q-alg↗

arXiv1997

Painlevé - Calogero correpondence

It is proved that the Painlevé VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{ν^2}{4},\be=-\frac{ν^2}{4}, \ga=\frac{ν^2}{4},\de=\f1{2}-\frac{ν^2}{4})$ is a reduced hamiltonian system. Its phase space is the set of flat $SL(2,C)$ connections over elliptic curves with a marked point and time of the system is given by the elliptic module. This equation can be derived via reduction procedure from the free infinite hamiltonian system. The phase space of later is the affine space of smooth connections and the "times are the Beltrami differentials. This approach allows to define the associate linear problem, whose isomonodromic deformations is provided by the Painlevé equation and the Lax pair. In addition, it leads to description of solutions by a linear procedure. This scheme can be generalized to $G$ bundles over Riemann curves with marked points, where $G$ is a simple complex Lie group. In some special limit such hamiltonian systems convert into the Hitchin systems. In particular, for $\SL$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N-body system. Relations to the classical limit of the Knizhnik- Zamolodchikov-Bernard equations is discussed.

alg-geom↗

arXiv1996

Double coset construction of moduli space of holomorphic bundles and Hitchin systems

We present a description of the moduli space of holomorphic vector bundles over Riemann curves as a double coset space which is differ from the standard loop group construction. Our approach is based on equivalent definitions of holomorphic bundles, based on the transition maps or on the first order differential operators. Using this approach we present two independent derivations of the Hitchin integrable systems. We define a "superfree" upstairs systems from which Hitchin systems are obtained by three step hamiltonian reductions. A special attention is being given on the Schottky parameterization of curves.

alg-geom↗

arXiv1996

Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras

In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple $Γ$-function factors over all positive roots of the corresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$). This is in nice correspondence with the recent results on 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("conformal") Toda type models are given by simple averages in the $d+0$ dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free-field wave functional, which is originally a Gaussian integral over interior of a $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-dimensional boundary itself. In the present paper we mostly concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions were known, and we present their survey. We also construct new "Gauss" Whittaker functions.

hep-th↗

arXiv1995

Generalized Hitchin systems and Knizhnik-Zamolodchikov-Bernard equation on elliptic curves

Knizhnik-Zamolodchikov-Bernard (KZB) equation on an elliptic curve with a marked point is derived by the classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on cotangent bundle to the loop group $L(GL(N,{\bf C}))$ extended by the shift operators, to be related to the elliptic module. After the reduction we obtain the Hamiltonian system on cotangent bundle to the moduli of holomorphic principle bundles and the elliptic module. It is a particular example of generalized Hitchin systems (GHS), which are defined as hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present the explicite form of higher quantum Hitchin integrals, which upon on reducing from GHS phase space to the Hitchin phase space gives a particular example of the Belinson-Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.

hep-th↗

arXiv1992

$c = r_g$ Theories of $W_G$-Gravity: The Set of Observables as a Model of Simply Laced $G$

We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the algebra of observables in the $c = 1$ string model to theories with $c > 1$. We emphasize the algebraic meaning of the KPW construction for $c = 1$ related to occurrence of a {\it model} of {\it SU}(2) as original structure on the algebra of observables. The attempts to preserve this structure in generalizations naturally leads to consideration of $W$-gravities. As a first step in the study of these generalized KPW constructions we design explicitly the subsector of the space of observables in appropriate $W_G$-string theory, which forms the {\it model} of $G$ for any simply laced {\it G}. The {\it model} structure is confirmed by the fact that corresponding one-loop Kac-Rocha-Caridi $W_G$-characters for $c = r_G$ sum into a chiral (open string) $k=1$ $G$-WZW partition function.

hep-th↗