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## Publications

Publications (140)

In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the characterization of Jacobians as ppav whose Kummer variety admits a trisecant line and the Pryms as ppav whose Kummer...

We give two characterizations of Jacobians of curves with involution having fixed points in the framework of two particular cases of Welter's trisecant conjecture. The geometric form of each of these characterizations is the statement that such Jacobians are exactly those containing a shifted Abelian subvariety whose image under the Kummer map is o...

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries of two dimensional O(N) sigma model. In the first part of the paper we use these symmetries to prove that the Fermi spectral curve for the double-periodic sigma model is algebraic. Thus, our previous construction of the complexified harmonic maps in the case o...

We consider the space of solutions of the Bethe ansatz equations of the $\widehat{\frak{sl}_N}$ XXX quantum integrable model, associated with the trivial representation of $\widehat{\frak{sl}_N}$. We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For tha...

We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN different...

New reductions of the 2D Toda equations associated with low-triangular difference operators are proposed. Their explicit Hamiltonian description is obtained.

We study the spectral theory of $n$-periodic strictly triangular difference
operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the
"superperiodic" operators for which all solutions of the equation $(L+1)\psi=0$
are (anti)periodic. We show that for a superperiodic operator $L$ there exists
a unique superperiodic operator ${\...

A generalization of the amoeba and the Ronkin function of a plane algebraic
curve for a pair of harmonic functions on an algebraic curve with punctures is
proposed. Extremal properties of $M$-curves are proved and connected with the
spectral theory of difference operators with positive coefficients.

Using constructions of the Whitham perturbation theory of integrable system
we prove a new sharp upper bound of $3g/2-2$ on the dimension of complete
subvarieties of $\M_g^{ct}$.

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M
g
of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.

Novikov's conjecture on the Riemann-Schottky problem: {\it the Jacobians of
smooth algebraic curves are precisely those indecomposable principally
polarized abelian varieties (ppavs) whose theta-functions provide solutions to
the Kadomtsev-Petviashvili (KP) equation}, was the first evidence of nowadays
well-established fact: connections between the...

Using meromorphic differentials with real periods, we show that a certain
tautological homology class on the moduli space of smooth algebraic curves of
genus g vanishes. The vanishing of the entire tautological ring for degree g-1
and higher, part of Faber's conjecture, is known in both homology and Chow ---
it was proven by Looijenga, Ionel, and G...

We consider the problem of extending the integrals of motion of soliton equations to the space of all finite-gap solutions.
We consider the critical points of these integrals on the moduli space of Riemann surfaces with marked points and jets of
local coordinates. We show that the solutions of the corresponding variational problem have an explicit...

We introduce the notion of abelian solutions of the 2D Toda lattice equations and the bilinear discrete Hirota equation, and show that all of them are algebro-geometric.
Mathematics Subject Classification (2000)Primary 37K10–Secondary 14H70

The present article is an exposition of the author's talk at the conference dedicated to the 70th birthday of S.P. Novikov. The talk contained the proof of Welters' conjecture which proposes a solution of the classical Riemann-Schottky problem of characterizing the Jacobians of smooth algebraic curves in terms of the existence of a trisecant of the...

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the...

We introduce the notion of abelian solutions of KP equations and show that all of them are algebro-geometric.

The Laplacian growth (the Hele-Shaw problem) of multiply-connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this...

Higher-rank solutions of the equations of the two-dimensionalized Toda lattice are constructed. The construction of these solutions is based on the theory of commuting difference operators, which is developed in the first part of the paper. It is shown that the problem of recovering the coefficients of commuting operators can be effectively solved...

The classical Weierstrass theorem claims that, among the analytic functions, the only functions admitting an algebraic addition theorem are the elliptic functions and their degenerations. This survey is devoted to far-reaching generalizations of this result that are motivated by the theory of integrable systems. The authors discovered a strong form...

In this paper, we develop the general approach, introduced in [l], to Lax operators on algebraic curves. We observe that the
space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct orthogonal
and symplectic analogs of Lax operators, prove that they form almost graded Lie algebras, and con...

The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems.

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hie...

We prove Welter's trisecant conjecture: an indecomposable principally polarized abelian variety $X$ is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety $K(X)$.

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hi...

We prove that Prym varieties of algebraic curves with two smooth fixed points of involution are exactly the indecomposable
principally polarized abelian varieties whose theta-functions provide explicit formulae for integrable 2D Schrödinger equation.

We prove that an indecomposable principally polarized abelian variety X is the Jacobain of a curve if and only if there exist vectors U ≠ 0, V such that the roots x
i(y) of the theta-functional equation θ(Ux + Vy + Z) = 0 satisfy the equations of motion of the formal infinite-dimensional Calogero-Moser system.

Given an abelian variety $X$ and a point $a\in X$ we denote by $ $ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $\kappa: X\to \kappa(X)\subset\mathbb P^N$ the map from $X$ to its Kummer variety. We prove that an indecomposable abelian variety $X$ is the Jacobian of a curve if and only if there exists a point $a=2...

A new approach to the analytic theory of difference equations with rational
and elliptic coefficients is proposed. It is based on the construction of
canonical meromorphic solutions which are analytical along "thick paths". The
concept of such solutions leads to a notion of local monodromies of difference
equations. It is shown that in the continuo...

The discrete Lax operators with the spectral parameter on an algebraic curve are defined. A hierarchy of commuting flows on the space of such operators is constructed. It is shown that these flows are linearized by the spectral transform and can be explicitly solved in terms of the theta-functions of the spectral curves. The Hamiltonian theory of t...

High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics of Tyurin Parameters characterizing the stable holomorphic vector bundles over the algebraic curves (Riemann...

This is an expanded version of lectures given in Hangzhou and Beijing, on the symplectic forms common to Seiberg-Witten theory and the theory of solitons. Methods for evaluating the prepotential are discussed. The construction of new integrable models arising from supersymmetric gauge theories are reviewed, including twisted Calogero-Moser systems...

The Hamiltonian theory of zero-curvature equations with spectral parameter on an arbitrary compact Riemann surface is constructed.
It is shown that the equations can be seen as commuting flows of an infinite-dimensional field generalization of the Hitchin
system. The field analog of the elliptic Calogero-Moser system is proposed. An explicit parame...

We present a Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation. Namely, we consider elliptic families of solutions of the KP equation such that their poles satisfy a constraint of being balanced. We show that the dynamics of these poles is described by a redu...

The Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves is constructed. An explicit formula for the symplectic structure on the space of monodromy and Stokes matrices is obtained. The Whitham equations for the isomonodromy equations are derived. It is shown that they provide a fl...

The integrable model corresponding to the $\mathcal{N}=2$ supersymmetric SU$(N)$ gauge theory with matter in the symmetric representation is constructed. It is a spin chain model, whose key feature is a new twisted monodromy condition.
AMS 2000 Mathematics subject classification: Primary 53D30; 53D20. Secondary 37J35

Integrable systems which do not have an “obvious“ group symmetry, beginning with the results of Poincaré and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were ba...

We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral problem for a finite Jacoby matrix. For the Toda lattice equations we obtain the explicit form of the equations...

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We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $\tau$-function also emerges i...

We review basic ideas and basic examples of the theory of the inverse spectral problems.

Commutative rings of one-dimensional difference operators of rank l>1 and their deformations are effectively constructed. Our analytical constructions are based on the so-called ''Tyurin parameters'' for the stable framed holomorphic vector bundles over algebraic curves of the genus equal to g and Chern number equal to lg. These parameters were hea...

We construct the integrable model corresponding to the $\N=2$ supersymmetric SU(N) gauge theory with matter in the antisymmetric representation, using the spectral curve found by Landsteiner and Lopez through M Theory. The model turns out to be the Hamiltonian reduction of a $N+2$ periodic spin chain model, which is Hamiltonian with respect to the...

An updated and detailed survey of basic ideas of the finite-gap theory is presented. That theory, developed to construct periodic and quasi-periodic solutions of the soliton equations, combines the Bloch-Floquet spectral theory of linear periodic operators, the theory of completely integrable Hamiltonian systems, the classical theory of Riemann sur...

It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear systems representible through the ''L-A-B-triple''. The Laplace transformations for these operators are also constr...

The action-angle variables for N-particle Hamiltonian system with the Hamiltonian $H=\sum_{n=0}^{N-1} \ln sh^{-2}(p_n/2)+\ln(\wp(x_n-x_{n+1})- \wp(x_n+x_{n+1})), x_N=x_0,$ are constructed, and the system is solved in terms of the Riemann $\theta$-functions. It is shown that this system describes pole dynamics of the elliptic solutions of 2D Toda la...

Discrete analogs of the Darboux-Egoroff metrics are considered. It is shown that the corresponding lattices in the Euclidean space are described by discrete analogs of the Lame equations. It is proved that up to a gauge transformation these equations are necessary and sufficient for discrete analogs of rotation coefficients to exist. Explicit examp...

An exact formula for the solutions to the WDVV equation in terms of horizontal sections of the corresponding flat connection is found.

We provide a unified construction of the symplectic forms which arise in the solution of both N=2 supersymmetric Yang-Mills theories and soliton equations. Their phase spaces are Jacobian-type bundles over the leaves of a foliation in a universal configuration space. On one hand, imbedded into finite-gap solutions of soliton equations, these symple...

We outline an approach to a theory of various generalizations of the elliptic Calogero-Moser (CM) and Ruijsenaars-Shneider (RS) systems based on a special inverse problem for linear operators with elliptic coefficients. Hamiltonian theory of such systems is developed with the help of the universal symplectic structure proposed by D.H. Phong and the...

An algebro-geometric approach to representations of Sklyanin algebra is proposed. To each 2 \times 2 quantum L-operator an algebraic curve parametrizing its possible vacuum states is associated. This curve is called the vacuum curve of the L-operator. An explicit description of the vacuum curve for quantum L-operators of the integrable spin chain o...

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $\omega={1\over 2}\r_{\infty} <\Psi_0^*\delta L\wedge\delta\Psi_0>\d k$. We...

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for τ-functions. Starting from a given algebraic curve, we express the τ-function and the Baker–Akhiezer function in terms of the Riemann theta function. We show that the elliptic soluti...

We study the asymptotic solutions of the Schrödinger equation for the color-singlet reggeon compound states in multi-color QCD. We show that in the leading order of asymptotic expansion, quasiclassical reggeon trajectories have the form of soliton waves propagating on the 2-dimensional plane of transverse coordinates. Applying the methods of the fi...

We determine the effective prepotential for N = 2 supersymmetric SU(Nc) gauge theories with an arbitrary number of flavors Nf < 2Nc from the exact solution constructed out of spectral curves. The prepotential is the same for the several models of spectral curves proposed in the literature. It has to all orders the logarithmic singularities of the o...

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely...

In spite of the diversity of solvable models of quantum field theory and the vast variety of methods, the final results display dramatic unification: the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies
$$ E = \sum\limits_i {\varepsilon \left( {{u_i}} \right)} $$ (1.1) where u
i
obey a system of alg...

Algebraic-geometrical n-orthogonal curvilinear coordinate systems in a flat space are constructed. They are expressed in terms of the Riemann theta function of auxiliary algebraic curves. The exact formulae for the potentials of algebraic geometrical Egoroff metrics and the partition functions of the corresponding topological field theories are obt...

Action-angle type variables for spin generalizations of the elliptic RuijsenaarsSchneider system are constructed. The equations of motion of these systems are solved in terms of Riemann theta-functions. It is proved that these systems are isomorphic to special elliptic solutions of the non-abelian 2D Toda chain. A connection between the finite gap...

We clarify the mass dependence of the effective prepotential in N = 2 supersymmetric SU(Nc) gauge theories with an arbitrary number Nf < 2Nc of flavors. The resulting differential equation for the prepotential extends the equations obtained previously for SU(2 and for zero masses.It can be viewed as an exact renormalization group equation for the p...

We calculate the effective prepotentials for N=2 supersymmetric SO(N_c) and Sp(N_c) gauge theories, with an arbitrary number of hypermultiplets in the defining representation, from restrictions of the prepotentials for suitable N=2 supersymmetric gauge theories with unitary gauge groups. (This extends previous work in which the prepotential for N=2...

A new general type of reductions of the hierarchy of Kadomtsev-Petviashvili equation is established. These reductions are equivalent to the Lax equations for pseudo-differential operators of the form L1−1L2, where L1, L2 are ordinary differential operators with coefficients that are functions of variables t1 = x, t2, …. It is shown that besides the...

The exact Seiberg-Witten (SW) description of the light sector in the N = 2 SUSY 4d Yang-Mills theory [N. Seiberg and E. Witten, Nucl. Phys. B 430 (1994) 485 (E); B 446 (1994) 19] is reformulated in terms of integrable systems and appears to be a Gurevich-Pitaevsky (GP) [A. Gurevich and L. Pitaevsky, JETP 65 (1973) 65; see also, S. Novikov, S. Manak...

The year of 1995 is not merely the centenary of the Korteweg—de Vries equation which we celebrate at this conference. It is also the year of the ‘majority’ of the finite-gap or algebraic—geometrical theory of integration of nonlinear equations — one of the most important components of the branch of modern mathematical physics, which is called the t...

The complete solutions of the spin generalization of the elliptic Calogero Moser systems are constructed. They are expressed in terms of Riemann theta-functions. The analoguous constructions for the trigonometric and rational cases are also presented.

The universal Whitham hierarchy is considered from the viewpoint of topological field theories. The τ-function is defined for this hierarchy. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form...

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of anN-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rankN for a function of ag-dimensional argument that generalize the classical functional Cauchy equation. It is shown tha...

The universal Witham hierarchy is considered from the point of view of topological field theories. The $\tau$-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity.

It is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless Lax equations. The exact formulae for the tree level partition functions ofA
n topological minimal models are found. The Virasoro constraints for the analogue of the -function of the dispersio...

During the last two years remarkable connections between the non-perturbative theory of two-dimensional gravity coupled with various matter fields, the theory of topological gravity coupled with topological matter fields, the theory of matrix models and, finally, the theory of integrable soliton equations with special Virasoro constraints have been...

The theorem is proved. THEOREM 2. Suppose P, q~2 are relatively prime integers and the sequence of natural numbers kv satisies the condition ~+i~2~ (v = 0,1,2 ....). Then the numbers a~ = ~=o P-~wq-~P~'V (k = i, 2, 3,...) are algebraically independent, normal in base q, and the continued fraction for a k can be given explicitly. Proof. The normalit...

Integrable systems which do not have an “obvious” group symmetry, beginning with the results of Poincaré und Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were ba...

CONTENTS Introduction Chapter I. The spectral theory of the non-stationary Schrödinger operator § 1. The perturbation theory for formal Bloch solutions § 2. The structure of the Riemann surface of Bloch functions § 3. The approximation theorem § 4. The spectral theory of finite-gap non-stationary Schrödinger operators § 5. The completeness theorem...

In this paper we construct the operator fields of the Riemann surfaces of arbitrary genus. The corresponding operator theory of interacting strings can be considered as the direct development of Virasoro-Mandelstam theory for g >= 0 and its unifacation with Polyakov-Belavin-Knizhnik theory.

dk with two simple poles at the points P±, with residues ±1 and purely imaginary periods with respect to all contours on . The real part of the corresponding integral (z) = Rek(z) is single-valued on and represents “time.” The level lines (z) = const represent the positions of the string at the present time. To the collection m = m+ + m of strings...

A general method of constructing the Wess-Zumino type lagrangians is proposed. The corresponding constants are shown to be quantized. Some examples in 1d, 3d and 4d are considered.

The main ideas of global “finite-zone integration” are presented, and a detailed analysis is given of applications of the technique developed to some problems based on the theory of elliptic functions. In the work the Peierls model is integrated as an important application of the algebrogeometric spectral theory of difference operators.

One constructs all the decreasing rational solutions of the Kadomtsev-Petviashvili equations. The presented method allows us to identify the motion of the poles of the obtained functions with the motion of a system of N particles on a line with a Hamiltonian of the Calogero-Moser type. Thus, this Hamiltonian system is imbedded in the theory of the...