Institute Theoretical and Experimental Physics natanzon@mccme.ru Theory of singularity constructs an algebra on the tangent space to a polynomial p(z) = zn+1 + a1zn-1 + a2zn-2 + · · · + an in the space P ol(n) of all polynomials of this type. This algebra Ap appears also in works of Vafa as (classical) topological Landau-Ginsburg model. The algebra Ap is associative, and has an unity. It is equipped with a linear functional lp : Ap C, such that the bilinear form (d1, d2) = lp(d1d2) is non degenerates. We call by Frobenius pairs all pairs (algebra, linear functional) with these algebraic properties. Algebra Ap is commutative for any classical topological Landau-Ginsburg model.

Commutative Frobenius pairs one-to-one correspond to topological field theories that appear from closed topological strings. These topological field theories naturally extend up to open-closed topological field theories, describing strings with boundary, and even up to Kleinian topological field theories, describing strings with arbitrary world sheets. In their turn, open-closed topological field theories one-to-one correspond to combinations from one commutative and one unrestricted Frobenius pairs, connected by Cardy condition [1]. We call such algebraic structure by Cardi-Frobenius algebra.

In the present paper we construct some extension of classical topological Landau-Ginsburg model to a Cardy-Frobenius algebra with a quaternion structure. Next we prove, that the set of such quaternion LandauGinsburg models over all polynomials p(z) = zn+1 + a1zn-1 + a2zn-2 + · · · + an form a non-commutative Frobenius manifold in means of [2].

Let us explain this result in more detail. The moduli space of the classical topological Landau-Ginsburg models coincides with the space P ol(n) of miniversale deformation for the singularity of type An. The metrics (d1, d2) = lp(d1d2) on the algebras Ap turn P ol(n) into a Riemannian manifold with some additional properties. The differential-geometric structure, arising here, is an important example of Frobenius manifolds in means of Dubrovin. The theory of Frobenius manifolds has a lot of applications in different areas of mathematics (integrable systems, singularity theory, topology of symplectic manifolds, geometry of moduli spaces of algebraic curves etc.).

The Dubrovin’s theory of Frobenius manifolds is a theory of flat deformations of commutative Frobenius pairs. As we discussed, Frobenius pairs are extended up to Cardy-Frobenius algebras. This suggests on extension of Frobenius manifolds to Cardy-Frobenius manifolds. An approach to this problem, is contained in [2]. It is based on Kontsevich–Manin cohomological field theory.

In this paper we define Cardy-Frobenius bundles as spaces of some flat deformations of Cardy–Frobenius algebras and prove, that they are Cardy-Frobenius (noncommutative) manifolds. Moreover we prove that the family of all the quaternion Landau–Ginsburg models form a CardyFrobenius bundle with quaternion structure.

References [1] Alekseevskii A., Natanzon S. Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves // Selecta Mathematica., New series. – 2006. – v. 12, n. 3. – P. 307– 377.

[2] Natanzon S. Extended cohomological field theories and noncommutative Frobenius manifolds // Geometry and Physics. – 2004. – / 4. – P. 387–403.

A criterion of the essential spectrum for elasticity and other self-adjoint systems on peak-shaped domains Nazarov S. A. (Russia) Institute of Mechanical Engineering Problems, Saint-Petersburg serna@snark.ipme.ru,srgnazarov@yahoo.co.uk Let Rn be a domain bounded by the compact surface which is Lipschitz everywhere except at the origin O of the Cartesian coordinate system x = (y, z) Rn-1 R. In a neighborhood U of the point O the domain is given by the relations z > 0, z-1-y, where Rn-is a domain with the Lipschitz surface and > 0 the peak sharpness exponent. We consider the spectral Neumann problem D(-x) A(x)D(x)u(x) = B(x)u(x), x, D((x)) A(x)D(x)u(x) = 0, x \ O.

Here stands for the outward normal, A and B are Hermitian matrices of sizes N N and k k, respectively, measurable, uniformly positive definite, and bounded for almost all x while N k. The N kmatrix D(x) consists of first-order homogeneous differential operators with constant coefficients and D is algebraically complete [1]. Then the sesquilinear form a(u, v) = (AD(x)u, D(x)v) possesses the polynomial property [2], i.e., the quadratic functional a(u, u) degenerates only on a finite dimensional space P of vector polynomials. The variational formulation of the Neumann spectral problem reads: To find C and u H \ {0} such that a(u, v) = b(u, v), v H, where b(u, v) = (Bu, v) and the function space H is obtained as the completion of Cc ( \ O) with respect to the norm generated by the inner product u, v = a(u, v)+b(u, v). Let K be a positive self-adjoint operator in H given by the formula Ku, v = b(u, v), u, v H.

Introducing the new spectral parameter µ = (1 + )-1, we find that the set C \ {µ C : Re µ [0, 1], Im µ = 0} belongs to the resolvent field of the operator K and µ = 1 is an eigenvalue of the multiplicity dimP <.

Theorem. 1) The spectrum of the operator K on the segment {µ C : Re µ (0, 1], Im µ = 0} is discrete for any > 0 if and only if any vector polynomial p in the linear space P does not depend on the variable z = xn.

2) If the linear space P contains the polynomial p(y, z) = zJp0(y) + · · · + zpJ-1(y) + pJ(y) with J 1 and p0 = 0, then the embedding H L2() is not compact for J-1 and, thus, the essential spectrum of the operator K contains a point different from µ = 0. In the case > J-1 the eigenvalue µ = 1 with the finite-dimensional eigenspace P belongs to the continuous spectrum of the operator K.

Theorem delivers only a sufficient condition of the existence of a point of the essential spectrum of the operator K on the segment {µ C :

Re µ (0, 1], Im µ = 0}. However, for a sharp ( 1) peak the condition becomes also necessary and, thus, Theorem gives a criterion of the discrete spectrum.

The change µ = µ-1 - 1 transmits all the properties of the spectrum of the operator K to the spectrum of the Neumann problem in the domain. In particular, for a scalar Neumann problem in a peakshaped domain, P = R and therefore the spectrum is always discrete.

The elasticity problem for a three-dimensional solid has the linear space of rigid motions a+bx as the space P so that it has non-empty essential spectrum for 1. In the case < 1 the spectrum is discrete. Much more information can be obtained for this particular problem that provoke to certain hypotheses for the problem in the general formulation.

References [1] Neas J. Les mthodes in thorie des quations elliptiques. ParisPrague: Masson-Academia, 1967.

[2] Nazarov S.A. The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes // Uspehi mat. nauk. – 1999. V. 54, N 5. P. 77–142. (English transl.: Russ. Math. Surveys. – 1999. V. 54, N 5. P. 947–1014).

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems Neishtadt A. I. (UK, Russia) Loughborough University Space Research Institute aneishta@iki.rssi.ru Simo C. (Spain) Universitat de Barcelona carles@maia.ub.es Treschev D. V. (Russia) Steklov Mathematical Institute;

Lomonosov Moscow State University treschev@mi.ras.ru Vasiliev A. A. (Russia) Space Research Institute valex@iki.rssi.ru Many problems in the theory of charged particles’ motion, the theory of propagation of short-wave excitations, and in celestial mechanics can be reduced to the analysis of 2 d.o.f. Hamiltonian systems with fast and slow variables. One degree of freedom corresponds to fast variables, and the other to slow variables. We assume that the ratio of time derivatives of slow and fast variables is of order 1. To describe the dynamics in such systems one can use the adiabatic approximation (see, e.g., [1]).

The system for the fast variables at frozen values of the slow variables is called the fast system. We shall consider the situation when on the fast system’s phase portrait there is a saddle point, and two separatrices passing through this point form an “8” figure. Assume that there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrices in the process of evolution of the slow variables. We shall assume that a certain symmetry condition holds: the areas inside the separatrix loops are equal.

We show that, under certain generality conditions, in the domain of separatrix crossings on every energy level there exist many, of order 1/, stable periodic trajectories of period 1/. Each one of them is surrounded by a stability island, and the measure of this island is estimated from below by a value of order. Thus, the total measure of the stability islands is estimated from below by a quantity that is independent of. A stability island is a domain on an energy level bounded by a twodimensional invariant torus. A stability island contains a discrete family of invariant tori contractible to the periodic trajectory. Let us introduce a “modified action” equal to the “action” for points inside the separatrix loops, and equal to one half of the “action” for the other points. This “modified action” is a perpetual adiabatic invariant of the motion inside a stability island: its value along a phase trajectory perpetually undergoes only oscillations with amplitude of order. Therefore, the stability islands are also islands of perpetual adiabatic invariance.

The existence of stability islands with total measure that is not small with in the domain of separatrix crossings is quite unexpected. Visually, in many problems, this domain looks like a region of dynamical chaos.

This result was established in [2] in the case of a Hamiltonian system with one degree of freedom such that the Hamiltonian function is slowly periodically depending on time. Here we generalize this result to 2 d.o.f.

systems. Like in [2], the proofs are based on the study of asymptotic formulas for the corresponding Poincar map. To find linearly stable periodic trajectories we look for linearly stable fixed points of the Poincar map.

The results on Lyapunov stability of the periodic trajectories (the fixed points) and on the existence of invariant tori surrounding the periodic trajectories (invariant curves around the fixed points) are provided by the Kolmogorov–Arnold–Moser theory.

References [1] Arnold V. I., Kozlov V. V., Neishtadt A. I. Mathematical aspects of classical and celestial mechanics, 3rd edition // Berlin: Springer, 2006.

[2] Neishtadt A. I., Sidorenko V. V., Treschev D. V. Stable periodic motions in the problem of passage through a separatrix // Chaos, 7, 2–11, 1997.

Fuzzy fractional monodromy Nekhoroshev Nikolai (Russia) Nikolai.Nekhoroshev@mat.unimi.it A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given.

Limit cycles appearing in polynomial perturbations of Darboux integrable systems Novikov D. (Israel) Weizmann Institute of Science dnovikov@wisdom.weizmann.ac.il We prove an existential finiteness result for integrals of rational oneforms over the level curves of Darbouxian integral.

Discrete Systems and Complex Analysis Novikov S. P. (Russia) novikov@landau.ac.ru University of Maryland, College Park Russian Academy of Sciences Our goal is to find realization of modern mathematical ideas (such as topology, symplectic geometry, algebra and algebraic geometry) in the problems of theoretical and mathematical physics. Three examples will be discussed in this talk: quantum scattering on the graphs with tails and symplectic geometry; completely integrable systems on a trivalent tree;

discretization of Complex Analysis on the equilateral triangle lattice and completely integrable systems. Some of our works in these areas are joint with I. Krichever and I. Dynnikov.

Isomonodromic deformations and special functions Novokshenov V. Yu. (Russia) Institute of Mathematics RAS, Ufa novokshenov@yahoo.com We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. In this sense, the special function provides isomonodromic deformation of some linear ODE with rational coefficients. This ODE plays a role of one of the two equations of the Lax pair. In its turn, this gives rise to an alternative definition: a matrix Riemann–Hilbert problem with a parameter, entering the conjugation matrix in a manner similar to the soliton theory. Thus the ODE for the special function appears to be integrable in the sense of Lioville–Arnold, i.e., it has the commuting integrals of motions, the invariant submanifolds and the corresponding angle variables.

We also show that our definition has not only a conceptual value: many well-known properties of the single-variable special functions can be rederived from the isomonodromy point of view. The examples of relevant Riemann–Hilbert (RH) problems are given for the Airy, Bessel, gamma and zeta functions. Those matrix RH problems are Abelian and exactly solvable, which provides the integral repesentations for these functions.

We show how to get the non-Abelian generalizations of the RH problems, leading to new examples of special functions, such as Painlev transcendents [1].

References [1] A. S. Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlev Transcendents. Riemann–Hilbert Approach, AMS Monographs, 2006, 560 p.

Givental integral representation for Classical groups Oblezin S. V. (Russia) Institute for Theoretical and Experimental Physics Sergey.Oblezin@itep.ru In 1996 Givental introduced a new representation for the gl+1 quantum Toda wave function (Whittaker function) in terms of stationary phase integral. The Givental formula admits a set of remarkable properties, particularly it has a very simple combinatorial description and admits a recursive structure with respect to the rank.

We propose integral representations for Whittaker functions for classical series sp2, so2, and so2+1 of Lie algebras. Constructed integral representations generalize Givental integral formula. Moreover, the proposed integral formulas also have simple combinatorial description, and they also admit the recursive structure. We show that the corresponding recursive operator is given by a degeneration of the Baxter Q-operator for gl+1-Toda chains. We generalize the construction of Q-operator to affine algebras sp2, so2, so2+1.

References [1] A. Givental, Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture, AMS Trans. (2) (1997), 103-115. Preprint 1996, [arXiv:alg-geom/9612001].

[2] A. Gerasimov, S. Kharchev, D. Lebedev, S. Oblezin, On a GaussGivental representation for quantum Toda chain wave function, Int.

Math. Res. Notices, 6 (2006), 23 pages.

Preprint 2005, [arXiv:math.RT/0505310].

[3] A. Gerasimov, D. Lebedev, S. Oblezin, Givental integral representation for classical groups, Preprint 2006, [math.RT/0608152].

Analyticity of formal normal forms of germs of generic dicritic foliations Ortiz-Bobadilla L. (Mexico) Instituto de Matemticas, Universidad Nacional Autnoma de Mxico laura@math.unam.mx Rosales-Gonzlez E. (Mexico) Instituto de Matemticas, Universidad Nacional Autnoma de Mxico ernesto@math.unam.mx Voronin S. M. (Russia) Departament of Mathematics, Chelyabinsk State University voron@csu.ru d We consider the class Vn of dicritic germs of holomorphic vector fields in (C2, 0) with vanishing n-jet at the origin, n 1 and their generated foliation. In [1] is proved that under some genericity assumptions, the formal orbital equivalence of two generic germs implies their analytic orbital d equivalence and orbitally formal normal forms of germs in Vn are given. In this work we give analytic normal forms of these generic germs of dicritic foliations in a neighborhood of the origin.

References [1] L. Ortiz-Bobadilla, E. Rosales-Gonzalez, S. M. Voronin. Rigidity theorem for degenerated singular points of germs of dicritic holomorphic vector fields in the complex plane. Mosc. Math. J. 5 (2005), no. 1, 171–206.

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