| Algebraic integrable systems: special Kähler geometry and Poisson structures | ||||||||||||||||||
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Scientific Coordinator of Research Unit Claudio Bartocci Location of the Research Unit University of Genova Scientific Sectors interested in the Research A03X, A01C Key Words
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Scientific Background |
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The notion of Poisson manifold, which generalizes that of symplectic manifold, naturally arises within the framework of analytical mechanics. Poisson geometry appears to be crucial in the study of integrable systems, both finite and infinite dimensional, and is the starting point in order to generalize the concept of integrabilty to quantum systems.
The Genova unit, mainly in collaboration with the University of Milano Bicocca, SISSA (Trieste) and the University of Salamanca, has carried out in the last years a research activity focused on several problems closely related to Poisson geometry. Bihamiltonian manifolds (which are differentiable manifolds endowed with a pencil of compatible Poisson structures [Ma]) provide a suitable framework to study completely integrable systems. The problem of separability for the Hamilton-Jacobi equation is linked to the problem of integrability, and can be tackled by means of different strategies [AHH, Sk, GNR]. The connection between bihamiltonian structures and the separation of variables has been recently explained in a general geometric setting, which allows us to interpret in a modern vein classical Levi-Civita's, StŠckel's and Eisenhart's results [FMT, FP, IMM]. Under suitable hypotheses (which are satisfied in the great majority of significant examples) a bihamiltonian structure induces hamiltonian integrable systems on the symplectic leaves of the foliation given by one Poisson bivector of the pencil. More precisely, each leaf admits a structure of "omega-N manifold", i.e. the pencil of Poisson tensors is generated by means of a recursion operator N from a nondegenerate Poisson tensor (i.e. corresponding to a symplectic structure S). On omega-N manifolds there exists a special class of coordinates, called Darboux-Nijenhuis coordinates, which are canonical w.r.t. the symplectic structure S and diagonalize N. In this framework, one can formulate necessary and sufficient conditions to separability in Darboux-Nijenhuis coordinates. These techniques have been exploited to investigate, from an alternative point of view, several examples, like the Toda lattice [FMP], stationary reductions of KdV equation [FMPZ], the Neumann-Rosochatius system [P, BFP]. The definition of Poisson structure extends in a natural fashion to complex manifolds. Complex Poisson surfaces play a major role in the theory of algebraically completely integrable systems. Indeed, the choice of a Poisson bivector on a surface X (in other terms, the choice of a global section of the anticanonical bundle) determines natural Poisson structures both on the moduli space of stable sheaves and on the Hilbert scheme of points of X [Bo, Mar]. By means of this construction, under suitable hypotheses, it is possible to associate an integrable system to a linear system defined on the Poisson surface [Hu]. Many important examples (for instance, the Neumann system, the Hitchin system, the Toda lattice) can be obtained in this way. Moreover, Sklyanin's Lie Poisson group structures associated to splittings of the loop group LGL(N) have been shown to be equivalent to Poisson structures defined on moduli spaces of sheaves on an associated Poisson surface [HM]. A theorem of classification of Poisson surfaces (refining previously known results) has been recently proved, together with a counting of independent Poisson structures on a given algebraic surface [BMa]. In order to establish a link between the bihamiltonian approach and the algebro-geometric construction associated to linear systems on Poisson surfaces, it seems to be helpful to make use of the notion of special Kaehler manifold. According to the terminology fixed in [Fr], a special Kaehler manifold is a Kaehler (or more generally, pseudo-Kaehler) manifold equipped with a flat torsionless connection such that the covariant derivative of the Kaehler form and the covariant differential of the complex structure are both equal to zero. These geometric structures establish a bridge between supersymmetric gauge theories and integrable systems [DW]. A very simple way to get a special Kaehler manifold is the following. It can be proved that a classical integrable system ˆ la Liouville-Arnol'd whose base B admits a symplectic form covariantly constant w.r.t. the Gauss-Manin connection carries a natural hypersymplectic structure; morevover, a special Kaehler structure is induced on B [BMe]. This construction (reminiscent of Hitchin's approach to special Kaehler geometry [Hi]) could be useful to implement in the study of integrable systems the use of relative Fourier-Mukai transforms [BBHM, BMP], which maps local systems whose support is contained in lagrangian submanifolds into holomorphic sheaves with connection on the dual fibration. These methods might be relevant to study dualities for integrable systems [FGNR] and to deal with questions related to matrix models in topological string theories [Ag, SS]. References[AHH] Adams, M. R., Harnad, J., Hurtubise, J., Darboux coordinates and Liouville-Arnol'd integration in loop algebras, Comm. Math. Phys. 155 (1993), 385-413.[Ag] Aganagic M. et al., Topological strings and Integrable hierarchies, hep-th/0312085. [BMa] Bartocci C., Macr“ E., Classification of Poisson surfaces, Comm. Contemporary Mathematics, to appear. [BMe] Bartocci C., Mencattini, I., Hyper-symplectic structures on integrable systems, J. Geom. Phys. 50 (2004), 339-344. [BBHM] Bartocci C., Bruzzo U., Hernandez Ruiperez D., Munoz Porras J., Relatively stable bundles over elliptic fibrations, Math. Nachr. 238 (2002), 23-36. [BFP] Bartocci C., Falqui G., Pedroni M., A geometric approach to the separability of the Neumann-Rosochatius system, nlin.SI/0307021. [Bo] Bottacin, F., Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math. 121 (1995), 421-436. [BMP] Bruzzo, U., Marelli, G., Pioli, F. A Fourier transform for sheaves on Lagrangian families of real tori, Parts I & II, J. Geom. Phys. 39 (2001), 174-182 and 41 (2002), 312-329. [DW] Donagi R., Witten, E., Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B460 (1996), 299-334. [FMP] Falqui, G., Magri, F., Pedroni, M., Bihamiltonian Geometry and Separation of Variables for Toda Lattices, Journal of Nonlinear Mathematical Physics 8 (2001) Supplement, Proceedings of the Workshop NEEDS'99 (B. Pelloni et al., eds.),pp. 118-127. [FMPZ] Falqui, G., Magri, F., Pedroni, M., Zubelli, J.P., A Bi Hamiltonian Theory for Stationary KdV Flows and their Separability, Regular and Chaotic Dynamics 5 (2000), 33-52. [FMT] Falqui, G., Magri, F., Tondo, G., Bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy, Theor. Math. Phys. 122 (2000), 176-192. [FP] Falqui, G., Pedroni, M., Separation of variables for bi Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179. [FGNR] Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V., Duality in integrable systems and gauge theories. J. High Energy Phys. 2000, no. 7, Paper 28, 40 pp. [Fr] Freed D.S., Special Kaehler manifolds, Comm. Math. Phys. 203 (1999), 31-52. [GNR] Gorsky, A., Nekrasov, N., Rubtsov, V., Hilbert schemes, separated variables, and D-branes, Comm. Math. Phys. 222 (2001), 299-318. [Hi] Hitchin, N., The moduli space of complex Lagrangian submanifolds, Asian J. Math. 3 (1999), 77-91. [Hu] Hurtubise J.C., Integrable systems and algebraic surfaces, Duke Math. J. 83 (1996), 19-50; Erratum, ibid., 84 (1996), 815. [HM] Hurtubise, J. C., Markman, E., Surfaces and the Sklyanin bracket, Comm. Math. Phys. 230 (2002), 485-502. [IMM] Ibort A., Magri F., Marmo G., Bihamiltonian structures and Staeckel separability, J.Geom.Phys. 33 (2000), 210-228. [Ma] Magri, F., Eight lectures on integrable systems, Lecture Notes in Physics 495 (Y. Kosmann-Schwarzbach et al. eds.), Springer, 1997, pp. 256-296. [Mar] Markman, E., Spectral curves and integrable systems, Comp. Math. 93 (1994), 255-290. [P] Pedroni, M., Bi-Hamiltonian aspects of the separability of the Neumann system, in: Proceedings of the NEEDS2001 Conference, Theor. Math. Phys. 133 (2002), 1720-1727. [SS] Seiberg N., Shih D., Branes, Rings and Matrix Models in Minimal (Super)string Theory, hep-th/0312170. [Sk] Sklyanin, E., Separation of variables: new trends, Progr. Theor. Phys. Suppl. 118 (1995), 35-60 |
Research Program Description | |||||||||||||||||
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The Genova research unit will mainly work in collaboration with the units of Milano and Trieste. The focus of the research activity will fall upon the following topic: classical integrable systems and special Kaehler geometry.
More precisely, in accordance with the guidelines of the research project "Geometric methods in the theory of nonlinear waves and their applications", the primary goals of the Genova unit will be the following.
1. In connection with the existence of a special Kahler structure on the base of hypersymplectic integrable systems (according to the construction we have briefly described in Part 2.4), it is most important to find out explicit examples, possibly among classical hamiltonian systems. On the other hand, special Kaehler geometry turns out to be helpful to study the structure of some moduli spaces (e.g. moduli space of complex lagrangian submanifolds): we aim at investigating the structure of the TeichmŸller space from this viewpoint. 2. In his description of moduli spaces of complex lagrangian submanifolds, Hitchin has provided a new characterization of special Kaehler manifolds as those admitting a bilagrangian local immersion in their cotangent spaces. On the other hand, the notion of bilagrangian distribution is crucial in the bihamiltonian approach to the separability of the Hamilton-Jacobi equation. So, it is natural to clarify the relationship between these two points of view 3. By the results described in the scientific background, the equation of the spectral curve associated to a separable system defined by an omega-N manifold (under suitable hypotheses) should be derived in a purely geometrical fashion from the separation coordinates (i.e. the so-called Darboux-Nijenhuis coordinates). Moreover, we expect that the equation of the Poisson surface underlying the associated Jacobian fibration can be derived from the separability conditions as well (in the Neumann-Rosochatius system this surface is a rational surface). 4. In close connection with point (3), our purpose is to prove a result of "reduction by restriction" for separable hamiltonian systems on omega-N manifolds. It should follow that the separability of the Neumann-Rosochatius system is a straightforward consequence of the separability (in cartesian coordinates) of the corresponding non-constrained point-particle in (n+1)-dimensional Euclidean space. In order to carry out its research work, the Genova unit will profit from the collaboration of some visiting scientists, which are recognized as leading experts in their own research field. |
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Last Update: February 22, 2005