Mathematical Sciences: Hamiltonian Theory of Soliton Equations and Geometry of Moduli Spaces

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AbstractProposal: DMS-9802577Principl Investigator: Igor KricheverThe main objective of the present project is a further development ofthe algebraic-geometric integration theory of non-linear equations,models of solid state physics, and models of quantum field theories.An immediate goal is a complete algebraic-geometric approach to theHamiltonian theory of integrable equations, applicable to 2D equationsas well as finite-dimensional models. Particular attention will bepaid to the investigation of the non-local symplectic structures for2D integrable equations which arise in this way, and to theHamiltonian theory of finite-dimensional systems equivalent to thepole dynamics of elliptic, trigonometric and rational solutions of 2Dsoliton equations. Among these systems are spin-generalizations ofCalogero-Moser and Ruijsenaars-Schneider systems. These systems arerelated to Seiberg-Witten solutions of N=2 supersymmetric gaugetheories, and have attracted recently considerable interest. Effortswill also be devoted to the clarification of some unexpected relationsbetween Seiberg-Witten solutions of N=2 supersymmetric gauge theoriesand topological field theories. The moduli spaces ofalgebraic-geometric solutions of soliton equations provide a unifyingframework for these problems. Seiberg-Witten solutions are related tothe symplectic geometry of Jacobian bundles over these moduli spaces,while topological field theories are related to their Riemanniangeometry. The effective Lagrangian in the first case and the freeenergy in the second case are just restrictions of the exponential ofthe tau-function of the universal Whitham hierarchy, which is itself acorner stone of the perturbation theory of soliton equations. It isvery important to determine whether these relations can be explainedfrom first principles.The algebraic-geometric theory of soliton equations developed in themiddle seventies has had enormous influence on many branches ofmathematics and theoretical physics. Originally it was mainly aimed toconstruct exact solutions of the wide variety of equations describingwave phenomena in the plasma physics, non-linear optics, oceanology,super-conductivity. In recent years the universality of the methodsand ideas developed has led to the outreach far beyond the initialframework. It includes applications to the string theory andsupersymmetric gauge theories. The new approach to the Hamiltoniantheory of soliton equations combines all these directions and shouldallow us to make the next important step. A development of theHamiltonian theory of difference equations as a Hamiltonian theory ofsystems with discrete time is a challenging problem which shouldprovide a bridge between classical and quantum integrable systems.