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RG equations from Whitham hierarchy. (English) Zbl 0951.37022

Summary: The second derivatives of the prepotential with respect to Whitham time variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of the algebraic ones for the Kontsevich matrix model. In particular the authors provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on Donaldson theory.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14K25 Theta functions and abelian varieties
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
81T13 Yang-Mills and other gauge theories in quantum field theory
81T17 Renormalization group methods applied to problems in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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