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Integrable hierarchies in Donaldson-Witten and Seiberg-Witten theories. (English) Zbl 0979.81057

Aratyn, Henrik (ed.) et al., Integrable hierarchies and modern physical theories. Proceedings of the NATO advanced research Workshop, Chicago, IL, USA, July 22-26, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 18, 15-32 (2001).
Summary: We review various aspects of integrable hierarchies appearing in \({\mathcal N}=2\) supersymmetric gauge theories. In particular we show that the blowup function in Donaldson-Witten theory, up to a redefinition of the fast times, is a \(\tau\)-function for a \(g\)-gap solution of the KdV hierarchy. In the case of four-manifolds of simple type, instead, the blowup function becomes a \(\tau\)-function corresponding to a multisoliton solution. We obtain a new expression for the contact terms that links these results to the Whitham hierarchy formulation of Seiberg-Witten theories.
For the entire collection see [Zbl 0964.00039].

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81R12 Groups and algebras in quantum theory and relations with integrable systems
81T60 Supersymmetric field theories in quantum mechanics
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