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Integrable systems and classification of 2-dimensional topological field theories. (English) Zbl 0824.58029

Babelon, Olivier (ed.) et al., Integrable systems: the Verdier memorial conference. Actes du colloque international de Luminy, France, July 1-5, 1991. Boston, MA: Birkhäuser. Prog. Math. 115, 313-359 (1993).
Topological field theory (TFT) is characterized by a peculiar principle which is invariance with respect to arbitrary changes of the metric of spacetime. The question addressed to in this article is: what makes a model within TFT integrable? The main tool in the present approach is the Hamiltonian formalism of integrable hierarchies of the so-called Korteweg-de Vries type and the analysis of semi-classical limits of such systems. The author classifies two-dimensional fields as solutions of the Witten-Dijkgraaf-Verlinde equations. He then constructs the partition function at tree level and this way obtains the genus-zero correlation functions. Topological gravity theory is described (at tree level) via certain integrable hierarchies of hydrodynamic type. As a byproduct, new examples of infinite-dimensional Lie algebras of Virasoro type are constructed.
For the entire collection see [Zbl 0807.00017].

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B68 Virasoro and related algebras
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