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Painlevé transcendents in two-dimensional topological field theory. (English) Zbl 1026.34095
Conte, Robert (ed.), The Painlevé property. One century later. New York, NY: Springer. CRM Series in Mathematical Physics. 287-412 (1999).
The paper is devoted to the theory of E. Witten-Dijkgraaf H. Verlinde Verlinde(WDVV)-equations of associativity. The relationships of WDVV in form of a two-way connection to the theory of Painlevé equations are mainly considered. First, any solution of WDVV satisfying semisimplicity conditions can be expressed via Painlevé-type transcendents. Conversely, theory of WDVV works as a source of remarkable particular solutions to the Painlevé equations.
The paper itself is an extended version of the lecture notes of a course given in [M. Francaviglia (ed.) et al., Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 14-22, 1993. Berlin: Springer-Verlag Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)]. WDVV are formulated and main examples of solutions are given. A coordinate-free reformulation of WDVV is introduced and a class of semisimple Frobenius manifolds is defined. Finally, the classification of semisimple Frobenius manifolds is completed and a nontrivial example is given.
For the entire collection see [Zbl 0989.00036].

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
81T10 Model quantum field theories