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Painlevé Transcendents in Two-Dimensional Topological Field Theory

  • Chapter
The Painlevé Property

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

This paper is devoted to the theory of WDVV equations of associativity. This remarkable system of nonlinear differential equations was discovered by E. Witten [85]and R. Dijkgraaf, E. Verlinde, and H. Verlinde [24]in the beginning of the 1990s. It was first derived as equations for the so-called primary free energy of a family of two-dimensional topological field theories. Later it proved to be an efficient tool in the solution of problems of the theory of Gromov-Witten invariants, reflection groups and singularities, and integrable hierarchies.

Here we mainly consider the relationships of WDVV to the theory of Pain-levé equations. This is a two-way connection. First, any solution to WDVV satisfying certain semisimplicity conditions can be expressed via Painle-vé-type transcendents. Conversely, theory of WDVV works as a source of remarkable particular solutions of the Painlevé equations.

This chapter is an extended version of the lecture notes of a course given at the 1996 Cargèse summer school, “The Painlevé Property: One Century Later.” It is organized as follows.

In Section 1 we give a sketch of the ideas of two-dimensional topological field theory, we formulate WDVV, and give the main examples of solutions coming from quantum cohomology and from singularity theory. In Section 2 we give a coordinate-free reformulation of WDVV introducing the notion of a Frobenius manifold. We also construct the first main geometrical object, namely the deformed affine connection on a Frobenius manifold. The monodromy of the deformed connection at the origin gives us the first set of important invariants of Frobenius manifolds. In Section 3 we define the class of semisimple Frobenius manifolds. In physics these correspond to two-dimensional topological field theories with all relevant perturbations. We construct the so-called canonical coordinates on such manifolds. In Section 4 we complete the classification of semisimple Frobenius manifolds in terms of monodromy data of a certain universal linear differential operator with rational coefficients. We give a nontrivial example of computation of the monodromy data in quantum cohomology. In the last section we develop a “mirror construction” representing the principal geometrical objects on a semisimple Frobenius manifold by residues and oscillatory integrals of a family of analytic functions on Riemann surfaces.

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  1. Boris Dubrovin
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  1. Service de Physique de l’etat Condensé, CEA Saclay, F91191, Gif-sur-Yvette Cedex, France

    Robert Conte

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Dubrovin, B. (1999). Painlevé Transcendents in Two-Dimensional Topological Field Theory. In: Conte, R. (eds) The Painlevé Property. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1532-5_6

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  • DOI: https://doi.org/10.1007/978-1-4612-1532-5_6

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