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- B Dubrovin
- 1992

In this paper we consider from the point of view of differential geometry and of the theory of integrable systems the so-called WDVV equations as defining relations of 2-dimensional topological field theory. A complete classification of massive topological con-formal field theories (TCFT) is obtained in terms of monodromy data of an auxillary linear… (More)

- Boris DUBROVIN
- 1998

Contents Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of… (More)

We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov-Witten invariants via tau-function of the… (More)

- B. Dubrovin
- 2008

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 2 and α arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary… (More)

- Boris Dubrovin
- 2006

Hamiltonian perturbations of the simplest hyperbolic equation u t + a(u)u x = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this… (More)

- Boris Dubrovin
- 1998

Main mathematical applications of Frobenius manifolds are in the theory of Gromov-Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius man-ifolds establishes remarkable relationships between these,… (More)

For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus ≤ 1 Virasoro conjecture of T. and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.