Mathematical Physics, Analysis and Geometry

, Volume 4, Issue 3, pp 245–291 | Cite as

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

  • Davide Guzzetti


We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.

WDVV equation Frobenius manifold isomonodromic deformation Painlevé equation monodromy boundary-value problem 


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© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Davide Guzzetti
    • 1
  1. 1.Research Institute for Mathematical Sciences (RIMS)Kyoto UniversityKitashirakawa, Sakyo-ku, KyotoJapan

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