Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 078, 15 pages      arXiv:1206.0372
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Frobenius 3-Folds via Singular Flat 3-Webs

Sergey I. Agafonov
Departmento de Matemática, Universidade Federal da Paraiba, João Pessoa, Brazil

Received May 28, 2012, in final form October 17, 2012; Published online October 21, 2012

We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ satisfying the following conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the Chern connection form is holomorphic, 3) the curvature form vanishes identically.

Key words: Frobenius manifold; hexagonal 3-web; Chern connection; infinitesimal symmetry.

pdf (492 kb)   tex (151 kb)


  1. Agafonov S.I., Flat 3-webs via semi-simple Frobenius 3-manifolds, J. Geom. Phys. 62 (2012), 361-367, arXiv:1108.1997.
  2. Agafonov S.I., Linearly degenerate reducible systems of hydrodynamic type, J. Math. Anal. Appl. 222 (1998), 15-37.
  3. Agafonov S.I., Local classification of singular hexagonal 3-webs with holomorphic Chern connection form and infinitesimal symmetries, arXiv:1105.1402.
  4. Agafonov S.I., On implicit ODEs with hexagonal web of solutions, J. Geom. Anal. 19 (2009), 481-508, arXiv:0808.0348.
  5. Akivis M.A., Goldberg V.V., Differential geometry of webs, in Handbook of Differential Geometry, Vol. I, North-Holland, Amsterdam, 2000, 1-152.
  6. Blaschke W., Einführung in die Geometrie der Waben, Birkhäuser Verlag, Basel und Stuttgart, 1955.
  7. Cartan E., Les sous-groupes des groupes continus de transformations, Ann. Sci. École Norm. Sup. (3) 25 (1908), 57-194.
  8. Dubrovin B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  9. Dubrovin B., Integrable systems in topological field theory, Nuclear Phys. B 379 (1992), 627-689.
  10. Grifone J., Salem E. (Editors), Web theory and related topics, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
  11. Hénaut A., On planar web geometry through abelian relations and connections, Ann. of Math. (2) 159 (2004), 425-445.
  12. Manin Yu.I., Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, Vol. 47, American Mathematical Society, Providence, RI, 1999.
  13. Mokhov O.I., Ferapontov E.V., Associativity equations of two-dimensional topological field theory as integrable Hamiltonian nondiagonalizable systems of hydrodynamic type, Funct. Anal. Appl. 30 (1996), 195-203, hep-th/9505180.

Previous article  Next article   Contents of Volume 8 (2012)