Dubrovin's duality for $F$-manifolds with eventual identities

Article, Preprint English OPEN
David, Liana ; Strachan, Ian A. B. (2010)
  • Publisher: Elsevier
  • Journal: Advances in Mathematics (issn: 0001-8708, vol: 226, pp: 4,031-4,060)
  • Related identifiers: doi: 10.1016/j.aim.2010.11.006
  • Subject: Mathematics(all) | Nonlinear Sciences - Exactly Solvable and Integrable Systems | Mathematics - Differential Geometry
    arxiv: Mathematics::Differential Geometry | Mathematics::Symplectic Geometry

A vector field <i>E</i> on an <i>F</i>-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on <i>M</i>. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between <i>F</i>-manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of <i>F</i>-manifolds and preserves the class of Riemannian <i>F</i>-manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChk-structures are preserved by our duality. We use eventual identities to construct compatible pair of metrics.
  • References (17)
    17 references, page 1 of 2

    [1] S. Cecotti, C. Vafa: Topological-antitopological fusion, Nuclear Physics B 367 (1991), p. 359-461.

    [2] L. David, I. A. B. Strachan: Compatible metrics on a manifold and Nonlocal Bi-Hamiltonian Structures, Internat. Math. Res. Notices, no. 66 (2004), p. 3533-3557.

    [3] L. David, I. A. B. Strachan: Conformal flat pencil of metrics, Frobenius structures and a modified Saito construction, J. Geom. Physics 56 (2006), p. 1561-1575.

    [4] B. Dubrovin: On almost duality for Frobenius manifolds, Geometry, topology and mathematical physics, 75-132, Amer. Math. Soc. Transl. Ser. 2, 212.

    [5] B. Dubrovin: Flat pencils of metrics and Frobenius manifolds in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 47-72, World Sci. Publishing, River Edge, NJ, 1998.

    [6] C. Hertling: tt∗-geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), p. 77-161.

    [7] C. Hertling: Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, Cambridge University Press 2002.

    [8] C. Hertling, Yuri I. Manin: Weak Frobenius Manifolds, Internat. Math. Res. Notices, no. 6 (1999), p. 277-286.

    [9] J. Lin: Some constraints on Frobenius manifolds with a tt∗-structure, arXiv:0904.3219v1.

    [10] P. Lorenzoni, M. Pedroni, A. Raimondo: F -manifolds and integrable systems of hydrodinamic type, arXiv:0905.4052.v2

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