Abstract
In this paper, we define the holomorphic Laplacian for a Frobenius manifold. We give the description of the Laplacian in terms of the prepotential. This will be used to characterize the flat coordinates for the universal unfolding of the function with a simple elliptic singularity. This characterization enables us to solve the so-called “Jacobi’s inversion problem,” i.e. the description of the flat coordinates as the automorphic functions on the period domain w.r.t. the period mapping for the primitive forms. More explicitly the Laplacian relates the flat coordinates with the theta functions on the period domain (see 10.3). About the description of flat coordinates as the automorphic functions, we shall write in a forthcoming paper. The author would like to thank Prof. Claus Hertling and Prof. Atsushi Takahashi for valuable discussions. The auther also thanks the referee for the valuable advices.
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© 2004 Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden
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Satake, I. (2004). The Laplacian for a Frobenius manifold. In: Hertling, K., Marcolli, M. (eds) Frobenius Manifolds. Aspects of Mathematics, vol 36. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-80236-1_12
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DOI: https://doi.org/10.1007/978-3-322-80236-1_12
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