Analytic Theory of Legendre-Type Transformations for a Frobenius Manifold

Let M be an n-dimensional Frobenius manifold. Fix kappa is an element of{1,& ctdot;,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa \in \{1,\dots ,n\}$$\end{document}. Assuming certain invertibility, Dubrovin introduced the Legendre-type transformation S kappa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\kappa $$\end{document}, which transforms M to an n-dimensional Frobenius manifold S kappa(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\kappa (M)$$\end{document}. In this paper, we show that these S kappa(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_\kappa (M)$$\end{document} share the same monodromy data at the Fuchsian singular point of the Dubrovin connection, and that for the case when M is semisimple they also share the same Stokes matrix and the same central connection matrix. A straightforward application of the monodromy identification is the following: if we know the monodromy data of some semisimple Frobenius manifold M, we immediately obtain those of its Legendre-type transformations. Another application gives the identification between the kappa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}th partition function of a semisimple Frobenius manifold M and the topological partition function of S kappa(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\kappa }(M)$$\end{document}.