Analytic Theory of Legendre-Type Transformations for a Frobenius Manifold

Let M be an n-dimensional Frobenius manifold. Fix \(\kappa \in \{1,\dots ,n\}\). Assuming certain invertibility, Dubrovin introduced the Legendre-type transformation \(S_\kappa \), which transforms M to an n-dimensional Frobenius manifold \(S_\kappa (M)\). In this paper, we show that these \(S_\kappa (M)\) 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 \(\kappa \)th partition function of a semisimple Frobenius manifold M and the topological partition function of \(S_{\kappa }(M)\).