Analytic theory of Legendre-type transformations for a Frobenius manifold

Abstract

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 $S_{\kappa}(M)$.