In the first part of this book, the author introduces the notion of cyclic stratum of a Frobenius manifold \(M\). This is the set of points of the extended manifold \(\mathbb{C}^* \times M\), at which the unit vector field is a cyclic vector for the isomonodromic system defined by the flatness condition of the extended deformed connection. The study of the geometry of the complement of the cyclic stratum is addressed. The author shows that at points of the cyclic stratum, the isomonodromic system attached to \(M\) can be reduced to a scalar differential equation, called the master differential equation of \(M\). In the case of Frobenius manifolds coming from Gromov–Witten theory, namely quantum cohomologies of smooth projective varieties, such a construction reproduces the notion of quantum differential equation.

In the second part of this book, the author introduces two multilinear transforms, called Borel–Laplace \((\alpha, \beta)\)-multi-transforms, on spaces of Ribenboim formal power series with exponents and coefficients in an arbitrary finite-dimensional \(\mathbb{C}\)-algebra \(A\).

In the third and final part of the book, as an application, the author shows how to use the new analytic tools, introduced in the previous parts, in order to study the quantum differential equations of Hirzebruch surfaces.