Generalized Mittag-Leffler functions, Borel-Laplace multitransforms, and quantum differential equations of Fano varieties

Abstract

This paper addresses the integration problem for the isomonodromic system of quantum differential equations associated with smooth projective Fano varieties. We begin by introducing a class of multivariable, multivalued analytic functions of Mittag-Leffler type. We study their analytic properties and provide explicit Mellin-Barnes integral representations. For any smooth Fano toric variety, we prove that suitable integral linear combinations of these generalized Mittag-Leffler functions and their derivatives form bases of solutions for the corresponding quantum differential equation. In the second part of the paper, we introduce two families of integral transforms -- referred to as Borel-Laplace multitransforms -- acting on tuples of functions. We show that the Laplace multitransform enables the reconstruction of the full solution space of the quantum differential equations for Fano complete intersections, starting from a basis of solutions associated with the ambient Fano variety. Dually, the Borel multitransform reconstructs the solution space for the quantum differential equations of Fano projective bundles of arbitrary rank from that of the base variety. These results both systematize and generalize the research directions initiated in arXiv:2005.08262 and arXiv:2210.05445. Applications include explicit examples involving Fano toric complete intersections, Fano bundles on toric varieties, and low-dimensional Fano varieties such as strict del Pezzo surfaces.