Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for P1-bundles

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDEs) associated with the quantum cohomology of $P^1$-bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the $P^1$-bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17], called Borel $(\alpha ,\beta )$-multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen $P^1$-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDEs. As an example, we show how to integrate the qDE of blow-up of $P^2$ at one point via Borel multitransforms of solutions of the qDE of $P^1$.