Borel $(\alpha,\beta)$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for $\mathbb P^1$-bundles

TL;DR

This paper develops integral transform methods to explicitly solve quantum differential equations for $P^1$-bundles, extending classical theorems into the quantum cohomology context with universal integral kernels.

Contribution

It introduces a quantum analog of the Leray-Hirsch theorem using Borel $(eta,eta)$-multitransforms to reconstruct solutions from base spaces, with explicit integral formulas involving special functions.

Findings

Derived integral representations for solutions of quantum differential equations.

Identified universal integral kernels independent of specific $P^1$-bundles.

Applied methods to compute solutions for blow-ups of projective planes.

Abstract

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations ($qDE$s) associated with the quantum cohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that bases of solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can be reconstructed from the datum of bases of solutions of the corresponding $qDE$ 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 arXiv:2005.08262, called $Borel$ $(\alpha,\beta)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the B\"ohmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen $\mathbb P^1$-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin-Barnes integral representations of solutions of $qDE$s. As an example, we show how to integrate the $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms of solutions of the $qDE$ of $\mathbb P^1$.