Borel $(α,β)$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for $\mathbb P^1$-bundles
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866913525735096320 |
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| author | Cotti, Giordano |
| author_facet | Cotti, Giordano |
| contents | 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 $Borel$ $(α,β)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer-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 $\mathbb P^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05445 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Borel $(α,β)$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for $\mathbb P^1$-bundles Cotti, Giordano Algebraic Geometry Mathematical Physics Differential Geometry 53D45, 44A20, 44A30, 33B20 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 $Borel$ $(α,β)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer-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 $\mathbb P^1$. |
| title | Borel $(α,β)$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for $\mathbb P^1$-bundles |
| topic | Algebraic Geometry Mathematical Physics Differential Geometry 53D45, 44A20, 44A30, 33B20 |
| url | https://arxiv.org/abs/2210.05445 |