Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals

Matsubara-Heo, Saiei-Jaeyeong; Takayama, Nobuki

doi:10.1007/978-3-030-52200-1_7

Cited by 11 publications

(23 citation statements)

References 40 publications

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“…Recent mathematical literature on intersection numbers of twisted cycles and co-cycles include application to Gel'fand-Kapranov-Zelevinski systems [52][53][54] and to quadratic relations [55,56]. In particular, the latter study have been stimulated by a conjecture on Feynman Integral relations [57][58][59][60][61].…”

Section: Linear and Quadratic Relationsmentioning

confidence: 99%

“…with on-shell internal lines, that admit a one-fold integral representations. As proposed in [2], applications of this novel method to the decomposition of full Feynman integrals in terms of a complete set of MIs, including the ones corresponding to subdiagrams, as well as deriving contiguity relations for special functions admitting multi-fold integral representation, required the use of multivariate intersection numbers [5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Vector Space of Feynman Integrals and Multivariate Intersection Numbers

et al. 2019

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Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for constructing multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the derivation of contiguity relations for special functions admitting multi-fold integral representations, and to the decomposition of a few Feynman integrals at one-and two-loops, as first steps towards potential applications to generic multi-loop integrals.

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Section: Linear and Quadratic Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vector Space of Feynman Integrals and Multivariate Intersection Numbers

et al. 2019

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“…Recent mathematical literature on intersection numbers of twisted cycles and co-cycles include application to Gel'fand-Kapranov-Zelevinski systems [13,66,67] and to quadratic relations [25][26][27][28][29][30][31].…”

Section: Linear and Quadratic Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Decomposition of Feynman integrals by multivariate intersection numbers

Frellesvig

Gasparotto

Laporta

et al. 2021

J. High Energ. Phys.

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We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

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“…

In this review article, we report on some recent advances on the computational aspects of cohomology intersection numbers of GKZ systems developed in [21], [30], [31] and [32]. We also discuss the relation between intersection theory and evaluation of an integral of a product of powers of absolute values of polynomials.

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confidence: 99%