Quadratic Relations for Generalized Hypergeometric Functions PFP-1

Ohara, Katsuyoshi; Sugiki, Yuichi; Takayama, Nobuki

doi:10.1619/fesi.46.213

Cited by 35 publications

(37 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They can be computed straightforwardly by considering all the places where C i and C j intersect geometrically (additional care needs to be taken when boundaries of C i and C j are non-normally crossing). In the current manuscript, we will not make use of intersection numbers for cycles: there exist numerous ways of evaluating them, and we refer the reader to, e.g., [31,43,[68][69][70][71][72][73][74][75][76][77][78][79][80][81][82]. In the example at hand, the 6 The latter is an equivalence class of cycles [C| :…”

Section: Basis Reduction With Intersection Numbersmentioning

confidence: 99%

“…Similarly, intersection numbers ϕ i |ϕ j can be evaluated in multiple different ways, see, e.g., [31,44,45,69,[71][72][73][74][75][76][77][78][79][80][82][83][84]. They are rational functions in kinematic invariants and the dimension D. It was recently found that for logarithmic forms ϕ i and ϕ j there exists a formula localizing on the critical points given by ω = 0 [45]:…”

Section: A Intersection Numbers Of Logarithmic Formsmentioning

confidence: 99%

See 1 more Smart Citation

Feynman integrals and intersection theory

Mastrolia

Mizera

2019

J. High Energ. Phys.

188

277

View full text Add to dashboard Cite

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop. * pierpaolo.mastrolia@pd.infn.it † smizera@pitp.ca arXiv:1810.03818v2 [hep-th]

show abstract

Section: Basis Reduction With Intersection Numbersmentioning

confidence: 99%

Section: A Intersection Numbers Of Logarithmic Formsmentioning

confidence: 99%

Feynman integrals and intersection theory

Mastrolia

Mizera

2019

J. High Energ. Phys.

188

277

View full text Add to dashboard Cite

show abstract

“…By using our results, we can reduce the twisted period relations (10) to quadratic relations between the m+1 F m . We write down one of them as a corollary.…”

Section: Proposition 42mentioning

confidence: 96%

“…In [10], twisted period relations for m+1 F m are obtained from the study of the intersection forms of the (co)homology groups with coefficients in the local system of rank m. Another integral representation of m+1 F m and its inductive structure are used. Our (co)homology groups have coefficients in the local system of rank 1 for the general m variable case.…”

Section: Introductionmentioning

confidence: 99%

“…Our (co)homology groups have coefficients in the local system of rank 1 for the general m variable case. The twisted period relations in [10] are transformed into quadratic relations between the m+1 F m and their derivatives. Since our quadratic relations consist of only the m+1 F m , the structures of twisted period relations in [10] and that of ours are completely different.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

INTERSECTION NUMBERS AND TWISTED PERIOD RELATIONS FOR THE GENERALIZED HYPERGEOMETRIC FUNCTION m+1Fm

Gotō¹

2015

Kyushu J. Math.

View full text Add to dashboard Cite

Abstract. We consider the generalized hypergeometric function m+1 F m and the differential equation m+1 E m that it satisfies. We use the twisted (co)homology groups associated with an Euler-type integral representation. We evaluate the intersection numbers of the twisted cocycles that are defined as the mth exterior products of logarithmic 1-forms. We also provide the twisted cycles corresponding to the local solutions to m+1 E m around the origin, and we evaluate their intersection numbers. The intersection numbers of the twisted (co)cycles lead to the twisted period relations between two fundamental systems of solutions to m+1 E m .

show abstract