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arxiv: 2604.09129 · v1 · submitted 2026-04-10 · 🧮 math.AG · hep-th

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Picard-Fuchs Equations of Twisted Differential forms associated to Feynman Integrals

Pierre Vanhove
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Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification 🧮 math.AG hep-th
keywords Feynman integralsPicard-Fuchs equationstwisted differential formsGriffiths-Dwork algorithmD-moduleshypergeometric motivesCalabi-Yau motivesperiod integrals
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The pith

An extension of the Griffiths-Dwork pole reduction produces Picard-Fuchs operators for twisted differential forms in Feynman integrals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an extension of the Griffiths-Dwork algorithm to handle twisted differential forms that arise from dimensionally or analytically regulated Feynman integrals. This extension allows the derivation of the D-module of differential operators that annihilate these forms. The method is demonstrated on families of integrals that produce hypergeometric, elliptic, and Calabi-Yau differential motives, yielding their twisted Picard-Fuchs operators. A sympathetic reader would care because this provides a systematic algebraic-geometric tool for finding differential equations satisfied by Feynman integrals, which are central to calculations in quantum field theory.

Core claim

The central claim is that the extended Griffiths-Dwork pole reduction algorithm can be applied directly to the twisted period integrals coming from Feynman integrals to obtain the D-module of differential operators acting on the associated twisted differential forms. This produces twisted Picard-Fuchs operators for the hypergeometric, elliptic, and Calabi-Yau cases arising from these integrals.

What carries the argument

The extended Griffiths-Dwork pole reduction algorithm, which reduces poles in twisted differential forms to derive the annihilating differential operators.

Load-bearing premise

The twisted period integrals from regulated Feynman integrals admit the extended Griffiths-Dwork reduction without extra obstructions or modifications.

What would settle it

Applying one of the derived Picard-Fuchs operators to a concrete twisted period integral from a known Feynman family and verifying whether the result is zero to high numerical precision would confirm or refute the claim.

Figures

Figures reproduced from arXiv: 2604.09129 by Pierre Vanhove.

Figure 1
Figure 1. Figure 1: The box graph with massless external and internal states [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A two-loop graphs with a = 4, b = 1 and c = 2. Two-loop graphs can be labelled by the number of edges (a, b, c) on each cycle, in figure 2 we have represented a graph with a = 4, b = 1, c = 2. Planar graphs are graph for which least one edge number is equal to 1, and their graph polynomials are given by [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two-loop sunset with a = b = c = 1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Multi-loop sunset with n edges We now turn to the n−1-loop sunset integral in D = 2−2ϵ dimensions attached to the graph in fig. 4 which reads I ϵ ⊖(n) (p 2 , ⃗m, t) = Z ∆n [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

Dimensionally or analytically regulated Feynman integrals lead to relative twisted period integrals. We present a recent extension of the Griffiths-Dwork pole reduction algorithm for deriving the D-module of differential operators acting on the twisted differential forms from Feynman integrals. We illustrate the application of this algorithm by providing twisted Picard-Fuchs operators for hypergeometric, elliptic and Calabi-Yau differential motives arising from families of Feynman integrals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends the Griffiths-Dwork pole reduction algorithm to incorporate twist factors from dimensional or analytic regularization of Feynman integrals, thereby deriving the D-module of differential operators (twisted Picard-Fuchs equations) acting on the corresponding twisted differential forms. Explicit operators are computed for families yielding hypergeometric, elliptic, and Calabi-Yau motives, with the algorithm applied step-by-step to produce operators whose orders and coefficients match the expected structure for twisted periods.

Significance. If the extension holds, the work supplies a concrete, algorithmic method for obtaining Picard-Fuchs operators in the twisted setting relevant to regulated Feynman integrals. The explicit examples across three distinct motive types provide verifiable output that can be cross-checked against known D-module structures, strengthening the utility for computations in algebraic geometry and mathematical physics.

minor comments (3)
  1. [§3] §3 (algorithm description): the precise definition of the twist factor in the pole-reduction step is stated but the transition from the untwisted Griffiths-Dwork operator to the twisted version would benefit from an explicit intermediate equation showing how the twist modifies the residue computation.
  2. [§4.2] §4.2 (elliptic example): the reported differential operator is given to order 2, but the coefficient of the first-derivative term is written without indicating whether it has been normalized by the leading coefficient; a brief remark on normalization would aid reproducibility.
  3. The bibliography omits the original Griffiths-Dwork reference (Griffiths 1969) and the Dwork reference on p-adic cohomology; both should be added for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. We are pleased that the extension of the Griffiths-Dwork algorithm to the twisted setting and the explicit examples for hypergeometric, elliptic, and Calabi-Yau motives are viewed as useful contributions.

Circularity Check

0 steps flagged

No significant circularity; algorithmic extension is constructive and example-driven

full rationale

The paper describes an explicit algorithmic extension of the Griffiths-Dwork pole reduction to twisted periods arising from regulated Feynman integrals. It then applies the procedure step-by-step to concrete families (hypergeometric, elliptic, Calabi-Yau), producing explicit differential operators whose orders and coefficients can be recomputed independently from the given data. No load-bearing step reduces to a fitted parameter renamed as a prediction, a self-definitional loop, or an unverified self-citation chain; the derivation remains constructive and externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

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Works this paper leans on

77 extracted references · 65 canonical work pages

  1. [1]

    Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report,

    J. R. Andersen, J. Bellm, J. Bendavid, N. Berger, D. Bhatia, B. Biedermann, S. Br¨ auer, D. Britzger, A. G. Buckley and R. Camacho,et al.“Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report,” [arXiv:1803.07977 [hep-ph]]

    work page arXiv 2017

  2. [2]

    Abreu, R

    S. Abreu, R. Britto and C. Duhr, “The SAGEX review on scattering amplitudes Chapter 3: Mathematical structures in Feynman integrals,” J. Phys. A55(2022) no.44, 443004 [arXiv:2203.13014 [hep-th]]

    work page arXiv 2022

  3. [3]

    Snowmass Theory Frontier Report,

    N. Craig, C. Cs´ aki, A. X. El-Khadra, Z. Bern, R. Boughezal, S. Catterall, Z. Davoudi, A. de Gouvˆ ea, P. Draper and P. J. Fox,et al.“Snowmass Theory Frontier Report,” [arXiv:2211.05772 [hep-ph]]

    work page arXiv

  4. [4]

    The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes,

    N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante and P. Vanhove, “The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes,” J. Phys. A55(2022) no.44, 443014 [arXiv:2203.13024 [hep-th]]

    work page arXiv 2022

  5. [5]

    The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes,

    D. A. Kosower, R. Monteiro and D. O’Connell, “The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes,” J. Phys. A55(2022) no.44, 443015 [arXiv:2203.13025 [hep-th]]

    work page arXiv 2022

  6. [6]

    Effective Field Theory and Applica- tions: Weak Field Observables from Scattering Amplitudes in Quantum Field Theory,

    N. E. J. Bjerrum-Bohr, L. Plant´ e and P. Vanhove, “Effective Field Theory and Applica- tions: Weak Field Observables from Scattering Amplitudes in Quantum Field Theory,” In : Handbook of Quantum Gravity. Singapore : Springer Nature Singapore, 2024. p. 85-124. [arXiv:2212.08957 [hep-th]]

    work page arXiv 2024

  7. [7]

    Baumann, D

    D. Baumann, D. Green, A. Joyce, E. Pajer, G. L. Pimentel, C. Sleight and M. Taronna, “Snowmass White Paper: The Cosmological Bootstrap,” [arXiv:2203.08121 [hep-th]]

    work page arXiv

  8. [8]

    Benincasa,Amplitudes meet Cosmology: A (Scalar) Primer,2203.15330

    P. Benincasa, “Amplitudes meet Cosmology: A (Scalar) Primer,” [arXiv:2203.15330 [hep-th]]

    work page arXiv

  9. [9]

    Brown,On the periods of some Feynman integrals,0910.0114

    F. C. S. Brown, “On the periods of some Feynman integrals,” [arXiv:0910.0114 [math.AG]]

    work page arXiv

  10. [10]

    A Feynman Integral via Higher Normal Functions,

    S. Bloch, M. Kerr and P. Vanhove, “A Feynman Integral via Higher Normal Functions,” Compos. Math.151(2015) no.12, 2329-2375 [arXiv:1406.2664 [hep-th]]

    work page arXiv 2015

  11. [11]

    Local mirror symmetry and the sunset Feynman integral,

    S. Bloch, M. Kerr and P. Vanhove, “Local mirror symmetry and the sunset Feynman integral,” Adv. Theor. Math. Phys.21(2017), 1373-1453 [arXiv:1601.08181 [hep-th]]

    work page arXiv 2017

  12. [12]

    Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms,

    J. L. Bourjaily, Y. H. He, A. J. Mcleod, M. Von Hippel and M. Wilhelm, “Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms,” Phys. Rev. Lett.121(2018) no.7, 071603 [arXiv:1805.09326 [hep-th]]

    work page arXiv 2018

  13. [13]

    Embed- ding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space,

    J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel and M. Wilhelm, “Embed- ding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space,” JHEP01 (2020), 078 [arXiv:1910.01534 [hep-th]]

    work page arXiv 2020

  14. [14]

    Bounded Collection of Feynman Integral Calabi-Yau Geometries,

    J. L. Bourjaily, A. J. McLeod, M. von Hippel and M. Wilhelm, “Bounded Collection of Feynman Integral Calabi-Yau Geometries,” Phys. Rev. Lett.122(2019) no.3, 031601 [arXiv:1810.07689 [hep-th]]

    work page arXiv 2019

  15. [15]

    Thel-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods,

    A. Klemm, C. Nega and R. Safari, “Thel-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods,” JHEP04(2020), 088 [arXiv:1912.06201 [hep-th]]

    work page arXiv 2020

  16. [16]

    Analytic structure of all loop banana integrals,

    K. B¨ onisch, F. Fischbach, A. Klemm, C. Nega and R. Safari, “Analytic structure of all loop banana integrals,” JHEP05(2021), 066 [arXiv:2008.10574 [hep-th]]

    work page arXiv 2021

  17. [17]

    Bönisch, C

    K. B¨ onisch, C. Duhr, F. Fischbach, A. Klemm and C. Nega, “Feynman integrals in di- mensional regularization and extensions of Calabi-Yau motives,” JHEP09(2022), 156 [arXiv:2108.05310 [hep-th]]

    work page arXiv 2022

  18. [18]

    Functions Beyond Multiple Polylogarithms for Precision Collider Physics,

    J. L. Bourjaily, J. Broedel, E. Chaubey, C. Duhr, H. Frellesvig, M. Hidding, R. Marzucca, A. J. McLeod, M. Spradlin and L. Tancredi,et al.“Functions Beyond Multiple Polylogarithms for Precision Collider Physics,” Contribution to Snowmass 2021 [arXiv:2203.07088 [hep-ph]]

    work page arXiv 2021

  19. [19]

    A symbol and coaction for higher-loop sunrise integrals,

    A. Forum and M. von Hippel, “A symbol and coaction for higher-loop sunrise integrals,” SciPost Phys. Core6(2023), 050 [arXiv:2209.03922 [hep-th]]

    work page arXiv 2023

  20. [20]

    Yangian-Invariant Fishnet Inte- grals in Two Dimensions as Volumes of Calabi-Yau Varieties,

    C. Duhr, A. Klemm, F. Loebbert, C. Nega and F. Porkert, “Yangian-Invariant Fishnet Inte- grals in Two Dimensions as Volumes of Calabi-Yau Varieties,” Phys. Rev. Lett.130(2023) no.4, 4 [arXiv:2209.05291 [hep-th]]

    work page arXiv 2023

  21. [21]

    Frellesvig, R

    H. Frellesvig, R. Morales and M. Wilhelm, “Calabi-Yau Meets Gravity: A Calabi-Yau Threefold at Fifth Post-Minkowskian Order,” Phys. Rev. Lett.132(2024) no.20, 201602 [arXiv:2312.11371 [hep-th]]

    work page arXiv 2024

  22. [22]

    Feynman integrals, geometries and differential equa- tions,

    S. P¨ ogel, X. Wang and S. Weinzierl, “Feynman integrals, geometries and differential equa- tions,” PoSRADCOR2023(2024), 007 [arXiv:2309.07531 [hep-th]]. TWISTED PICARD-FUCHS EQUATIONS 19

    work page arXiv 2024

  23. [23]

    Klemm, C

    A. Klemm, C. Nega, B. Sauer and J. Plefka, “Calabi-Yau periods for black hole scattering in classical general relativity,” Phys. Rev. D109(2024) no.12, 124046 [arXiv:2401.07899 [hep-th]]

    work page arXiv 2024

  24. [24]

    Driesse, G

    M. Driesse, G. U. Jakobsen, A. Klemm, G. Mogull, C. Nega, J. Plefka, B. Sauer and J. Uso- vitsch, “Emergence of Calabi–Yau manifolds in high-precision black-hole scattering,” Nature 641(2025) no.8063, 603-607 [arXiv:2411.11846 [hep-th]]

    work page arXiv 2025

  25. [25]

    Calabi-Yau Feynman integrals in gravity:ε-factorized form for apparent singularities,

    H. Frellesvig, R. Morales, S. P¨ ogel, S. Weinzierl and M. Wilhelm, “Calabi-Yau Feynman integrals in gravity:ε-factorized form for apparent singularities,” JHEP02(2025), 209 [arXiv:2412.12057 [hep-th]]

    work page arXiv 2025

  26. [26]

    Aspects of canon- ical differential equations for Calabi-Yau geometries and beyond,

    C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi and F. J. Wagner, “Aspects of canon- ical differential equations for Calabi-Yau geometries and beyond,” JHEP06(2025), 128 [arXiv:2503.20655 [hep-th]]

    work page arXiv 2025

  27. [27]

    Smirnov and A

    A. V. Smirnov and A. V. Petukhov, “The Number of Master Integrals is Finite,” Lett. Math. Phys.97(2011), 37-44 [arXiv:1004.4199 [hep-th]]

    work page arXiv 2011

  28. [28]

    Lee and A.A

    R. N. Lee and A. A. Pomeransky, “Critical points and number of master integrals,” JHEP 11(2013), 165 [arXiv:1308.6676 [hep-ph]]

    work page arXiv 2013

  29. [29]

    On Motives associated to graph polynomials,

    S. Bloch, H. Esnault and D. Kreimer, “On Motives associated to graph polynomials,” Com- mun. Math. Phys.267(2006), 181-225 [arXiv:math/0510011 [math.AG]]

    work page arXiv 2006

  30. [30]

    Motives associated to sums of graphs

    S. Bloch, “Motives associated to sums of graphs”, [arXiv:0810.1313 [math.AG]]

    work page arXiv

  31. [31]

    The elliptic dilogarithm for the sunset graph,

    S. Bloch and P. Vanhove, “The elliptic dilogarithm for the sunset graph,” J. Number Theor. 148(2015), 328-364 [arXiv:1309.5865 [hep-th]]

    work page arXiv 2015

  32. [32]

    Periods and Hodge structures in perturbative quantum field theory,

    S. Weinzierl, “Periods and Hodge structures in perturbative quantum field theory,” Contemp. Math.648(2015), 249-260 [arXiv:1302.0670 [hep-th]]

    work page arXiv 2015

  33. [33]

    Brown,Notes on Motivic Periods,1512.06410

    F. Brown, “Notes on Motivic Periods,” [arXiv:1512.06410 [math.NT]]

    work page arXiv

  34. [34]

    Single-valued multiple polylogarithms and a proof of the zig-zag conjecture,

    F. Brown and O. Schnetz, “Single-valued multiple polylogarithms and a proof of the zig-zag conjecture,” J. Number Theor.148(2015), 478-506

    2015

  35. [35]

    Feynman quadrics-motive of the massive sunset graph,

    M. Marcolli and G. Tabuada, “Feynman quadrics-motive of the massive sunset graph,” J. Number Theor.195(2019), 159-183 [arXiv:1705.10307]

    work page arXiv 2019

  36. [36]

    Vanhove,The physics and the mixed Hodge structure of Feynman integrals,Proc

    P. Vanhove, “The physics and the mixed Hodge structure of Feynman integrals,” Proc. Symp. Pure Math.88(2014), 161-194 [arXiv:1401.6438 [hep-th]]

    work page arXiv 2014

  37. [37]

    Motivic Geometry of two-Loop Feynman Integrals,

    C. F. Doran, A. Harder, P. Vanhove and E. Pichon-Pharabod, “Motivic Geometry of two-Loop Feynman Integrals,” Quart. J. Math. Oxford Ser.75(2024) no.3, 901-967 [arXiv:2302.14840 [math.AG]]

    work page arXiv 2024

  38. [38]

    Algorithm for differential equations for Feynman integrals in general dimensions,

    L. de la Cruz and P. Vanhove, “Algorithm for differential equations for Feynman integrals in general dimensions,” Lett. Math. Phys.114(2024) no.3, 89 [arXiv:2401.09908 [hep-th]]

    work page arXiv 2024

  39. [39]

    Routledge, 1971

    Noboru Nakanishi,Graph theory and Feynman integrals, volume 11. Routledge, 1971

    1971

  40. [40]

    Feynman Integrals. A Comprehensive Treatment for Students and Re- searchers,

    S. Weinzierl, “Feynman Integrals. A Comprehensive Treatment for Students and Re- searchers,” Springer, 2022, ISBN 978-3-030-99557-7, 978-3-030-99560-7, 978-3-030-99558-4 [arXiv:2201.03593 [hep-th]]

    work page arXiv 2022

  41. [41]

    Choice of Invariant Variables for the

    V. E. Asribekov, “Choice of Invariant Variables for the ”Many-Point” Functions,” J. Exp. Theor. Phys.15(1962) no.2, 394

    1962

  42. [42]

    Quantum Field Theory,

    C. Itzykson and J. B. Zuber, “Quantum Field Theory,” McGraw-Hill, 1980, ISBN 978-0-486- 44568-7

    1980

  43. [43]

    An Introduction to quantum field theory,

    M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory,” Addison- Wesley, 1995, ISBN 978-0-201-50397-5, 978-0-429-50355-9, 978-0-429-49417-8

    1995

  44. [44]

    Generalized Feynman Amplitudes,

    E. R. Speer, “Generalized Feynman Amplitudes,” vol. 62 of Annals of Mathematics Studies. Princeton University Press, New Jersey, Apr., 1969

    1969

  45. [45]

    Les ´ equations aux diff´ erences lin´ eaires et les int´ egrales des fonctions multi- formes

    K. Aomoto, “Les ´ equations aux diff´ erences lin´ eaires et les int´ egrales des fonctions multi- formes”, J. Fac. Sci. Univ. Tokyo, 22(3), 271-297 (1975)

    1975

  46. [46]

    On vanishing of cohomology attached to certain many valued meromorphic functions

    K. Aomoto, “On vanishing of cohomology attached to certain many valued meromorphic functions”, J. Math. Soc. Japan 27(2): 248-255 (1975)

    1975

  47. [47]

    Configurations and Invariant Gauss-Manin Connections of Integrals I

    K. Aomoto, “Configurations and Invariant Gauss-Manin Connections of Integrals I.” Tokyo Journal of Mathematics 5, 249-287

  48. [48]

    Theory of Hypergeometric Functions,

    K. Aomoto, K. and M. Kita, , “Theory of Hypergeometric Functions,” Springer Monographs in Mathematics, Springer-Verlag, Tokyo, 2011

    2011

  49. [49]

    Mizera,Scattering Amplitudes from Intersection Theory,Phys

    S. Mizera, “Scattering Amplitudes from Intersection Theory,” Phys. Rev. Lett.120(2018) no.14, 141602 [arXiv:1711.00469 [hep-th]]. 20 PIERRE VANHOVE

    work page arXiv 2018

  50. [50]

    Vector Space of Feynman Integrals and Multivariate Intersection Numbers,

    H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, “Vector Space of Feynman Integrals and Multivariate Intersection Numbers,” Phys. Rev. Lett.123 (2019) no.20, 201602 [arXiv:1907.02000 [hep-th]]

    work page arXiv 2019

  51. [51]

    Status of Intersection Theory and Feynman Integrals,

    S. Mizera, “Status of Intersection Theory and Feynman Integrals,” PoSMA2019(2019), 016 [arXiv:2002.10476 [hep-th]]

    work page arXiv 2019

  52. [52]

    Algorithms for minimal Picard–Fuchs operators of Feynman integrals,

    P. Lairez and P. Vanhove, “Algorithms for minimal Picard–Fuchs operators of Feynman integrals,” Lett. Math. Phys.113(2023) no.2, 37 [arXiv:2209.10962 [hep-th]]

    work page arXiv 2023

  53. [53]

    Picard-Fuchs equations for Feynman inte- grals,

    S. M¨ uller-Stach, S. Weinzierl and R. Zayadeh, “Picard-Fuchs equations for Feynman inte- grals,” Commun. Math. Phys.326(2014), 237-249 [arXiv:1212.4389 [hep-ph]]

    work page arXiv 2014

  54. [54]

    Feynman integrals, toric geometry and mirror symmetry,

    P. Vanhove, “Feynman integrals, toric geometry and mirror symmetry,” in “Elliptic integrals, elliptic functions and modular forms in quantum field theory” editors Johannes Bl¨ umlein, Carsten Schneider, Peter Paule, (Springer, 2019) p. 415–458, [arXiv:1807.11466 [hep-th]]

    work page arXiv 2019

  55. [55]

    Feynman integrals as A-hypergeometric functions,

    L. de la Cruz, “Feynman integrals as A-hypergeometric functions,” JHEP12(2019), 123 [arXiv:1907.00507 [math-ph]]

    work page arXiv 2019

  56. [56]

    Hypergeometric Series Representations of Feynman Integrals by GKZ Hy- pergeometric Systems,

    R. P. Klausen, “Hypergeometric Series Representations of Feynman Integrals by GKZ Hy- pergeometric Systems,” JHEP04(2020), 121 [arXiv:1910.08651 [hep-th]]

    work page arXiv 2020

  57. [57]

    GKZ-hypergeometric systems for Feynman integrals,

    T. F. Feng, C. H. Chang, J. B. Chen and H. B. Zhang, “GKZ-hypergeometric systems for Feynman integrals,” Nucl. Phys. B953(2020), 114952 [arXiv:1912.01726 [hep-th]]

    work page arXiv 2020

  58. [58]

    Ananthanarayan, S

    B. Ananthanarayan, S. Banik, S. Bera and S. Datta, “FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems,” Comput. Phys. Commun. 287(2023), 108699 [arXiv:2211.01285 [hep-th]]

    work page arXiv 2023

  59. [59]

    Vector spaces of generalized Euler in- tegrals,

    D. Agostini, C. Fevola, A. L. Sattelberger and S. Telen, “Vector spaces of generalized Euler in- tegrals,” Commun. Num. Theor. Phys.18(2024) no.2, 327-370 [arXiv:2208.08967 [math.AG]]

    work page arXiv 2024

  60. [60]

    Four lectures on Euler integrals,

    S. J. Matsubara-Heo, S. Mizera and S. Telen, “Four lectures on Euler integrals,” SciPost Phys. Lect. Notes75(2023), 1 [arXiv:2306.13578 [math-ph]]

    work page arXiv 2023

  61. [61]

    Feynman Integral Relations from GKZ Hypergeometric Systems,

    H. J. Munch, “Feynman Integral Relations from GKZ Hypergeometric Systems,” PoS LL2022(2022), 042 [arXiv:2207.09780 [hep-th]]

    work page arXiv 2022

  62. [62]

    Kinematic singularities of Feynman integrals and principal A-determinants,

    R. P. Klausen, “Kinematic singularities of Feynman integrals and principal A-determinants,” JHEP02(2022), 004 [arXiv:2109.07584 [hep-th]]

    work page arXiv 2022

  63. [63]

    Restrictions of Pfaffian systems for Feynman integrals,

    V. Chestnov, S. J. Matsubara-Heo, H. J. Munch and N. Takayama, “Restrictions of Pfaffian systems for Feynman integrals,” JHEP11(2023), 202 [arXiv:2305.01585 [hep-th]]

    work page arXiv 2023

  64. [64]

    Dlapa, M

    C. Dlapa, M. Helmer, G. Papathanasiou and F. Tellander, “Symbol alphabets from the Landau singular locus,” JHEP10(2023), 161 [arXiv:2304.02629 [hep-th]]

    work page arXiv 2023

  65. [65]

    Presented at the (1966)

    Griffiths, P.A.: The Residue Calculus and Some Transcendental Results in Algebraic Geom- etry, I. Presented at the (1966)

    1966

  66. [66]

    The Residue Calculus And Some Transcendental Results In Algebraic Ge- ometry, II

    Griffiths, P.A.: “The Residue Calculus And Some Transcendental Results In Algebraic Ge- ometry, II”. Proceedings of the National Academy of Sciences. 55, 1392-1395 (1966)

    1966

  67. [67]

    Multiple polylogarithms and mixed Tate motives,

    A. B. Goncharov, “Multiple polylogarithms and mixed Tate motives,” [arXiv:math/0103059 [math.AG]]

    work page arXiv

  68. [68]

    Single-valued Motivic Periods and Multiple Zeta Values,

    F. Brown, “Single-valued Motivic Periods and Multiple Zeta Values,” SIGMA2(2014), e25 [arXiv:1309.5309 [math.NT]]

    work page arXiv 2014

  69. [69]

    Schouten identities for Feynman graph amplitudes; The Mas- ter Integrals for the two-loop massive sunrise graph,

    E. Remiddi and L. Tancredi, “Schouten identities for Feynman graph amplitudes; The Mas- ter Integrals for the two-loop massive sunrise graph,” Nucl. Phys. B880(2014), 343-377 [arXiv:1311.3342 [hep-ph]]

    work page arXiv 2014

  70. [70]

    Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral,

    E. Remiddi and L. Tancredi, “Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral,” Nucl. Phys. B907(2016), 400-444 [arXiv:1602.01481 [hep-ph]]

    work page arXiv 2016

  71. [71]

    The two-loop sunrise graph with arbitrary masses,

    L. Adams, C. Bogner and S. Weinzierl, “The two-loop sunrise graph with arbitrary masses,” J. Math. Phys.54(2013), 052303 [arXiv:1302.7004 [hep-ph]]

    work page arXiv 2013

  72. [72]

    Candelas, X

    P. Candelas, X. de la Ossa, P. Kuusela and J. McGovern, “Mirror symmetry for five-parameter Hulek-Verrill manifolds,” SciPost Phys.15(2023) no.4, 144 [arXiv:2111.02440 [hep-th]]

    work page arXiv 2023

  73. [73]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl, “The three-loop equal-mass banana integral inε- factorised form with meromorphic modular forms,” JHEP09(2022), 062 [arXiv:2207.12893 [hep-th]]

    work page arXiv 2022

  74. [74]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl, “Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral,” Phys. Rev. Lett.130(2023) no.10, 101601 [arXiv:2211.04292 [hep-th]]. TWISTED PICARD-FUCHS EQUATIONS 21

    work page arXiv 2023

  75. [75]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl, “Bananas of equal mass: any loop, any order in the dimensional regularisation parameter,” JHEP04(2023), 117 [arXiv:2212.08908 [hep-th]]

    work page arXiv 2023

  76. [76]

    Position space equations for banana Feynman diagrams,

    V. Mishnyakov, A. Morozov and P. Suprun, “Position space equations for banana Feynman diagrams,” Nucl. Phys. B992(2023), 116245 [arXiv:2303.08851 [hep-th]]

    work page arXiv 2023

  77. [77]

    On factorization hierarchy of equations for banana Feynman integrals,

    V. Mishnyakov, A. Morozov and M. Reva, “On factorization hierarchy of equations for banana Feynman integrals,” Nucl. Phys. B1010(2025), 116746 [arXiv:2311.13524 [hep-th]]. Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS, F-91191 Gif- sur-Yvette Cedex, France

    work page arXiv 2025