Feynman Integrals, Toric Geometry and Mirror Symmetry

  • Pierre VanhoveEmail author
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)


This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals. We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series. We explain how this allows to determine the minimal differential operator acting on the Feynman integrals. We illustrate the method on sunset integrals in two dimensions at various loop orders. The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces. Therefore the sunset family is a natural home for mirror symmetry techniques. We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm. The equivalence between these two expressions is a consequence of (1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and (2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface.



It is a pleasure to thank Charles Doran and Albrecht Klemm for discussions. The research of P. Vanhove has received funding the ANR grant “Amplitudes” ANR-17- CE31-0001-01, and is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N\(^\circ \) 14.641.31.0001.


  1. 1.
    J.R. Andersen et al., Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report, arXiv:1803.07977 [hep-ph]
  2. 2.
    D. Neill, I.Z. Rothstein, Classical space-times from the S matrix. Nucl. Phys. B 877, 177 (2013)., arXiv:1304.7263 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    N.E.J. Bjerrum-Bohr, J.F. Donoghue, P. Vanhove, On-shell techniques and universal results in quantum gravity. JHEP 1402, 111 (2014)., arXiv:1309.0804 [hep-th]
  4. 4.
    F. Cachazo, A. Guevara, Leading Singularities and Classical Gravitational Scattering, arXiv:1705.10262 [hep-th]
  5. 5.
    A. Guevara, Holomorphic classical limit for spin effects in gravitational and electromagnetic scattering, arXiv:1706.02314 [hep-th]
  6. 6.
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, G. Festuccia, L. Planté, P. Vanhove, General relativity from scattering amplitudes, arXiv:1806.04920 [hep-th]
  7. 7.
    Phys. Lett. B Hypergeometric representation of the two-loop equal mass sunrise diagram. 638, 195 (2006)., arXiv:0603227 [hep-ph/0603227]MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    S. Bauberger, F.A. Berends, M. Bohm, M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams. Nucl. Phys. B 434, 383 (1995)., arXiv:9409388 [hep-ph/9409388]CrossRefGoogle Scholar
  9. 9.
    D.H. Bailey, J.M. Borwein, D. Broadhurst, M.L. Glasser, ElliptiC Integral Evaluations Of Bessel Moments. J. Phys. A 41, 205203 (2008)., arXiv:0801.0891 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D. Broadhurst, Elliptic Integral Evaluation of a Bessel Moment by Contour Integration of a Lattice Green Function, arXiv:0801.4813 [hep-th]
  11. 11.
    D. Broadhurst, Feynman integrals, L-series and Kloosterman moments. Commun. Num. Theor. Phys. 10, 527 (2016)., arXiv:1604.03057 [physics.gen-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    M. Caffo, H. Czyz, E. Remiddi, The pseudothreshold expansion of the two loop sunrise selfmass master amplitudes. Nucl. Phys. B 581, 274 (2000)., arXiv:9912501 [hep-ph/9912501]CrossRefGoogle Scholar
  13. 13.
    S. Laporta, E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph. Nucl. Phys. B 704, 349 (2005), arXiv:0406160 [hep-ph/0406160]MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    L. Adams, C. Bogner, S. Weinzierl, The Two-Loop Sunrise Graph with Arbitrary Masses in Terms of Elliptic Dilogarithms, arXiv:1405.5640 [hep-ph]
  15. 15.
    L. Adams, C. Bogner, S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case. J. Math. Phys. 56(7), 072303 (2015)., arXiv:1504.03255 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    L. Adams, C. Bogner, S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral. J. Math. Phys. 57(3), 032304 (2016)., arXiv:1512.05630 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    L. Adams, C. Bogner, S. Weinzierl, A walk on sunset boulevard. PoS RADCOR 2015, 096 (2016)., arXiv:1601.03646 [hep-ph]
  18. 18.
    L. Adams, S. Weinzierl, On a class of feynman integrals evaluating to iterated integrals of modular forms, arXiv:1807.01007 [hep-ph]
  19. 19.
    L. Adams, E. Chaubey, S. Weinzierl, From Elliptic Curves to Feynman Integrals, arXiv:1807.03599 [hep-ph]
  20. 20.
    S. Bloch, M. Kerr, P. Vanhove, Local mirror symmetry and the sunset feynman integral. Adv. Theor. Math. Phys. 21, 1373 (2017)., arXiv:1601.08181 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    V.V. Batyrev, Dual polyhedra and mirror symmetry for CalabiYau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994)zbMATHGoogle Scholar
  22. 22.
    S. Hosono, A. Klemm, S. Theisen, S.T. Yau, Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces. Commun. Math. Phys. 167, 301 (1995)., arXiv:9308122 [hep-th/9308122]MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    T.-M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow, Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999)., arXiv:9903053 [hep-th/9903053]MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    M.X. Huang, A. Klemm, M. Poretschkin, Refined stable pair invariants for E-, M- and \([p, Q]\)-strings. JHEP 1311, 112 (2013)., arXiv:1308.0619 [hep-th]
  25. 25.
    C.F. Doran, M. Kerr, Algebraic K-theory of toric hypersurfaces. Commun. Number Theory Phys. 5(2), 397–600 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    P. Vanhove, The physics and the mixed hodge structure of Feynman integrals. Proc. Symp. Pure Math. 88, 161 (2014)., arXiv:1401.6438 [hep-th]
  27. 27.
    C. Bogner, S. Weinzierl, Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585 (2010), arXiv:1002.3458 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    P. Tourkine, Tropical Amplitudes, arXiv:1309.3551 [hep-th]
  29. 29.
    O. Amini, S. Bloch, J.I.B. Gil, J. Fresan, Feynman amplitudes and limits of heights. Izv. Math. 80, 813 (2016)., arXiv:1512.04862 [math.AG]MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    E.R. Speer, Generalized Feynman Amplitudes, vol. 62 of Annals of Mathematics Studies (Princeton University Press, New Jersey,1969)Google Scholar
  31. 31.
    A. Primo, L. Tancredi, On the maximal cut of feynman integrals and the solution of their differential equations. Nucl. Phys. B 916, 94 (2017)., arXiv:1610.08397 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    A. Primo, L. Tancredi, Maximal cuts and differential equations for feynman integrals. an application to the three-loop massive banana. Graph. Nucl. Phys. B 921, 316 (2017)., arXiv:1704.05465 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    J. Bosma, M. Sogaard, Y. Zhang, Maximal cuts in arbitrary dimension. JHEP 1708, 051 (2017)., arXiv:1704.04255 [hep-th]
  34. 34.
    H. Frellesvig, C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation. JHEP 1704, 083 (2017)., arXiv:1701.07356 [hep-ph]
  35. 35.
    K.G. Chetyrkin, F.V. Tkachov, Integration by parts: the algorithm to calculate beta functions in 4 loops. Nucl. Phys. B 192, 159 (1981). Scholar
  36. 36.
    O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses. Nucl. Phys. B 502, 455 (1997)., arXiv:9703319 [hep-ph/9703319]CrossRefGoogle Scholar
  37. 37.
    O.V. Tarasov, Methods for deriving functional equations for Feynman integrals. J. Phys. Conf. Ser. 920(1), 012004 (2017)., arXiv:1709.07058 [hep-ph]Google Scholar
  38. 38.
    P. Griffiths, On the periods of certain rational integrals: I. Ann. Math. 90, 460 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    D. Cox, S. Katz, Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, vol,. 68 (American Mathematical Society, Providence, 1999).
  40. 40.
    T. Bitoun, C. Bogner, R.P. Klausen, E. Panzer, Feynman Integral Relations from Parametric Annihilators, arXiv:1712.09215 [hep-th]
  41. 41.
    W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 4-1-1 — A computer algebra system for polynomial computations (2018).
  42. 42.
    I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Generalized euler integrals and A-hypergeometric functions. Adv. Math. 84, 255–271 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants (Birkhäuser, Boston, 1994)zbMATHCrossRefGoogle Scholar
  44. 44.
    V.V. Batyrev, Variations of the mixed hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J. 69(2), 349–409 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    V.V. Batyrev, D.A. Cox, On the hodge structure of projective hypersurfaces in toric varieties. Duke Math. J. 75(2), 293–338 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    S. Hosono, A. Klemm, S. Theisen, Lectures on mirror symmetry. Lect. Notes Phys. 436, 235 (1994)., arXiv:9403096 [hep-th/9403096]
  47. 47.
    C. Closset, Toric geometry and local Calabi-Yau varieties: An Introduction to toric geometry (for physicists), arXiv:0901.3695 [hep-th]
  48. 48.
    J. Stienstra, Jan, GKZ hypergeometric structures, arXiv:math/0511351
  49. 49.
    V.V. Batyrev, D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. Commun. Math. Phys. 168, 493 (1995)., arXiv:9307010 [alg-geom/9307010]MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    S. Hosono, G.K.Z. Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces (Birkhäuser, Boston, 1998)zbMATHGoogle Scholar
  51. 51.
    S. Hosono, A. Klemm, S. Theisen, S.T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces. Nucl. Phys. B 433, 501 (1995). [AMS/IP Stud. Adv. Math. 1, 545 (1996)]., arXiv:9406055 [hep-th/9406055]MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    E. Cattani, Three lectures on hypergeometric functions (2006)Google Scholar
  53. 53.
    F. Beukers, Monodromy of A-hypergeometric functions. J. für die Reine und Angewandte Mathematik 718, 183–206 (2016)MathSciNetzbMATHGoogle Scholar
  54. 54.
    J. Stienstra, Resonant hypergeometric systems and mirror symmetry, arXiv:alg-geom/9711002
  55. 55.
    C. Doran, A.Y. Novoseltsev, P. Vanhove, work in progressGoogle Scholar
  56. 56.
    L. Adams, C. Bogner, S. Weinzierl, The two-loop sunrise graph with arbitrary masses, arXiv:1302.7004 [hep-ph]
  57. 57.
    P. Candelas, X.C. de la Ossa, P.S. Green, L. Parkes, A pair of calabi-yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21 (1991). [AMS/IP Stud. Adv. Math. 9, 31 (1998)]. Scholar
  58. 58.
    D.R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces. AMS/IP Stud. Adv. Math. 9, 185 (1998), arXiv:9111025 [hep-th/9111025]
  59. 59.
    H.A. Verrill, Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations, arXiv:0407327
  60. 60.
    S. Bloch, P. Vanhove, The elliptic dilogarithm for the sunset graph. J. Number Theor. 148, 328 (2015)., arXiv:1309.5865 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    S. Bloch, M. Kerr, P. Vanhove, A Feynman integral via higher normal functions. Compos. Math. 151(12), 2329 (2015)., arXiv:1406.2664 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    F.C.S. Brown, A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917
  63. 63.
    J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism. JHEP 1805(093) (2018)., arXiv:1712.07089 [hep-th]
  64. 64.
    J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves Ii: an application to the sunrise integral. Phys. Rev. D 97(11), 116009 (2018)., arXiv:1712.07095 [hep-ph]
  65. 65.
    J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, Elliptic Symbol Calculus: from Elliptic Polylogarithms to Iterated Integrals of Eisenstein Series, arXiv:1803.10256 [hep-th]
  66. 66.
    J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, From Modular Forms to Differential Equations for Feynman Integrals, arXiv:1807.00842 [hep-th]
  67. 67.
    J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, Elliptic Polylogarithms and Two-Loop Feynman Integrals, arXiv:1807.06238 [hep-ph]
  68. 68.
    E. Remiddi, L. Tancredi, An elliptic generalization of multiple polylogarithms. Nucl. Phys. B 925, 212 (2017)., arXiv:1709.03622 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies (Princeton University Press, Princeton, 1993)Google Scholar
  70. 70.
    D.A. Cox, J.B. Little, H.K. Schenck, Toric Varieties, Graduate Studies in Mathematics (Book 124) (American Mathematical Society, 2011)Google Scholar
  71. 71.
    S. Bloch, H. Esnault, D. Kreimer, On motives associated to graph polynomials. Commun. Math. Phys. 267, 181 (2006)., arXiv:0510011 [math/0510011 [math-ag]]MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    S. Hosono, Central charges, symplectic forms, and hypergeometric series in local mirror symmetry, in Mirror Symmetry V, ed. by N. Yui, S.-T. Yau, J. Lewis (American Mathematical Society, Providence, 2006), pp. 405–439Google Scholar
  73. 73.
    S.H. Katz, A. Klemm, C. Vafa, Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997)., arXiv:9609239 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    J. Stienstra, Mahler measure variations, eisenstein series and instanton expansions, in Mirror symmetry V, AMS/IP Studies in Advanced Mathematics, ed. by N. Yui, S.-T. Yau, J.D. Lewis, vol. 38 (International Press & American Mathematical Society, Providence, 2006), pp. 139–150, arXiv:math/0502193
  75. 75.
    L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms. J. Math. Phys. 57(12), 122302 (2016)., arXiv:1607.01571 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    C. Bogner, A. Schweitzer, S. Weinzierl, Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral. Nucl. Phys. B 922, 528 (2017)., arXiv:1705.08952 [hep-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    C. Bogner, A. Schweitzer, S. Weinzierl, Analytic Continuation of the Kite Family, arXiv:1807.02542 [hep-th]

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Authors and Affiliations

  1. 1.CEA, DSM, Institut de Physique ThéoriqueIPhT, CNRS, MPPU, URA2306SaclayFrance
  2. 2.National Research University Higher School of EconomicsMoscowRussian Federation

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