Skip to main content
SpringerLink
Log in

Feynman Integrals, Toric Geometry and Mirror Symmetry

  • Chapter
  • First Online:
Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

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.

IPHT-t18/096.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    An ideal I of a ring R, is the subset \(I\subset R\), such that 1) \(0\in I\), 2) for all \(a, b\in I\) then \(a+b\in I\), 3) for \(a\in I\) and \(b\in R\), \(a\cdot b\in R\). For \(P(x_1,\dots ,x_n)\) an homogeneous polynomial in \(R=\mathbb C[x_1,\dots ,x_n]\) the Jacobian ideal of P is the ideal generated by the first partial derivative \(\{\partial _{x_i} P(x_1,\dots ,x_n)\}\) [39]. Given a multivariate polynomial \(P(x_1,\dots ,x_n)\) its Jacobian ideal is easily evaluated using Singular command jacob(P). The hypersuface \(P(x_1,\cdots ,x_n)=0\) for an homogeneous polynomial, like the Symanzik polynomials, is of codimension 1 in the projective space \(\mathbb P^{n-1}\). The singularities of the hypersurface are determined by the irreducible factors of the polynomial. This determines the cohomology of the complement of the graph hypersurface and the number of independent master integrals as shown in [40].

  2. 2.

    Consider an homogeneous polynomial of degree d

    $$P(\underline{z},\underline{x})=\sum _{0\le r_i\le n\atop r_1+\cdots +r_n=d}z_{i_1,\dots ,i_n}\prod _{i=1}^n x_i^{r_i}$$

    this is called a toric polynomial if it is invariant under the following actions

    $$ z_i \rightarrow \prod _{j=1}^n t_i^{\alpha _{ij}} z_i; \qquad x_i\rightarrow \prod _{j=1}^n t_i^{\beta _{ij}} x_i $$

    for \((t_1,\dots ,t_n)\in \mathbb C^n\) and \(\alpha _{ij}\) and \(\beta _{ij}\) integers. The second Symanzik polynomial have a natural torus action acting on the mass parameters and the kinematic variables as we will see on some examples below. We refer to the book [39] for more details.

  3. 3.

    The convergence of these series is discussed in [50, §3–2] and [48, §5.2].

  4. 4.

    This quantity is the usual Yukawa coupling of particle physics and string theory compactification. The Yukawa coupling is determined geometrically by the integral of the wedge product of differential forms over particular cycles [57]. The Yukawa couplings which depend non-trivially on the internal geometry appear naturally in the differential equations satisfied by the periods of the underlying geometry as explained for instance in these reviews [46, 58].

  5. 5.

    The Jacobi theta functions are defined by \(\theta _2(q):=2q^{1\over 8}\prod _{n\ge 1} (1-q^n)(1+q^n)^2\), \(\theta _3(q):= \prod _{n\ge 1}(1-q^n)(1+q^{n-\frac{1}{2}})^2\) and \(\theta _4(q):=\prod _{n\ge 1}(1-q^n)(1-q^{n-\frac{1}{2}})^2\).

  6. 6.

    A del Pezzo surface is a two-dimensional Fano variety. A Fano variety is a complete variety whose anti-canonical bundle is ample. The anti-canonical bundle of a non-singular algebraic variety of dimension n is the line bundle defined as the nth exterior power of the inverse of the cotangent bundle. An ample line bundle is a bundle with enough global sections to set up an embedding of its base variety or manifold into projective space.

  7. 7.

    The graph polynomial (47) for higher loop sunset graphs defines Fano variety, which is as well a Calabi-Yau manifold.

  8. 8.

    The fan of a toric variety is defined in the standard reference [69] and the review oriented to a physicts audience in [47].

  9. 9.

    Feynman integrals are period integrals of mixed Hodge structures [26, 71]. At a singular point some cycles of integration vanish, the so-called vanishing cycles, and the limiting behaviour of the period integral is captured by the asymptotic behaviour of the cohomological Hodge theory. The asymptotic Hodge theory inherit some filtration and weight structure of the original Hodge theory.

  10. 10.

    It has been already noticed in [74] the special role played by the Mahler measure and mirror symmetry.

  11. 11.

    We would like to thank Albrecht Klemm for discussions and communication that helped clarifying the link between the work in [20] and the analysis in [24].

References

  1. J.R. Andersen et al., Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report, arXiv:1803.07977 [hep-ph]

  2. D. Neill, I.Z. Rothstein, Classical space-times from the S matrix. Nucl. Phys. B 877, 177 (2013). https://doi.org/10.1016/j.nuclphysb.2013.09.007, arXiv:1304.7263 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  3. N.E.J. Bjerrum-Bohr, J.F. Donoghue, P. Vanhove, On-shell techniques and universal results in quantum gravity. JHEP 1402, 111 (2014). https://doi.org/10.1007/JHEP02(2014)111, arXiv:1309.0804 [hep-th]

  4. F. Cachazo, A. Guevara, Leading Singularities and Classical Gravitational Scattering, arXiv:1705.10262 [hep-th]

  5. A. Guevara, Holomorphic classical limit for spin effects in gravitational and electromagnetic scattering, arXiv:1706.02314 [hep-th]

  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. Phys. Lett. B Hypergeometric representation of the two-loop equal mass sunrise diagram. 638, 195 (2006). https://doi.org/10.1016/j.physletb.2006.05.033, arXiv:0603227 [hep-ph/0603227]

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/0550-3213(94)00475-T, arXiv:9409388 [hep-ph/9409388]

    Article  Google Scholar 

  9. D.H. Bailey, J.M. Borwein, D. Broadhurst, M.L. Glasser, ElliptiC Integral Evaluations Of Bessel Moments. J. Phys. A 41, 205203 (2008). https://doi.org/10.1088/1751-8113/41/20/205203, arXiv:0801.0891 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Broadhurst, Elliptic Integral Evaluation of a Bessel Moment by Contour Integration of a Lattice Green Function, arXiv:0801.4813 [hep-th]

  11. D. Broadhurst, Feynman integrals, L-series and Kloosterman moments. Commun. Num. Theor. Phys. 10, 527 (2016). https://doi.org/10.4310/CNTP.2016.v10.n3.a3, arXiv:1604.03057 [physics.gen-ph]

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Caffo, H. Czyz, E. Remiddi, The pseudothreshold expansion of the two loop sunrise selfmass master amplitudes. Nucl. Phys. B 581, 274 (2000). https://doi.org/10.1016/S0550-3213(00)00274-1, arXiv:9912501 [hep-ph/9912501]

    Article  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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. 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). https://doi.org/10.1063/1.4926985, arXiv:1504.03255 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1063/1.4944722, arXiv:1512.05630 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Adams, C. Bogner, S. Weinzierl, A walk on sunset boulevard. PoS RADCOR 2015, 096 (2016). https://doi.org/10.22323/1.235.0096, arXiv:1601.03646 [hep-ph]

  18. L. Adams, S. Weinzierl, On a class of feynman integrals evaluating to iterated integrals of modular forms, arXiv:1807.01007 [hep-ph]

  19. L. Adams, E. Chaubey, S. Weinzierl, From Elliptic Curves to Feynman Integrals, arXiv:1807.03599 [hep-ph]

  20. S. Bloch, M. Kerr, P. Vanhove, Local mirror symmetry and the sunset feynman integral. Adv. Theor. Math. Phys. 21, 1373 (2017). https://doi.org/10.4310/ATMP.2017.v21.n6.a1, arXiv:1601.08181 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  21. V.V. Batyrev, Dual polyhedra and mirror symmetry for CalabiYau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994)

    MATH  Google Scholar 

  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). https://doi.org/10.1007/BF02100589, arXiv:9308122 [hep-th/9308122]

    Article  MathSciNet  MATH  Google Scholar 

  23. T.-M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow, Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999). https://doi.org/10.4310/ATMP.1999.v3.n3.a3, arXiv:9903053 [hep-th/9903053]

    Article  MathSciNet  MATH  Google Scholar 

  24. M.X. Huang, A. Klemm, M. Poretschkin, Refined stable pair invariants for E-, M- and \([p, Q]\)-strings. JHEP 1311, 112 (2013). https://doi.org/10.1007/JHEP11(2013)112, arXiv:1308.0619 [hep-th]

  25. C.F. Doran, M. Kerr, Algebraic K-theory of toric hypersurfaces. Commun. Number Theory Phys. 5(2), 397–600 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Vanhove, The physics and the mixed hodge structure of Feynman integrals. Proc. Symp. Pure Math. 88, 161 (2014). https://doi.org/10.1090/pspum/088/01455, arXiv:1401.6438 [hep-th]

  27. C. Bogner, S. Weinzierl, Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585 (2010), arXiv:1002.3458 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Tourkine, Tropical Amplitudes, arXiv:1309.3551 [hep-th]

  29. O. Amini, S. Bloch, J.I.B. Gil, J. Fresan, Feynman amplitudes and limits of heights. Izv. Math. 80, 813 (2016). https://doi.org/10.1070/IM8492, arXiv:1512.04862 [math.AG]

    Article  MathSciNet  MATH  Google Scholar 

  30. E.R. Speer, Generalized Feynman Amplitudes, vol. 62 of Annals of Mathematics Studies (Princeton University Press, New Jersey,1969)

    Google Scholar 

  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). https://doi.org/10.1016/j.nuclphysb.2016.12.021, arXiv:1610.08397 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.nuclphysb.2017.05.018, arXiv:1704.05465 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Bosma, M. Sogaard, Y. Zhang, Maximal cuts in arbitrary dimension. JHEP 1708, 051 (2017). https://doi.org/10.1007/JHEP08(2017)051, arXiv:1704.04255 [hep-th]

  34. H. Frellesvig, C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation. JHEP 1704, 083 (2017). https://doi.org/10.1007/JHEP04(2017)083, arXiv:1701.07356 [hep-ph]

  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). https://doi.org/10.1016/0550-3213(81)90199-1

    Article  Google Scholar 

  36. O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses. Nucl. Phys. B 502, 455 (1997). https://doi.org/10.1016/S0550-3213(97)00376-3, arXiv:9703319 [hep-ph/9703319]

    Article  Google Scholar 

  37. O.V. Tarasov, Methods for deriving functional equations for Feynman integrals. J. Phys. Conf. Ser. 920(1), 012004 (2017). https://doi.org/10.1088/1742-6596/920/1/012004, arXiv:1709.07058 [hep-ph]

    Google Scholar 

  38. P. Griffiths, On the periods of certain rational integrals: I. Ann. Math. 90, 460 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Cox, S. Katz, Mirror symmetry and algebraic geometry. Mathematical Surveys and Monographs, vol,. 68 (American Mathematical Society, Providence, 1999). https://doi.org/10.1090/surv/068

  40. T. Bitoun, C. Bogner, R.P. Klausen, E. Panzer, Feynman Integral Relations from Parametric Annihilators, arXiv:1712.09215 [hep-th]

  41. W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 4-1-1 — A computer algebra system for polynomial computations (2018). http://www.singular.uni-kl.de

  42. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Generalized euler integrals and A-hypergeometric functions. Adv. Math. 84, 255–271 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  43. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants (Birkhäuser, Boston, 1994)

    Chapter  MATH  Google Scholar 

  44. V.V. Batyrev, Variations of the mixed hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J. 69(2), 349–409 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Hosono, A. Klemm, S. Theisen, Lectures on mirror symmetry. Lect. Notes Phys. 436, 235 (1994). https://doi.org/10.1007/3-540-58453-6_13, arXiv:9403096 [hep-th/9403096]

  47. C. Closset, Toric geometry and local Calabi-Yau varieties: An Introduction to toric geometry (for physicists), arXiv:0901.3695 [hep-th]

  48. J. Stienstra, Jan, GKZ hypergeometric structures, arXiv:math/0511351

  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). https://doi.org/10.1007/BF02101841, arXiv:9307010 [alg-geom/9307010]

    Article  MathSciNet  MATH  Google Scholar 

  50. S. Hosono, G.K.Z. Systems, Gröbner Fans, and Moduli Spaces of Calabi-Yau Hypersurfaces (Birkhäuser, Boston, 1998)

    MATH  Google Scholar 

  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)]. https://doi.org/10.1016/0550-3213(94)00440-P, arXiv:9406055 [hep-th/9406055]

    Article  MathSciNet  MATH  Google Scholar 

  52. E. Cattani, Three lectures on hypergeometric functions (2006)

    Google Scholar 

  53. F. Beukers, Monodromy of A-hypergeometric functions. J. für die Reine und Angewandte Mathematik 718, 183–206 (2016)

    MathSciNet  MATH  Google Scholar 

  54. J. Stienstra, Resonant hypergeometric systems and mirror symmetry, arXiv:alg-geom/9711002

  55. C. Doran, A.Y. Novoseltsev, P. Vanhove, work in progress

    Google Scholar 

  56. L. Adams, C. Bogner, S. Weinzierl, The two-loop sunrise graph with arbitrary masses, arXiv:1302.7004 [hep-ph]

  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)]. https://doi.org/10.1016/0550-3213(91)90292-6

    Article  MathSciNet  MATH  Google Scholar 

  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. H.A. Verrill, Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations, arXiv:0407327

  60. S. Bloch, P. Vanhove, The elliptic dilogarithm for the sunset graph. J. Number Theor. 148, 328 (2015). https://doi.org/10.1016/j.jnt.2014.09.032, arXiv:1309.5865 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  61. S. Bloch, M. Kerr, P. Vanhove, A Feynman integral via higher normal functions. Compos. Math. 151(12), 2329 (2015). https://doi.org/10.1112/S0010437X15007472, arXiv:1406.2664 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  62. F.C.S. Brown, A. Levin, Multiple Elliptic Polylogarithms, arXiv:1110.6917

  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). https://doi.org/10.1007/JHEP05(2018)093, arXiv:1712.07089 [hep-th]

  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). https://doi.org/10.1103/PhysRevD.97.116009, arXiv:1712.07095 [hep-ph]

  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. 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. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, Elliptic Polylogarithms and Two-Loop Feynman Integrals, arXiv:1807.06238 [hep-ph]

  68. E. Remiddi, L. Tancredi, An elliptic generalization of multiple polylogarithms. Nucl. Phys. B 925, 212 (2017). https://doi.org/10.1016/j.nuclphysb.2017.10.007, arXiv:1709.03622 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  69. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies (Princeton University Press, Princeton, 1993)

    Google Scholar 

  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. S. Bloch, H. Esnault, D. Kreimer, On motives associated to graph polynomials. Commun. Math. Phys. 267, 181 (2006). https://doi.org/10.1007/s00220-006-0040-2, arXiv:0510011 [math/0510011 [math-ag]]

    Article  MathSciNet  MATH  Google Scholar 

  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. 405–439

    Google Scholar 

  73. S.H. Katz, A. Klemm, C. Vafa, Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997). https://doi.org/10.1016/S0550-3213(97)00282-4, arXiv:9609239 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  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. 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). https://doi.org/10.1063/1.4969060, arXiv:1607.01571 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.nuclphysb.2017.07.008, arXiv:1705.08952 [hep-ph]

    Article  MathSciNet  MATH  Google Scholar 

  77. C. Bogner, A. Schweitzer, S. Weinzierl, Analytic Continuation of the Kite Family, arXiv:1807.02542 [hep-th]

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. CEA, DSM, Institut de Physique Théorique, IPhT, CNRS, MPPU, URA2306, 91191, Saclay, Gif-sur-Yvette, France

    Pierre Vanhove

  2. National Research University Higher School of Economics, Moscow, Russian Federation

    Pierre Vanhove

Authors
  1. Pierre Vanhove
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Vanhove .

Editor information

Editors and Affiliations

  1. Deutsches Elektronen-Synchrotron, DESY, Zeuthen, Germany

    Johannes Blümlein

  2. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Carsten Schneider

  3. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Peter Paule

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Vanhove, P. (2019). Feynman Integrals, Toric Geometry and Mirror Symmetry. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-04480-0_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-04479-4

  • Online ISBN: 978-3-030-04480-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation