Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 103, 54 pages      arXiv:2101.04419      https://doi.org/10.3842/SIGMA.2021.103
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

Francis Brown
All Souls College, University of Oxford, Oxford, OX1 4AL, UK

Received March 04, 2021, in final form November 14, 2021; Published online November 23, 2021

Abstract
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups and quantum field theory.

Key words: graph complexes; Outer space; tropical curves; motives; multiple zeta values; Feynman integrals; quantum field theory.

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References

  1. Alm J., The Grothendieck-Teichmueller Lie algebra and Brown's dihedral moduli spaces, arXiv:1805.06684.
  2. Baker O., The Jacobian map on Outer space, Ph.D. Thesis, Cornell University, 2011, available at https://ecommons.cornell.edu/handle/1813/30769.
  3. Bar-Natan D., McKay B., Graph cohomology: an overview and some computations, available at http://www.math.toronto.edu/ drorbn/papers/GCOC/GCOC.ps.
  4. Belkale P., Brosnan P., Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116 (2003), 147-188.
  5. Berghoff M., Kreimer D., Graph complexes and Feynman rules, arXiv:2008.09540.
  6. Bloch S., Esnault H., Kreimer D., On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), 181-225, arXiv:math.AG/0510011.
  7. Bogner C., MPL - a program for computations with iterated integrals on moduli spaces of curves of genus zero, Comput. Phys. Comm. 203 (2016), 339-353, arXiv:1510.04562.
  8. Borel A., Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235-272.
  9. Borinsky M., Graphs, automorphisms and clockworks, in preparation.
  10. Borinsky M., Schnetz O., Graphical functions in even dimensions, arXiv:2105.05015.
  11. Borinsky M., Vogtmann K., The Euler characteristic of ${\rm Out}(F_n)$, Comment. Math. Helv. 95 (2020), 703-748, arXiv:1907.03543.
  12. Brannetti S., Melo M., Viviani F., On the tropical Torelli map, Adv. Math. 226 (2011), 2546-2586, arXiv:0907.3324.
  13. Broadhurst D.J., Kreimer D., Knots and numbers in $\phi^4$ theory to $7$ loops and beyond, Internat. J. Modern Phys. C 6 (1995), 519-524, arXiv:hep-ph/9504352.
  14. Brown F., The massless higher-loop two-point function, Comm. Math. Phys. 287 (2009), 925-958, arXiv:0804.1660.
  15. Brown F., On the periods of some Feynman integrals, arXiv:0910.0114.
  16. Brown F., Mixed Tate motives over $\mathbb Z$, Ann. of Math. 175 (2012), 949-976, arXiv:1102.1312.
  17. Brown F., Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Number Theory Phys. 11 (2017), 453-556, arXiv:1512.06409.
  18. Brown F., Notes on motivic periods, Commun. Number Theory Phys. 11 (2017), 557-655, arXiv:1512.06410.
  19. Brown F., Doryn D., Framings for graph hypersurfaces, arXiv:1301.3056.
  20. Brown F., Kreimer D., Angles, scales and parametric renormalization, Lett. Math. Phys. 103 (2013), 933-1007, arXiv:1112.1180.
  21. Brown F., Schnetz O., A K3 in $\phi^4$, Duke Math. J. 161 (2012), 1817-1862, arXiv:1006.4064.
  22. Bux K.-U., Smillie P., Vogtmann K., On the bordification of Outer space, J. Lond. Math. Soc. 98 (2018), 12-34, arXiv:1709.01296.
  23. Caporaso L., Viviani F., Torelli theorem for graphs and tropical curves, Duke Math. J. 153 (2010), 129-171, arXiv:0901.1389.
  24. Cartier P., A primer of Hopf algebras, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 537-615.
  25. Chan M., Combinatorics of the tropical Torelli map, Algebra Number Theory 6 (2012), 1133-1169, arXiv:1012.4539.
  26. Chan M., Galatius S., Payne S., Tropical curves, graph complexes, and top weight cohomology of $\mathcal{M}_g$, J. Amer. Math. Soc. 34 (2021), 565-594, arXiv:1805.10186.
  27. Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203-242, arXiv:hep-th/9808042.
  28. Culler M., Vogtmann K., Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91-119.
  29. Deligne P., Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5-57.
  30. Deligne P., Le groupe fondamental de la droite projective moins trois points, in Galois Groups over ${\bf Q}$ (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., Vol. 16, Springer, New York, 1989, 79-297.
  31. Denham G., Schulze M., Walther U., Matroid connectivity and singularities of configuration hypersurfaces, Lett. Math. Phys. 111 (2021), 11, 67 pages, arXiv:1902.06507.
  32. Drinfel'd V.G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with ${\rm Gal}(\overline{\bf Q}/{\bf Q})$, Leningrad Math. J. 2 (1990), 829-860.
  33. Furusho H., Pentagon and hexagon equations, Ann. of Math. 171 (2010), 545-556, arXiv:math.QA/0702128.
  34. Getzler E., Kapranov M.M., Modular operads, Compositio Math. 110 (1998), 65-126, arXiv:dg-ga/9408003.
  35. Golz M., Dodgson polynomial identities, Commun. Number Theory Phys. 13 (2019), 667-723, arXiv:1810.06220.
  36. Grothendieck A., On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95-103.
  37. Kaufmann R.M., Ward B.C., Feynman categories, Astérisque 387 (2017), vii+161 pages.
  38. Khoroshkin A., Willwacher T., Zivković M., Differentials on graph complexes, Adv. Math. 307 (2017), 1184-1214, arXiv:1411.2369.
  39. Kontsevich M., Formal (non)commutative symplectic geometry, in The Gelfand Mathematical Seminars, 1990-1992, Birkhäuser Boston, Boston, MA, 1993, 173-187.
  40. Kontsevich M., Derived Grothendieck-Teichmüller group and graph complexes [after T. Willwacher], Astérisque 407 (2019), Exp. No. 1126, 183-211.
  41. Looijenga E., Cohomology of ${\mathcal M}_3$ and ${\mathcal M}^1_3$, in Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math., Vol. 150, Amer. Math. Soc., Providence, RI, 1993, 205-228.
  42. Maurer S.B., Matrix generalizations of some theorems on trees, cycles and cocycles in graphs, SIAM J. Appl. Math. 30 (1976), 143-148.
  43. Nagnibeda T., The Jacobian of a finite graph, in Harmonic Functions on Trees and Buildings (New York, 1995), Contemp. Math., Vol. 206, Amer. Math. Soc., Providence, RI, 1997, 149-151.
  44. Panzer E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Comm. 188 (2015), 148-166, arXiv:1403.3385.
  45. Rosset S., A new proof of the Amitsur-Levitski identity, Israel J. Math. 23 (1976), 187-188.
  46. Schnetz O., Quantum periods: a census of $\phi^4$-transcendentals, Commun. Number Theory Phys. 4 (2010), 1-47, arXiv:0801.2856.
  47. Siegel C.L., The volume of the fundamental domain for some infinite groups, Trans. Amer. Math. Soc. 39 (1936), 209-218.
  48. Whitney H., On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), 509-533.
  49. Willwacher T., M. Kontsevich's graph complex and the Grothendieck-Teichmüller Lie algebra, Invent. Math. 200 (2015), 671-760, arXiv:1009.1654.
  50. Willwacher T., Zivković M., Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics, Adv. Math. 272 (2015), 553-578, arXiv:1401.4974.

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