Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

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.

[1]  A. Ferretti.  Algebra & Number Theory , 2012 .

[2]  Mathias Schulze,et al.  Matroid connectivity and singularities of configuration hypersurfaces , 2019, Letters in mathematical physics.

[3]  P. Deligné.  Le Groupe Fondamental de la Droite Projective Moins Trois Points , 1989 .

[4]  Margarida Melo,et al.  On the tropical Torelli map , 2009, 0907.3324.

[5]  Francis Brown,et al.  The Massless Higher-Loop Two-Point Function , 2008, 0804.1660.

[6]  Benjamin C. Ward,et al.  Feynman Categories , 2013, Astérisque.

[7]  Christian Bogner,et al.  MPL - A program for computations with iterated integrals on moduli spaces of curves of genus zero , 2015, Comput. Phys. Commun..

[8]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[9]  Philip G. Johnson.  Cornell University , 1897, The Journal of comparative medicine and veterinary archives.

[10]  F. Brown.  Notes on Motivic Periods , 2015, 1512.06410.

[11]  MATT STEVENSON,et al.  ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES , 2015 .

[12]  Karen Vogtmann,et al.  On the bordification of Outer space , 2017, J. Lond. Math. Soc..

[13]  D. Kreimer,et al.  Graph complexes and Feynman rules , 2020, Communications in Number Theory and Physics.

[14]  Pierre Cartier,et al.  A Primer of Hopf Algebras , 2007 .

[15]  T. Willwacher.  M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra , 2010, 1009.1654.

[16]  Pentagon and hexagon equations , 2007, math/0702128.

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Erik Panzer,et al.  Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals , 2014, Comput. Phys. Commun..

[19]  P. Deligne.  Theorie de Hodge I , 1970 .

[20]  F. Brown,et al.  Framings for graph hypersurfaces , 2013, 1301.3056.

[21]  D. J. Broadhurst,et al.  Knots and Numbers in ϕ4 Theory to 7 Loops and Beyond , 1995 .

[22]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[23]  Marcel Golz.  Dodgson polynomial identities , 2018, Communications in Number Theory and Physics.

[24]  Adv , 2019, International Journal of Pediatrics and Adolescent Medicine.

[25]  J. Alm.  The Grothendieck-Teichmueller Lie algebra and Brown's dihedral moduli spaces , 2018, 1805.06684.

[26]  O. Schnetz.  Quantum periods : A census of φ 4-transcendentals , 2010 .

[27]  C. Siegel.  The volume of the fundamental domain for some infinite groups , 1936 .

[28]  L. Caporaso,et al.  TORELLI THEOREM FOR GRAPHS AND TROPICAL CURVES , 2009, 0901.1389.

[29]  F. Brown.  On the periods of some Feynman integrals , 2009, 0910.0114.

[30]  Francis Brown,et al.  Mixed Tate motives over $\Z$ , 2011, 1102.1312.

[31]  M. Kontsevich.  FORMAL (NON)-COMMUTATIVE SYMPLECTIC GEOMETRY , 1993 .

[32]  I. Gelfand,et al.  The Gelfand Mathematical Seminars, 1990–1992 , 1993 .

[33]  B. M. Fulk.  MATH , 1992 .

[34]  H. Whitney.  On the Abstract Properties of Linear Dependence , 1935 .

[35]  A new proof of the Amitsur-Levitski identity , 1976 .

[36]  Melody Chan,et al.  Combinatorics of the tropical Torelli map , 2010, 1012.4539.

[37]  Michael Borinsky,et al.  Graphical functions in even dimensions , 2021 .

[38]  Matroids motives, and a conjecture of Kontsevich , 2000, math/0012198.

[39]  M. M. KAPRANOVwhere.  Modular Operads , 1994 .

[40]  K. Vogtmann,et al.  Moduli of graphs and automorphisms of free groups , 1986 .

[41]  THE JACOBIAN MAP ON OUTER SPACE , 2011 .

[42]  Sammie Bae,et al.  Graphs , 2020, Algorithms.

[43]  Einzelwerken Muster,et al.  Invent , 2021, Encyclopedic Dictionary of Archaeology.

[44]  F. Brown,et al.  Angles, Scales and Parametric Renormalization , 2011, 1112.1180.

[45]  T. Willwacher,et al.  Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics , 2015 .

[46]  野村栄一,et al.  2 , 1900, The Hatak Witches.

[47]  A. Borel.  Stable real cohomology of arithmetic groups , 1974 .

[48]  J. B. Ferguson,et al.  Motives , 1983 .

[49]  W. Gillam,et al.  ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES , 2012 .

[50]  Stephen B. Maurer.  Matrix Generalizations of Some Theorems on Trees, Cycles and Cocycles in Graphs , 1976 .

[51]  Adam Korányi.  Harmonic Functions on Trees and Buildings , 1997 .

[52]  A. Kolla.  Angles , 2020, Encyclopedic Dictionary of Archaeology.

[53]  F. Brown.  Feynman amplitudes, coaction principle, and cosmic Galois group , 2017 .

[54]  On Motives Associated to Graph Polynomials , 2005, math/0510011.

[55]  T. Willwacher,et al.  Differentials on graph complexes , 2014, 1411.2369.