Kontsevich Graphons
Download PDF
Authors: A. SHOJAEI-FARD
DOI: 10.46793/KgJMat2302.213S
Abstract:
The article applies graph functions to extend the Kontsevich differential graded Lie algebraic formalism (in Deformation Quantization) to infinite Kontsevich graphs on the basis of the Connes-Kreimer Hopf algebraic renormalization and the theory of noncommutative differential geometry.
Keywords:
Graphons, Kontsevich’s admissible graphs, renormalization Hopf algebra, noncommutative differential calculus, Maurer-Cartan equations.
References:
[1] B. Bollobas and O. Riordan, Metrics for Sparse Graphs, Surveys in Combinatorics 2009, LMS Lecture Notes Series 365, Cambridge Univ. Press, Cambridge, 2009, 211–288. https://doi.org/10.1017/CBO9781107325975.009
[2] C. Borgs, J. T. Chayes, H. Cohn and N. Holden, Sparse exchangeable graphs and their limits via graphon processes, J. Mach. Learn. Res. 18(210), 1–71, 2018. https://www.jmlr.org/papers/v18/16-421.html
[3] C. Borgs, J. T. Chayes, H. Cohn and Y. Zhao, An Lp theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions, Trans. Amer. Math. Soc. 372(5) (2019), 3019–3062. https://doi.org/10.1090/tran/7543
[4] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199(1) (1998), 203–242. https://doi.org/10.1007/s002200050499
[5] C. Chryssomalakos, H. Quevedo, M. Rosenbaum and J. D. Vergara, Normal coordinates and primitive elements in the Hopf algebra of renormalization, Commun. Math. Phys. 225(3) (2002), 465–485. https://doi.org/10.1007/s002200100588
[6] M. Dubois-Violette, Lectures on graded differential algebras and noncommutative geometry, in: Noncommutative Differential Geometry and Its Applications to Physics, Mathematical Physics Studies 23, Springer, Dordrecht, 2001, 245–306. https://doi.org/10.1007/978-94-010-0704-7\_15
[7] D. Fiorenza and L. M. Ionescu, Graph complexes in deformation quantization, Lett. Math. Phys. 73(3) (2005), Article ID 193. https://doi.org/10.1007/s11005-005-0017-7
[8] L. M. Ionescu, Cohomology of Feynman graphs and perturbative quantum field theory, in: O. Kovras (Ed.), Focus on Quantum Field Theory, Nova Publishers Inc., 2004.
[9] S. Janson, Graphons, cut norm and distance, couplings and rearrangements, NYJM Monographs 4 (2013), 76 pages.
[10] D. Kreimer, Factorization in quantum field theory: an exercise in Hopf algebras and local singularities, in: Cartier P., Moussa P., Julia B., Vanhove P. (eds) Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, 715–736. https://doi.org/10.1007/978-3-540-30308-4\_14
[11] D. Kreimer, Unique factorization in pertubative QFT, Nuclear Physics B - Proceedings Supplements 116 (2003), 392–396. https://doi.org/10.1016/S0920-5632(03)80206-2
[12] V. Kathotia, Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, Internat. J. Math. 11(4) (2000), 523–551. https://doi.org/10.1142/S0129167X0000026X
[13] M. Kontsevich, Formal (non)commutative symplectic geometry, in: The Gelfand Mathematical Seminars 1990–1992, Birkhauser, Boston, Basel, Berlin, 1993, 173–187. https://doi.org/10.1007/978-1-4612-0345-2\_11
[14] L. Lovasz, Large Networks and Graph Limits, American Mathematical Society Colloquium Publications 60, Amer. Math. Soc., Providence, RI, 2012. https://doi.org/10.1090/coll/060
[15] J. Nesetril and P. Ossona de Mendez, Sparsity-Graphs, Structures, and Algorithms, Algorithms and Combinatorics 28, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-27875-4
[16] M. Rosenbaum and J. D. Vergara, The Hopf algebra of renormalization, normal coordinates and Kontsevich deformation quantization, J. Phys. A 37 (2004), 7939–7954. https://doi.org/10.1088/0305-4470/37/32/008
[17] A. Shojaei-Fard, Formal aspects of non-perturbative quantum field theory via an operator theoretic setting, Int. J. Geom. Methods Mod. Phys. 16(12) (2019), Paper ID 1950192, 23 pages. https://doi.org/10.1142/S0219887819501925
[18] A. Shojaei-Fard, Non-perturbative β-functions via Feynman graphons, Modern Phys. Lett. A 34(14) (2019), Paper ID 1950109, 10 pages. https://doi.org/10.1142/S0217732319501098
[19] A. Shojaei-Fard, Graphons and renormalization of large Feynman diagrams, Opuscula Math. 38(3) (2018), 427–455. https://doi.org/10.7494/OpMath.2018.38.3.427
[20] A . Shojaei-Fard, From Dyson-Schwinger equations to the Riemann-Hilbert correspondence, Int. J. Geom. Methods Mod. Phys. 7(4) (2010), 519–538. https://doi.org/10.1142/S0219887810004427
[21] W. D. van Suijlekom, Renormalization of gauge fields: a Hopf algebra approach, Commun. Math. Phys. 276(3) (2007), 773–798. https://doi.org/10.1007/s00220-007-0353-9
Login:
