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Lectures on Feynman Categories

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2016 MATRIX Annals

Part of the book series: MATRIX Book Series ((MXBS,volume 1))

Abstract

These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical way to encode operations and relations. This includes the aspects of operad-like theories such as PROPs, modular operads, twisted (modular) operads, properads, hyperoperads and their colored versions. There is more depth to the general theory as it applies as well to algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI-modules. Through decorations and transformations the theory is also related to the geometry of moduli spaces. Furthermore the morphisms in a Feynman category give rise to Hopf- and bi-algebras with examples coming from topology, number theory and quantum field theory. All these aspects are covered.

Workshop on higher structures at MATRIX in Creswick, June 7–9, 2016.

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References

  1. Barannikov, S.: Modular operads and Batalin-Vilkovisky geometry. Int. Math. Res. Not. IMRN (19), Art. ID rnm075, 31 (2007)

    Google Scholar 

  2. Batanin, M., Berger, C.: Homotopy theory for algebras over polynomial monads (2013). http://arxiv.org/abs/1305.0086

  3. Baues, H.J.: The double bar and cobar constructions. Compos. Math. 43(3), 331–341 (1981)

    Google Scholar 

  4. Berger, C., Kaufmann, R.M.: Comprehensive factorisation systems. Special issue in honor of Professors Peter J. Freyd and F. William Lawvere on the occasion of their 80th birthdays. Tbillisi Math. J. 10(3), 255–277 (2017)

    Google Scholar 

  5. Berger, C., Kaufmann, R.M.: Derived Feynman categories and modular geometry, (2018)

    Google Scholar 

  6. Borisov, D.V., Manin, Y.I.: Generalized operads and their inner cohomomorphisms. In: Geometry and Dynamics of Groups and Spaces. Progress in Mathematics, vol. 265 pp. 247–308. Birkhäuser, Basel (2008)

    Google Scholar 

  7. Brown, F.: Feynman amplitudes and cosmic Galois group (2015). Preprint. arxiv:1512.06409

    Google Scholar 

  8. Chapoton, F.: On some anticyclic operads. Algebr. Geom. Topol. 5, 53–69 (electronic) (2005)

    Article  MathSciNet  Google Scholar 

  9. Church, T., Ellenberg, J.S., Farb, B.: FI-modules: a new approach to stability for S n -representations (2012). arXiv:1204.4533

    Google Scholar 

  10. Conant, J., Vogtmann, K.: On a theorem of Kontsevich. Algebr. Geom. Topol. 3, 1167–1224 (2003)

    Article  MathSciNet  Google Scholar 

  11. Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199(1), 203–242 (1998)

    Article  MathSciNet  Google Scholar 

  12. Foissy, L.: Les algèbres de Hopf des arbres enracinés décorés. II. Bull. Sci. Math. 126(4), 249–288 (2002)

    Google Scholar 

  13. Fresse, B.: Props in model categories and homotopy invariance of structures. Georgian Math. J. 17(1), 79–160 (2010)

    Google Scholar 

  14. Gálvez-Carrillo, I., Kaufmann, R.M., Tonks, A.: Hopf algebras from cooperads and Feynman categories (2016). arXiv:1607.00196

    Google Scholar 

  15. Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. of Math. (2) 78, 267–288 (1963)

    Article  MathSciNet  Google Scholar 

  16. Getzler, E.: Operads revisited. In: Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. I. Progress in Mathematics, vol. 269 pp. 675–698. Birkhäuser, Boston, MA (2009)

    Chapter  Google Scholar 

  17. Getzler, E., Jones J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces (1994). http://arxiv.org/abs/hep-th/9403055

  18. Getzler, E., Kapranov, M.M.: Cyclic operads and cyclic homology. In: Geometry, Topology, & Physics, Conference Proceedings and Lecture Notes in Geometry and Topology, vol. IV, pp. 167–201. International Press, Cambridge, MA (1995)

    Google Scholar 

  19. Getzler, E., Kapranov, M.M.: Modular operads. Compos. Math. 110(1), 65–126 (1998)

    Article  MathSciNet  Google Scholar 

  20. Goncharov, A.B.: Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128(2), 209–284 (2005)

    Article  MathSciNet  Google Scholar 

  21. Harrelson, E., Voronov, A.A., Javier Zúñiga, J.: Open-closed moduli spaces and related algebraic structures. Lett. Math. Phys. 94(1), 1–26 (2010)

    Article  MathSciNet  Google Scholar 

  22. Joni, S.A., Rota, G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61(2), 93–139 (1979)

    Article  MathSciNet  Google Scholar 

  23. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993)

    Article  MathSciNet  Google Scholar 

  24. Kapranov, M., Manin, Yu.: Modules and Morita theorem for operads. Am. J. Math. 123(5), 811–838 (2001)

    Article  MathSciNet  Google Scholar 

  25. Kaufmann, R.M.: Feynman categories in quantum field theory (in progress)

    Google Scholar 

  26. Kaufmann, R.M.: Operads, moduli of surfaces and quantum algebras. In: Woods Hole Mathematics. Ser. Knots Everything, vol. 34, pp. 133–224. World Scientific Publishing, Hackensack, NJ (2004)

    Google Scholar 

  27. Kaufmann, R.M.: Moduli space actions on the Hochschild co-chains of a Frobenius algebra. I. Cell operads. J. Noncommut. Geom. 1(3), 333–384 (2007)

    Article  MathSciNet  Google Scholar 

  28. Kaufmann, R.M.: Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II. Correlators. J. Noncommut. Geom. 2(3), 283–332 (2008)

    Article  MathSciNet  Google Scholar 

  29. Kaufmann, R.M.: Dimension vs. genus: a surface realization of the little k-cubes and an E ∞ operad. In: Algebraic Topology—Old and New. Banach Center Publications, vol. 85, pp. 241–274. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2009)

    Google Scholar 

  30. Kaufmann, R., Lucas, J.: Decorated Feynman categories. J. Noncommut. Geom. 11(4), 1437–1464 (2017)

    Article  MathSciNet  Google Scholar 

  31. Kaufmann, R.M., Penner, R.C.: Closed/open string diagrammatics. Nuclear Phys. B 748(3), 335–379 (2006)

    Article  MathSciNet  Google Scholar 

  32. Kaufmann, R.M., Schwell, R.: Associahedra, cyclohedra and a topological solution to the A ∞ Deligne conjecture. Adv. Math. 223(6), 2166–2199 (2010)

    Article  MathSciNet  Google Scholar 

  33. Kaufmann, R.M., Ward, B.C.: Feynman categories. Astérisque 387, x+161 pp. (2017)

    Google Scholar 

  34. Kaufmann, R.M., Livernet, M., Penner, R.C.: Arc operads and arc algebras. Geom. Topol. 7, 511–568 (electronic) (2003)

    Article  MathSciNet  Google Scholar 

  35. Kaufmann, R.M., Ward, B.C., Javier Zúñiga, J.: The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations. J. Math. Phys. 56(10), 103504, 40 (2015)

    Article  MathSciNet  Google Scholar 

  36. Kimura, T., Stasheff, J., Voronov, A.A.: On operad structures of moduli spaces and string theory. Commun. Math. Phys. 171(1), 1–25 (1995)

    Article  MathSciNet  Google Scholar 

  37. Kontsevich, M.: Formal (non)commutative symplectic geometry. In: The Gel’fand Mathematical Seminars, 1990–1992, pp. 173–187. Birkhäuser, Boston, MA (1993)

    Chapter  Google Scholar 

  38. Leroux, P.: Les catégories de Möbius. Cahiers Topologie Géom. Différentielle 16(3), 280–282 (1975)

    MathSciNet  MATH  Google Scholar 

  39. Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, 2nd edn., vol. 5. Springer, New York (1998)

    Google Scholar 

  40. Manin, Y.I.: Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society Colloquium Publications, vol. 47. American Mathematical Society, Providence, RI (1999)

    Google Scholar 

  41. Markl, M.: Operads and PROPs. In: Handbook of Algebra, vol. 5, pp. 87–140. Elsevier/North-Holland, Amsterdam (2008)

    Google Scholar 

  42. Markl, M.: Modular envelopes, OSFT and nonsymmetric (non-σ) modular operads (2014). arXiv:1410.3414

    Google Scholar 

  43. Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol. 96. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  44. Markl, M., Merkulov, S., Shadrin, S.: Wheeled PROPs, graph complexes and the master equation. J. Pure Appl. Algebra 213(4), 496–535 (2009)

    Article  MathSciNet  Google Scholar 

  45. Merkulov, S., Vallette, B.: Deformation theory of representations of prop(erad)s. I. J. Reine Angew. Math. 634, 51–106 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Munkres, J.R.: Topology: A First Course. Prentice-Hall, Englewood Cliffs, NJ (1975)

    MATH  Google Scholar 

  47. Schwarz, A.: Grassmannian and string theory. Commun. Math. Phys. 199(1), 1–24 (1998)

    Article  MathSciNet  Google Scholar 

  48. Sen, A., Zwiebach, B.: Quantum background independence of closed-string field theory. Nucl. Phys. B 423(2–3), 580–630 (1994)

    Article  MathSciNet  Google Scholar 

  49. Vallette, B.: A Koszul duality for PROPs. Trans. Am. Math. Soc. 359(10), 4865–4943 (2007)

    Article  MathSciNet  Google Scholar 

  50. Ward, B.C.: Six operations formalism for generalized operads. Preprint. arxiv.org/abs/1701.01374

    Google Scholar 

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Acknowledgements

I thankfully acknowledge my co-authors with whom it has been a pleasure to work. I furthermore thank the organizers of the MATRIX workshop for providing the opportunity to give these lectures and for arranging the special issue.

The work presented here has at various stages been supported by the Humboldt Foundation, the Institute for Advanced Study, the Max–Planck Institute for Mathematics, the IHES and by the NSF. Current funding is provided by the Simons foundation.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907, USA

    Ralph M. Kaufmann

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  1. Ralph M. Kaufmann
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Correspondence to Ralph M. Kaufmann .

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Editors and Affiliations

  1. University of Melbourne, Melbourne, Australia

    Jan de Gier

  2. University of Western Australia, Crawley, Australia

    Cheryl E. Praeger

  3. UCLA, Los Angeles, USA

    Terence Tao

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Kaufmann, R.M. (2018). Lectures on Feynman Categories. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_19

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