Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

  • Mauro Carfora
  • Claudio DappiaggiEmail author
  • Nicolò Drago
  • Paolo Rinaldi


The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati, and Moretti for vector valued free fields. As a by-product of such classification, we provide a mathematically rigorous proof that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.



The work of C. D. was supported by the University of Pavia, while that of N. D. was supported in part by a research fellowship of the University of Pavia. We are grateful to Federico Faldino, Igor Khavkine, Alexander Schenkel and Jochen Zahn for the useful discussions. We are especially grateful to Klaus Fredenhagen for the enlightening discussions on the rôle of the algebra of functionals. This work is based partly on the MSc thesis of P. R. .


  1. [BRZ14]
    Bahns, D., Rejzner, K., Zahn, J.: The effective theory of strings. Commun. Math. Phys. 327, 779 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [BDLR18]
    Brouder, C., Dang, N.V., Laurent-Gengoux, C., Rejzner, K.: Properties of field functionals and characterization of local functionals. J. Math. Phys. 59(2), 023508 (2018). arXiv:1705.01937 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [BDFY15]
    Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J.: Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer (2015)Google Scholar
  4. [BDF09]
    Brunetti, R., Duetsch, M., Fredenhagen, K.: Perturbative algebraic quantum field theory and the renormalization groups. Adv. Theor. Math. Phys. 13(5), 1541 (2009). arXiv:0901.2038 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BF99]
    Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000). arXiv:math-ph/9903028 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [BFV03]
    Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: a new paradigm for local quantum field theory. Commun. Math. Phys. 237, 31 (2003). arXiv:math-ph/0112041 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [Car10]
    Carfora, M.: Renormalization group and the Ricci flow. Milan J. Math. 78, 319 (2010). arXiv:1001.3595 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Car14]
    Carfora, M.: The Wasserstein geometry of nonlinear \(\sigma \) models and the Hamilton-Perelman Ricci flow. Rev. Math. Phys. 29(01), 1750001 (2016). arXiv:1405.0827 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar
  9. [CG18]
    Carfora, M., Guenther, C.: Scaling and entropy for the RG-2 flow. arXiv:1805.09773v1 [math.DG]
  10. [CM17]
    Carfora, M., Marzuoli, A.: Quantum Triangulations. Lecture Notes in Physics, vol. 942. Springer, Berlin (2017)Google Scholar
  11. [Da14]
    Dang, N.V.: Extension of distributions, scalings and renormalization of QFT on Riemannian manifolds. arXiv:1411.3670 [math-ph]
  12. [DD16]
    Dappiaggi, C., Drago, N.: Constructing Hadamard states via an extended Møller operator. Lett. Math. Phys. 106(11), 1587 (2016). arXiv:1506.09122 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [DDR19]
    Dappiaggi, C., Drago, N., Rinaldi, P.: The algebra of Wick polynomials of a scalar field on a Riemannian manifold. arXiv:1903.01258 [math-ph]
  14. [DHP16]
    Drago, N., Hack, T.-P., Pinamonti, N.: The generalized principle of perturbative agreement and the thermal mass. Ann. Henri Poinc. 18(3), 807 (2017). arXiv:1502.02705 [math-ph]ADSCrossRefzbMATHGoogle Scholar
  15. [FR12]
    Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun. Math. Phys. 314, 93 (2012). arXiv:1101.5112 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [FR13]
    Fredenhagen, K., Rejzner, K.: Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory. Commun. Math. Phys. 317, 697 (2013). arXiv:1110.5232 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [Fri80]
    Friedan, D.: Nonlinear models in two epsilon dimensions. Phys. Rev. Lett. 45, 1057 (1980)ADSMathSciNetCrossRefGoogle Scholar
  18. [Fri85]
    Friedan, D.H.: Nonlinear models in two + epsilon dimensions. Ann. Phys. 163, 318 (1985)ADSCrossRefzbMATHGoogle Scholar
  19. [G98]
    Garabedian, P.R.: Partial Differential Equations. Chelsea Pub Co (1998)Google Scholar
  20. [Gaw99]
    Gawedzki, K.: Conformal field theory. In: Deligne, P., Etingof, P., Freed, D.D., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E. (eds.) Quantum Fields and Strings: A Course for Mathematicians, vol. 2. Institute for Advanced Study, AMS (1999)Google Scholar
  21. [HK63]
    Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [Her19]
    Herscovich, E.: Renormalization in Quantum Field Theory (after R. Borcherds). Astérisque (to appear).
  23. [Ham82]
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [HW01]
    Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved space–time. Commun. Math. Phys. 223, 289 (2001). arXiv:gr-qc/0103074 ADSCrossRefzbMATHGoogle Scholar
  25. [HW02]
    Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved space–time. Commun. Math. Phys. 231, 309 (2002). arXiv:gr-qc/0111108 ADSCrossRefzbMATHGoogle Scholar
  26. [HW03]
    Hollands, S., Wald, R.M.: On the renormalization group in curved space–time. Commun. Math. Phys. 237, 123 (2003). arXiv:gr-qc/0209029 ADSCrossRefzbMATHGoogle Scholar
  27. [HW05]
    Hollands, S., Wald, R.M.: Conservation of the stress tensor in interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227 (2005). arXiv:gr-qc/0404074 MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Hö03]
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I-Distribution Theory and Fourier Analysis. Springer, Berlin (2003)zbMATHGoogle Scholar
  29. [Hus94]
    Husemöller, D.: Fibre Bundles. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  30. [Kel10]
    Keller, K.J.: Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein–Glaser Renormalization. Ph.D. thesis, U. Hamburg (2009). arXiv:1006.2148 [math-ph]
  31. [Kel09]
    Keller, K.J.: Euclidean Epstein–Glaser renormalization. J. Math. Phys. 50, 103503 (2009). arXiv:0902.4789 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [KMM17]
    Khavkine, I., Melati, A., Moretti, V.: On Wick polynomials of boson fields in locally covariant algebraic QFT. Ann. Henri Poincaré 20, 929 (2019)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [KM16]
    Khavkine, I., Moretti, V.: Analytic dependence is an unnecessary requirement in renormalization of locally covariant QFT. Commun. Math. Phys. 344(2), 581 (2016). arXiv:1411.1302 [gr-qc]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [KMS93]
    Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  35. [Lin13]
    Lindner, F.: Perturbative Algebraic Quantum Field Theory at Finite Temperature. Ph.D. thesis, U. Hamburg (2013)Google Scholar
  36. [MS76]
    Milnor, J., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton (1974)zbMATHGoogle Scholar
  37. [Mor99a]
    Moretti, V.: Local zeta function techniques versus point splitting procedure: a few rigorous results. Commun. Math. Phys. 201, 327 (1999). arXiv:gr-qc/9805091 ADSCrossRefzbMATHGoogle Scholar
  38. [Mor99b]
    Moretti, V.: One loop stress tensor renormalization in curved background: the relation between zeta function and point splitting approaches, and an improved point splitting procedure. J. Math. Phys. 40, 3843 (1999). arXiv:gr-qc/9809006 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [NS14]
    Navarro, J., Sancho, J.B.: Peetre-Slovák Theorem Revisited. arXiv:1411.7499 [math.DG]
  40. [OS75]
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. 2. Commun. Math. Phys. 42, 281 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [OS73]
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s Functions. Commun. Math. Phys. 31, 83 (1973)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [Per02]
    Perelman, G.: The Entropy Formula for the Ricci Flow and Its Geometric Applications. arXiv:math/0211159 [math-dg]
  43. [Per03]
    Perelman, G.: Ricci Flow with Surgery on Three-Manifolds. arXiv:math/0303109 [math-dg]
  44. [PPV11]
    Poisson, E., Pound, A., Vega, I.: The motion of point particles in curved spacetime. Living Rev. Relativ 14, 7 (2011). arXiv:1102.0529 [gr-qc]ADSCrossRefzbMATHGoogle Scholar
  45. [Rej16]
    Rejzner, K.: Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer (2016)Google Scholar
  46. [Sch98]
    Schlingemann, D.: From Euclidean field theory to quantum field theory. Rev. Math. Phys. 11, 1151 (1999). arXiv:hep-th/9802035 MathSciNetCrossRefzbMATHGoogle Scholar
  47. [Shu87]
    Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  48. [Thu97]
    Thurston, W.P.: Three-dimensional geometry and topology, vol. 1. In: Levy, S. (ed.) Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)Google Scholar
  49. [Wa79]
    Wald, R.M.: On the Euclidean approach to quantum field theory in curved space–time. Commun. Math. Phys. 70, 221 (1979)ADSCrossRefGoogle Scholar
  50. [Wel08]
    Wells, R.O.: Differential Analysis on Complex Manifolds. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  51. [Za15]
    Zahn, J.: Locally covariant charged fields and background independence. Rev. Math. Phys. 27(07), 1550017 (2015). arXiv:1311.7661 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità di PaviaPaviaItaly
  2. 2.INFN, Sezione di PaviaPaviaItaly
  3. 3.Istituto Nazionale di Alta Matematica – Sezione di PaviaPaviaItaly

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