Skip to main content
SpringerLink
Log in

Renormalization Group and the Ricci Flow

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

We discuss from a geometric point of view the connection between the renormalization group flow for non–linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Anderson and B. Chow, A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds,Calculus of Variations and Partial Differential Equations, Volume 23, 1–12 (2005).

  2. I. Bakas, Geometric flows and (some of) their physical applications, AvH conference Advances in Physics and Astrophysics of the 21st Century, 6–11 September 2005, Varna, Bulgaria, hep-th/0511057.

  3. I. Bakas and C. Sourdis, Dirichlet sigma models and mean curvature flow, JHEP0706:057, (2007), arXiv:0704.3985

  4. I. Bakas, Renormalization group equations and geometric flows, arXiv:hep-th/0702034.

  5. I. Bakas, S. Kong and L. Ni, Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations, arXiv:0906.0589.

  6. H.-D. Cao, Recent progress on Ricci solitons, to appear in Proceedings of International Conference on Geometric Analysis (Taipei, July 2007), arXiv:0908.2006.

  7. Cappelli A., Friedan D., Latorre J.I.: c-theorem and spectral representation. Nuclear Physics B, 352, 616–670 (1991)

    Article  MathSciNet  Google Scholar 

  8. M. Carfora, The Conjugate Linearized Ricci Flow on Closed 3-Manifolds, Ann. Scuola Normale Superiore di Pisa (in press), arXiv:0710.3342

  9. Carfora M., Romano S.: Quantum fluctuations and geometry: from graph counting to Ricci flow. Reports on Mth. Phys. 64, 185–203 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Chow and D. Knopf, The Ricci Flow: An Introduction, Math. Surveys and Monographs 110, (2004) Am. Math. Soc.

  11. B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, Vol. 77, (2006) Am. Math. Soc.

  12. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and L. Ni, The Ricci Flow: Techniques and Applications: PartI: Geometric Aspects, Math. Surveys and Monographs Vol. 135, (2007) Am. Math. Soc.

  13. DeTurck D.: Deforming metrics in the direction of their Ricci tensor. J. Differential Geom. 18, 157–162 (1983)

    MATH  MathSciNet  Google Scholar 

  14. M. R. Douglas, Spaces of quantum field theories, Lectures at Erice 2009, unpublished.

  15. Driver B.K.: A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fateev V.A., Onofri E., Zamolodchikov A.B.: The sausage model. Nucl. Phys. B 406, 521 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Friedan, Nonlinear Models in \({2,\,\pm\epsilon}\) Dimensions, Ph. D. Thesis (Berkeley) LBL-11517, UMI-81-13038, Aug 1980. 212pp.

  18. Friedan D., Nonlinear Models in \({2\,\pm\,\epsilon}\) Dimensions, Phys.Rev.Lett. 45 1057 (1980).

    Google Scholar 

  19. Friedan D.: Nonlinear Models in \({2\,\pm\,\epsilon}\) Dimensions. Ann. of Physics 163, 318–419 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. K. Gawedzki, Lectures on conformal field theory, available at http://www.math.ias.edu/QFT/fall/, or see pp. 727–805 in P. Deligne, D. Freed, (eds.), Classical field theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Providence, American Mathematical Society (1999).

  21. J. Gegenberg and V. Suneeta, The Fixed Points of RG Flow with a Tachyon, JHEP 0609 (2006) 045, arXiv:hep-th/0605230.

  22. Gilkey P.B.: Invariance Theory, The Heat Equation, and The Atiyah–Singer Index Theorem, (2nd. ed.). CRC Press, Boca Raton, Florida (1994)

    Google Scholar 

  23. Guenther C., Oliynyk T.A.: Renormalization Group Flow for Nonlinear Sigma Models. Lett.Math.Phys. 84, 149–157 (2008) arXiv:0810.3954

    Article  MATH  MathSciNet  Google Scholar 

  24. Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)

    MATH  MathSciNet  Google Scholar 

  25. R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry 2, International Press, (1995) 7–136.

  26. D. Knopf, Estimating the trace-free Ricci tensor in Ricci flow, ArXiv:0711.1153.

  27. Leandre R.: Stochastic Wess–Zumino–Novikov–Witten model on the torus. J. Math. Phys. 44, 5530–5568 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lichnerowicz A.: Propagateurs et commutateurs en relativité générale. Pub. Math. de l’I.E.H.S. 10, 5 (1961)

    Google Scholar 

  29. Ni L.: Closed type-I ancient solutions to Ricci flow, Recent Advances in Geometric Analysis. Advanced Lectures in Mathematics 11, 147–150 (2009)

    Google Scholar 

  30. Ni L., Wallach N.: On a classification of gradient shrinking solitons. Math. Res. Lett. 15, 10001–10015 (2008)

    MathSciNet  Google Scholar 

  31. T. Oliynyk, V. Suneeta and E. Woolgar, A Gradient Flow for Worldsheet Nonlinear Sigma Models, Nucl.Phys. B739 (2006) 441–458, hep-th/0510239.

  32. G. Perelman, The entropy formula for the Ricci flow and its geometric applications math.DG/0211159

  33. G. Perelman, Ricci flow with surgery on Three-Manifolds math.DG/0303109

  34. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain threemanifolds math.DG/0307245

  35. N. Sesum, Curvature tensor under the Ricci flow, arXiv:math.DG/0311397.

  36. Shore G.M.: A local renormalization group equation, diffeomorphisms, and conformal invariance in sigma models. Nucl. Phys. B 286, 349 (1987)

    Article  MathSciNet  Google Scholar 

  37. M. Simon, Deformation of C 0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no.5, 1033–1074.

  38. Taubes C.H.: Constructions of measures and quantum field theories on mapping spaces. J. Differential Geometry 70, 23–58 (2005)

    MATH  MathSciNet  Google Scholar 

  39. Thurston W.P.: Three-dimensiional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6, 357–381 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  40. W. P. Thurston, Three-dimensiional geometry and topology Vol. 1. Edited by S. Levy. Princeton Math. Series, 35 Princeton Univ. Press, Princeton NJ, (1997).

  41. A.A. Tseytlin, On sigma model RG flow, “central charge” action and Perelman’s entropy, Phys.Rev. D75:064024,(2007), arXiv:hep-th/0612296.

  42. Fradkin E.S., Tseytlin A.A.: Effective field theory from quantized strings. Phys. Lett. B 158, 316 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  43. Fradkin E.S., Tseytlin A.A.: Quantum string theory effective action. Nucl. Phys. B 261, 1 (1985)

    Article  MathSciNet  Google Scholar 

  44. A.A. Tseytlin, Conformal anomaly in two-dimensional sigma model on curved background and strings, Phys. ¡lett. 178B, (1986) 34.

    Google Scholar 

  45. Tseytlin A.A.: Sigma model Weyl invariance conditions and string equations of motion. Nucl. Phys. B 294, 383 (1987)

    Article  MathSciNet  Google Scholar 

  46. Weitsman J.: Measures on Banach manifolds and supersymmetric quantum field theories. Commun. Math. Phys. 277, 101–125 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  47. Zamolodchikov A.B.: Irreversibility of the flux of the renormalization group in a 2-D field theory. JEPT Lett. 43, 730 (1986)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica Nucleare e Teorica and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Bassi 6, 27100, Pavia, Italy

    Mauro Carfora

Authors
  1. Mauro Carfora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mauro Carfora.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carfora, M. Renormalization Group and the Ricci Flow. Milan J. Math. 78, 319–353 (2010). https://doi.org/10.1007/s00032-010-0110-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-010-0110-y

Mathematics Subject Classification (2010)

Keywords

Search

Navigation