Skip to main content
SpringerLink
Log in

Renormalization group flow, entropy and eigenvalues

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

The irreversibility of the renormalization group flow is conjectured to be closely related to the concept of entropy. In this paper, the variation of eigenvalues of the Laplacian in the Polyakov action under the renormalization group flow will be studied. Based on the one-loop approximation to the effective field theory, we will use the heat kernel method and zeta function regularization. In even dimensions, the variation of eigenvalues is given by the top heat kernel coefficient, and the conformal anomaly is relevant. In odd dimensions, we will conjecture a formula for the variation of eigenvalues through the holographic renormalization in the setting of geometric AdS/CFT correspondence.

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. Anderson, M.T.: Geometrical aspects of the AdS/CFT correspondence. In: Biquard, O. (ed), AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries, IRMA Lectures in Mathematics and Theoretical Physics 8, pp. 1–31. EMS (2005)

  2. Cao, X.D.: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136(11), 4075–4078 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cardy, J.L.: Is there a c-theorem in four dimensions? Phys. Lett. B 215(4), 749–752 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  4. Carfora, M.: Renormalization group and the Ricci flow. Milan J. Math. 78, 319–353 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Casini, H., Huerta, M., Myers, R.C.: Towards a derivation of holographic entanglement entropy. JHEP 05, 36 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. de Boer, J., Verlinde, E., Verlinde, H.: On the holographic renormalization group. JHEP 08, 03 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Di Cerbo, L.F.: Eigenvalues of the Laplacian under the Ricci flow. Rend. Mat. 27, 183–195 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Duff, M.J.: Twenty years of the Weyl anomaly. Class. Quantum Gravity 11, 1387 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Friedan, D.: Nonlinear models in 2 + \(\epsilon \) dimensions. Phys. Rev. Lett. 45, 1057 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  10. Friedan, D.: Nonlinear models in 2 + \(\epsilon \) dimensions. Ann. Phys. 163, 318–419 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Gawedzki, K.: Lectures on conformal field theory. In: Deligne, P. (ed.) Quantum Fields and Strings: A Course for Mathematicians, pp. 727–805. AMS, Providence (2000)

    Google Scholar 

  12. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B 428, 105–114 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Hawking, S.W.: Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys. 55(2), 133–148 (1977)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Komargodski, Z., Schwimmer, A.: On renormalization group flows in four dimensions. JHEP 12, 99 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Li, J.F.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338(4), 927–946 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, X.D.: From the Boltzmann H-theorem to Perelman’s W-entropy formula for the Ricci flow. In: Chen, H., Long, Y.M., Nishiura, Y. (eds.) Emerging Topics on Differential Equations and Their Applications, pp. 68–74. World Scientific, Singapore (2012)

    Google Scholar 

  17. Liu, H., Mezei, M.: A refinement of entanglement entropy and the number of degrees of freedom. JHEP 04, 162 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Ma, Li: Eigenvalue monotonicity for the Ricci-Hamilton flow. Ann. Glob. Anal. Geom. 29(3), 287–292 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Maldacena, J.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Morgan, J., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs. American Mathematical Society, Providence (2007)

    Google Scholar 

  21. Nakayama, Y.: Scale invariance vs conformal invariance. Phys. Rep. 569, 1–93 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. Papadimitriou, I., Skenderis, K.: AdS/CFT correspondence and geometry. In: Biquard O (ed) AdS/CFT Correspondence: Einstein Metrics and their Conformal Boundaries. IRMA Lectures in Mathematics and Theoretical Physics vol. 8, pp. 73–101. EMS (2005)

  23. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002). arXiv:math/0211159v1

  24. Skenderis, K.: Lecture notes on holographic renormalization. Class. Quantum Gravity 19, 5849–5876 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Vassilevich, D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388, 279–360 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Zamolodchikov, A.: Irreversibility of the flux of the renormalization group in a 2D field theory. JETP Lett. 43, 730–732 (1986)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Dan Li

Authors
  1. Dan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dan Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, D. Renormalization group flow, entropy and eigenvalues. Lett Math Phys 107, 2333–2357 (2017). https://doi.org/10.1007/s11005-017-0987-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-017-0987-2

Keywords

Mathematics Subject Classification

Search

Navigation