Skip to main content
SpringerLink
Log in

On the Periodicity Conjecture for Y-systems

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

Abstract

The conjecture in question (due ultimately to Alexei Zamolodchikov) asserts the periodicity of all the solutions to the so-called Y-systems. Those systems are naturally associated to pairs of indecomposable Cartan matrices of finite type, and the conjectured period is equal to twice the sum of the respective Coxeter numbers. This conjecture has so far been proven only if one of the ranks equals one, in which case the Y-systems are intrinsically related to Fomin-Zelevinsky’s cluster algebras. In this paper, I use elementary projective geometry to prove the case when the two Cartan matrices involved are of type A with both ranks arbitrary.

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. Zamolodchikov Al.B. (1991). On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253: 391–394

    Article  ADS  MathSciNet  Google Scholar 

  2. Kuniba A. and Nakanishi T. (1992). Spectra in Conformal Field Theories from the Rogers Dilogarithm. Mod. Phys. Lett. A 7: 3487–3494

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Ravanini F., Valleriani A. and Tateo R. (1993). Dynkin TBAs. Int. J. Mod. Phys. A 8: 1707–1727

    Article  ADS  MathSciNet  Google Scholar 

  4. Frenkel E. and Szenes A. (1995). Thermodynamic Bethe ansatz and dilogarithm identities. I. Math. Res. Lett. 2(6): 677–693

    MATH  MathSciNet  Google Scholar 

  5. Gliozzi F. and Tateo R. (1996). Thermodynamic Bethe ansatz and three-fold triangulations. Int. J. Mod. Phys. A 11(22): 4051–4064

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Fomin S. and Zelevinsky A. (2003). Y-systems and generalized associahedra. Ann. of Math. 158(3): 977–1018

    Article  MATH  MathSciNet  Google Scholar 

  7. Volkov, A.Yu.: On Zamolodchikov’s Periodicity Conjecture. http://arxive.org/list/hep-th/0606094, 2006

  8. Henriques, A.: A Periodicity Theorem for the Octahedron Recurrence. http://arxive.org/list/math.CO/0604289, 2006

Download references

Author information

Authors and Affiliations

  1. Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050, Brussels, Belgium

    Alexandre Yu. Volkov

  2. International Solvay Institutes, Campus Plaine (ULB) – CP 231, Bd du Triomphe, B-1050, Brussels, Belgium

    Alexandre Yu. Volkov

Authors
  1. Alexandre Yu. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre Yu. Volkov.

Additional information

Communicated by L. Takhtajan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Volkov, A.Y. On the Periodicity Conjecture for Y-systems. Commun. Math. Phys. 276, 509–517 (2007). https://doi.org/10.1007/s00220-007-0343-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0343-y

Keywords

Search

Navigation