Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 209–224 | Cite as

Hardy spaces and boundary conditions from the Ising model

Article

Abstract

Functions in Hardy spaces on multiply-connected domains in the plane are given an explicit characterization in terms of a boundary condition inspired by the two-dimensional Ising model. The key underlying property is the positivity of a certain operator constructed inductively on the number of components of the boundary.

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References

  1. 1.
    Belavin A., Polyakov A., Zamolodchikov A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333 (1984)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D’Hoker E., Phong D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917–1065 (1988)CrossRefMathSciNetGoogle Scholar
  3. 3.
    D’Hoker E., Phong D.H.: Conformal scalar fields and chiral splitting on super Riemann surfaces. Commun. Math. Phys 125, 469–513 (1989)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Date, E., Jimbo, M., Kashiwara, M., Miwa, T. Transformation groups for soliton equations. In: Jimbo, M., Miwa, T. (eds.) Proceedings of RIMS Symposium, pp. 39–120, World Scientific, Singapore (1983)Google Scholar
  5. 5.
    Di Francesco P., Mathieu P., Senechal D.: Conformal Field Theory. Springer, New York (1997)MATHCrossRefGoogle Scholar
  6. 6.
    Earle C.J., Marden A.: On Poincaré Series with application to H p spaces on bordered Riemann surfaces. Ilinois J. Math. 13, 202–219 (1969)MATHMathSciNetGoogle Scholar
  7. 7.
    Fay, J.: Theta functions on Riemann surfaces. Lecture Notes Series, vol. 352, Springer, New York (1973)Google Scholar
  8. 8.
    Fisher S.D.: Function Theory on Planar Domains: A Second Course in Complex Analysis. Dover Publications, New York (2007)MATHGoogle Scholar
  9. 9.
    Hongler, C.: Conformal invariance of Ising model correlations, PhD thesis, University of Geneva (2010)Google Scholar
  10. 10.
    Hongler, C., Kytölä, K.: Ising interfaces and free boundary conditions. arXiv:1108.0643Google Scholar
  11. 11.
    Hongler, C., Smirnov, S.: The energy density in the planar Ising model. Acta Math. arXiv:1008.2645 (to appear)Google Scholar
  12. 12.
    Izyurov, K., Holomorphic spinor observables and interfaces in the critical Ising model, PhD thesis, University of Geneva, (2011)Google Scholar
  13. 13.
    Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (based on the second Russian edition published in 1946) (1953)Google Scholar
  14. 14.
    Schramm, O.: Conformally invariant scaling limits: an overview and a collection of problems. In: International Congress of Mathematicians, vol. I, pp. 513–554. European Mathematical Society, Zurich (2007)Google Scholar
  15. 15.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Sanz-Solé, M., et al. (eds.) Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, Vol. II: Invited lectures, pp. 1421–1451 European Mathematical Society (EMS), Zurich (2006)Google Scholar
  16. 16.
    Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172(2), 1435–1467 (2010)MATHCrossRefGoogle Scholar
  17. 17.
    Smirnov, S.: Discrete complex analysis and probability. Proceedings of the ICM, Hyderabad, India (to appear)Google Scholar
  18. 18.
    Zygmund A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1968)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

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