Letters in Mathematical Physics

, Volume 97, Issue 3, pp 243–261 | Cite as

On SLE Martingales in Boundary WZW Models

  • Anton AlekseevEmail author
  • Andrei Bytsko
  • Konstantin Izyurov


Following Bettelheim et al. (Phys Rev Lett 95:251601, 2005), we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm–Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik–Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G = SU(2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G = SU(n) and the level k = 1, there are several situations when boundary one-point correlators are SLE κ -martingales. If the boundary field is labelled by the defining n-dimensional representation of SU(n), we obtain \({\varkappa=2}\) . For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get \({\varkappa=8/(n {+} 2)}\) . We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the \({{\rm SLE}_\varkappa}\) evolution is replaced by \({{\rm SLE}_{\varkappa,\rho}}\) with \({\rho=\varkappa -6}\) .

Mathematics Subject Classification (2000)

60J67 81T40 


SLE boundary conformal field theory WZW model 


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Copyright information

© Springer 2011

Authors and Affiliations

  • Anton Alekseev
    • 1
    Email author
  • Andrei Bytsko
    • 2
    • 3
  • Konstantin Izyurov
    • 1
    • 3
  1. 1.Section of MathematicsUniversity of GenevaGeneva 4Switzerland
  2. 2.Steklov Mathematics InstituteSt. PetersburgRussia
  3. 3.Chebyshev LaboratorySt.-Petersburg State UniversitySaint-PetersburgRussia

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