Advertisement

Schramm–Loewner-evolution-type growth processes corresponding to Wess–Zumino–Witten theories

  • Shinji KoshidaEmail author
Article
  • 31 Downloads

Abstract

A group theoretical formulation of Schramm–Loewner-evolution-type growth processes corresponding to Wess–Zumino–Witten theories is developed that makes it possible to construct stochastic differential equations associated with more general null vectors than the ones considered in the most fundamental example in Alekseev et al. (Lett Math Phys 97:243–261, 2011). Also given are examples of Schramm–Loewner-evolution-type growth processes associated with null vectors of conformal weight 4 in the basic representations of \(\widehat{\mathfrak {sl}}_{2}\) and \(\widehat{\mathfrak {sl}}_{3}\).

Keywords

Schramm–Loewner evolution Wess–Zumino–Witten theory Conformal field theory 

Mathematics Subject Classification

60J67 81T40 17B67 

Notes

Acknowledgements

The author is grateful to K. Sakai and R. Sato for fruitful discussions. This work was supported by a Grant-in-Aid for JSPS Fellows (Grant No. 17J09658).

References

  1. 1.
    Alekseev, A., Bytsko, A., Izyurov, K.: On SLE martingales in boundary WZW models. Lett. Math. Phys. 97, 243–261 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Applebaum, D.: Probability on Compact Lie Groups. Probability Theory and Stochastic Modeling, vol. 70. Springer, Berlin (2014)zbMATHGoogle Scholar
  3. 3.
    Bauer, M., Bernard, D.: SLE\(_{\kappa }\) growth processes and conformal field theories. Phys. Lett. B 543, 135–138 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, M., Bernard, D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239, 493–521 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bauer, M., Bernard, D.: SLE martingales and the Virasoro algebra. Phys. Lett. B 557, 309–316 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bauer, M., Bernard, D.: Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5, 289–326 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120, 1125–1163 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bettelheim, E., Gruzberg, I.A., Ludwig, A.W.W., Wiegmann, P.: Stochastic Loewner evolution for conformal field theories with Lie group symmetries. Phys. Rev. Lett. 95, 251,601 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cardy, J.L.: Effect of boundary conditions on the operator center of two-dimensional conformally invariant theories. Nucl. Phys. B 275, 200–218 (1986)ADSCrossRefGoogle Scholar
  11. 11.
    Cardy, J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324, 581–596 (1989)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A Math. Gen. 25, L201–L206 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, S.: Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Math. 352, 157–161 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York (1997)Google Scholar
  15. 15.
    Dubédat, J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. 336, 761–809 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336, 695–760 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Methematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
  18. 18.
    Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Friedrich, R.: On connections of conformal field theory and stochastic Loewner evolution. ArXiv:math-ph/0410029 (2004)
  21. 21.
    Friedrich, R., Kalkkinen, J.: On conformal field theory and stochastic Loewner evolution. Nucl. Phys. B 687, 279–302 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fukusumi, Y.: Multiple Schramm–Loewner evolutions for coset Wess–Zumino–Witten models. ArXiv:1704.06006 (2017)
  23. 23.
    Kac, V.: Vertex Algebras for Beginners, University Lecture Series, vol. 10, 2nd edn. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  24. 24.
    Kontsevich, M.: CFT, SLE and phase boundaries. Oberwolfach Arbeitstagung (2003)Google Scholar
  25. 25.
    Koshida, S.: Local martingales associated with Schramm-Loewner evolutions with internal symmetry. J. Math. Phys. 59, 101,703 (2018). ArXiv:1803.06808
  26. 26.
    Kytölä, K.: Virasoro module structure of local martingales of SLE variants. Rev. Math. Phys. 5, 455–509 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lawler, G.F.: An introduction to the stochastic Loewner evolution. In: Kaimanovich, V.A. (ed.) Random Walks and Geometry. De Gruyter, Berlin (2004)Google Scholar
  28. 28.
    Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)Google Scholar
  29. 29.
    Lesage, F., Rasmussen, J.: SLE-type growth processes and the Yang–Lee singularity. J. Math. Phys. 45, 3040–3048 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Moghimi-Araghi, A., Rajabpour, M.A., Rouhani, S.: Logarithmic conformal null vectors and SLE. Phys. Lett. B 600, 298–301 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Nagi, J., Rasmussen, J.: On stochastic evolutions and superconformal field theory. Nucl. Phys. B 704, 475–489 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nazarov, A.: Schramm–Loewner evolution martingales in coset conformal field theory. JETP Lett. 96, 90–93 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    Rasmussen, J.: Note on stochastic Löwner evolutions and logarithmic conformal field theory. J. Stat. Mech. 2004, P09007 (2004)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rasmussen, J.: Stochastic evolutions in superspace and superconformal field theory. Lett. Math. Phys. 68, 41–52 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rasmussen, J.: On \(SU(2)\) Wess–Zumino–Witten models and stochastic evolutions. Afr. J. Math. Phys. 4, 1–9 (2007)zbMATHGoogle Scholar
  36. 36.
    Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sakai, K.: Multiple Schramm–Loewner evolutions for conformal field theories with Lie algebra symmetries. Nucl. Phys. B 867, 429–447 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333, 239–244 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Proceedings of the International Congress of Mathematicians (Madrid, August 22–30, 2006), pp. 1421–1451. European Mathematical Society, Zülich (2006)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Basic ScienceThe University of TokyoTokyoJapan

Personalised recommendations