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

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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 \(\widehat{\mathfrak {sl}}_{3}\).

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References

  1. Alekseev, A., Bytsko, A., Izyurov, K.: On SLE martingales in boundary WZW models. Lett. Math. Phys. 97, 243–261 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Applebaum, D.: Probability on Compact Lie Groups. Probability Theory and Stochastic Modeling, vol. 70. Springer, Berlin (2014)

    MATH  Google Scholar 

  3. Bauer, M., Bernard, D.: SLE\(_{\kappa }\) growth processes and conformal field theories. Phys. Lett. B 543, 135–138 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bauer, M., Bernard, D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239, 493–521 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bauer, M., Bernard, D.: SLE martingales and the Virasoro algebra. Phys. Lett. B 557, 309–316 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bauer, M., Bernard, D.: Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5, 289–326 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120, 1125–1163 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  11. Cardy, J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324, 581–596 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  12. Cardy, J.L.: Critical percolation in finite geometries. J. Phys. A Math. Gen. 25, L201–L206 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York (1997)

    Google Scholar 

  15. Dubédat, J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. 336, 761–809 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336, 695–760 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Methematical Society, Providence (2004)

    Book  MATH  Google Scholar 

  18. Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedrich, R.: On connections of conformal field theory and stochastic Loewner evolution. ArXiv:math-ph/0410029 (2004)

  21. Friedrich, R., Kalkkinen, J.: On conformal field theory and stochastic Loewner evolution. Nucl. Phys. B 687, 279–302 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Fukusumi, Y.: Multiple Schramm–Loewner evolutions for coset Wess–Zumino–Witten models. ArXiv:1704.06006 (2017)

  23. Kac, V.: Vertex Algebras for Beginners, University Lecture Series, vol. 10, 2nd edn. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  24. Kontsevich, M.: CFT, SLE and phase boundaries. Oberwolfach Arbeitstagung (2003)

  25. Koshida, S.: Local martingales associated with Schramm-Loewner evolutions with internal symmetry. J. Math. Phys. 59, 101,703 (2018). ArXiv:1803.06808

  26. Kytölä, K.: Virasoro module structure of local martingales of SLE variants. Rev. Math. Phys. 5, 455–509 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)

    Google Scholar 

  29. Lesage, F., Rasmussen, J.: SLE-type growth processes and the Yang–Lee singularity. J. Math. Phys. 45, 3040–3048 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Moghimi-Araghi, A., Rajabpour, M.A., Rouhani, S.: Logarithmic conformal null vectors and SLE. Phys. Lett. B 600, 298–301 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Nagi, J., Rasmussen, J.: On stochastic evolutions and superconformal field theory. Nucl. Phys. B 704, 475–489 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Nazarov, A.: Schramm–Loewner evolution martingales in coset conformal field theory. JETP Lett. 96, 90–93 (2012)

    Article  ADS  Google Scholar 

  33. Rasmussen, J.: Note on stochastic Löwner evolutions and logarithmic conformal field theory. J. Stat. Mech. 2004, P09007 (2004)

    Article  MATH  Google Scholar 

  34. Rasmussen, J.: Stochastic evolutions in superspace and superconformal field theory. Lett. Math. Phys. 68, 41–52 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Rasmussen, J.: On \(SU(2)\) Wess–Zumino–Witten models and stochastic evolutions. Afr. J. Math. Phys. 4, 1–9 (2007)

    MATH  Google Scholar 

  36. Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sakai, K.: Multiple Schramm–Loewner evolutions for conformal field theories with Lie algebra symmetries. Nucl. Phys. B 867, 429–447 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333, 239–244 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

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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).

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  1. Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, Japan

    Shinji Koshida

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  1. Shinji Koshida
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Correspondence to Shinji Koshida.

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Koshida, S. Schramm–Loewner-evolution-type growth processes corresponding to Wess–Zumino–Witten theories. Lett Math Phys 109, 1397–1413 (2019). https://doi.org/10.1007/s11005-018-01150-y

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  • DOI: https://doi.org/10.1007/s11005-018-01150-y

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