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SLE and Virasoro Representations: Localization

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Abstract

We consider some probabilistic and analytic realizations of Virasoro highest-weight representations. Specifically, we consider measures on paths connecting points marked on the boundary of a (bordered) Riemann surface. These Schramm–Loewner evolution-type measures are constructed by the method of localization in path space. Their partition function (total mass) is the highest-weight vector of a Virasoro representation, and the action is given by Virasoro uniformization.

We review the formalism of Virasoro uniformization, which allows to define a canonical action of Virasoro generators on functions (or sections) on a—suitably extended—Teichmüller space. Then we describe the construction of families of measures on paths indexed by marked bordered Riemann surfaces. Finally we relate these two notions by showing that the partition functions of the latter generate a highest-weight representation—the quotient of a reducible Verma module—for the former.

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References

  1. Alvarez O.: Theory of strings with boundaries: fluctuations, topology and quantum geometry. Nucl. Phys. B 216(1), 125–184 (1983)

    Article  ADS  Google Scholar 

  2. Alvarez-Gaumé L., Bost J.-B., Moore G., Nelson P., Vafa C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112(3), 503–552 (1987)

    Article  ADS  MATH  Google Scholar 

  3. Bauer M., Bernard D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239(3), 493–521 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Bauer M., Bernard D.: SLE martingales and the Virasoro algebra. Phys. Lett. B 557(3–4), 309–316 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Bauer M., Bernard D.: Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5(2), 289–326 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Bauer M., Bernard D., Kytölä K.: Multiple Schramm–Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6), 1125–1163 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Bauer R.O., Friedrich R.M.: On chordal and bilateral SLE in multiply connected domains. Math. Z. 258(2), 241–265 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Beffara, V.: Mouvement Brownien plan, SLE, invariance conforme et dimensions fractales. PhD thesis, Université Paris-Sud (2003)

  9. Beĭlinson A.A., Schechtman V.V.: Determinant bundles and Virasoro algebras. Commun. Math. Phys. 118(4), 651–701 (1988)

    Article  ADS  MATH  Google Scholar 

  10. Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Bony, J.-M.: Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble), 19(fasc. 1):277–304 xii (1969)

  12. Burghelea D., Friedlander L., Kappeler T.: Meyer–Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107(1), 34–65 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cardy J.L.: Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories. Nucl. Phys. B 275(2), 200–218 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  14. Cardy J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324(3), 581–596 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  15. Chandrasekharan K.: Elliptic Functions. Grundlehren der Mathematischen Wissenschaften, vol. 281. Springer, Berlin (1985)

    Google Scholar 

  16. Doyon, B.: Higher conformal variations and the Virasoro vertex operator algebra. ArXiv e-prints (2011)

  17. Doyon, B.: Calculus on manifolds of conformal maps and CFT. J. Phys. A 45(31):315202 (2012)

  18. Doyon B.: Conformal loop ensembles and the stress–energy tensor. Lett. Math. Phys. 103(3), 233–284 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Doyon, B.: Hypotrochoids in conformal restriction systems and Virasoro descendants. J. Stat. Mech. Theory Exp. (9):P09008 (2013)

  20. Doyon, B.: Random loops and conformal field theory. J. Stat. Mech. Theory Exp (2):P02015 (2014)

  21. Doyon B., Riva V., Cardy J.: Identification of the stress–energy tensor through conformal restriction in SLE and related processes. Commun. Math. Phys. 268(3), 687–716 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Dubédat J.: Critical percolation in annuli and SLE6. Commun. Math. Phys. 245(3), 627–637 (2004)

    Article  ADS  MATH  Google Scholar 

  23. Dubédat J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Dubédat J.: Commutation relations for Schramm–Loewner evolutions. Commun. Pure Appl. Math. 60(12), 1792–1847 (2007)

    Article  MATH  Google Scholar 

  25. Dubédat J.: Duality of Schramm–Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 697–724 (2009)

    MATH  Google Scholar 

  26. Dubédat J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22(4), 995–1054 (2009)

    Article  MATH  Google Scholar 

  27. Dubédat, J.: SLE and Virasoro representations: fusion. Commun. Math. Phys. (2015). doi:10.1007/s00220-014-2283-7

  28. Duplantier B., David F.: Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice. J. Stat. Phys. 51(3–4), 327–434 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Farkas H.M., Kra I.: Riemann Surfaces. Graduate Texts in Mathematics, 2nd edition, vol. 71. Springer, New York (1992)

    Google Scholar 

  30. Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973)

  31. Forman R.: Functional determinants and geometry. Invent. Math. 88(3), 447–493 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81. Springer, New York. Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation (1991)

  33. Friedrich R., Kalkkinen J.: On conformal field theory and stochastic Loewner evolution. Nucl. Phys. B 687(3), 279–302 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Friedrich R., Werner W.: Conformal restriction, highest-weight representations and SLE. Commun. Math. Phys. 243(1), 105–122 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Friedrich, R.M.: On connections of conformal field theory and stochastic Loewner evolution. hep-lat/0410029 (2004, preprint)

  36. Gardiner, F.P.: Teichmüller Theory and Quadratic Differentials. Pure and Applied Mathematics (New York). Wiley, New York (1987). A Wiley-Interscience Publication

  37. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994). Reprint of the 1978 original

  38. Hagendorf C., Bernard D., Bauer M.: The Gaussian free field and SLE4 on doubly connected domains. J. Stat. Phys. 140(1), 1–26 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)

    Article  MATH  Google Scholar 

  40. Huang Y.-Z.: Two-Dimensional Conformal Geometry and Vertex Operator Algebras. Progress in Mathematics, vol. 148. Birkhäuser Boston, Inc., Boston (1997)

    Google Scholar 

  41. Iohara K., Koga Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London (2011)

    Book  Google Scholar 

  42. Jost J.: Bosonic Strings: A Mathematical Treatment. AMS/IP Studies in Advanced Mathematics, vol. 21. American Mathematical Society, Providence (2001)

    Google Scholar 

  43. Kac M.: Can one hear the shape of a drum?. Am. Math. Mon. 73(4, part II), 1–23 (1966)

    Article  MATH  Google Scholar 

  44. Kac V.G., Raina A.K.: Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras. Advanced Series in Mathematical Physics, vol. 2. World Scientific Publishing Co. Inc., Teaneck (1987)

    Google Scholar 

  45. Kadanoff L.P., Ceva H.: Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3, 3918–3939 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  46. Kenyon R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185(2), 239–286 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  47. Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics, English edition. Springer, Berlin (2005). Translated from the 1981 Japanese original by Kazuo Akao

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

  49. Kontsevich, M., Suhov, Y.: On Malliavin measures, SLE, and CFT. Tr. Mat. Inst. Steklova 258(Anal. i Osob. Ch. 1):107–153 (2007)

  50. Kontsevich M.L.: The Virasoro algebra and Teichmüller spaces. Funct. Anal. Appl. 21(2), 156–157 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  51. Kozdron, M.J., Lawler, G.F.: The configurational measure on mutually avoiding SLE paths. In: Universality and Renormalization. Fields Institute Communications, vol. 50, pp. 199–224. American Mathematical Society, Providence (2007)

  52. Kytölä K.: Virasoro module structure of local martingales of SLE variants. Rev. Math. Phys. 19(5), 455–509 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003, electronic)

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

    Google Scholar 

  55. Lawler, G.F.: Defining SLE in multiply connected domains with the Brownian loop measure. arXiv:1108.4364 (2011, preprint)

  56. Lawler G.F., Werner W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  57. Le Jan, Y.: Markov Paths, Loops and Fields. Lecture Notes in Mathematics, vol. 2026. Springer, Heidelberg (2011). Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, école d’été de Probabilités de Saint-Flour [Saint-Flour Probability Summer School]

  58. McKean H.P. Jr, Singer I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1(1), 43–69 (1967)

    MATH  MathSciNet  Google Scholar 

  59. Mumford, D.: Tata Lectures on Theta. I. Progress in Mathematics, vol. 28. Birkhäuser Boston Inc., Boston. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman (1983)

  60. Mumford, D.: Tata Lectures on Theta. II. Progress in Mathematics, vol. 43. Birkhäuser Boston Inc., Boston (1984). Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura

  61. Polyakov A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  62. Rogers, L.C.G., Williams, D.: Diffusions, Markov processes, and martingales, vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000). Itô calculus, Reprint of the second 1994 edition

  63. Rohde S., Schramm O.: Basic properties of SLE. Ann. Math. (2) 161(2), 883–924 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  64. Rosenberg, S.: The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds. London Mathematical Society Student Texts, vol. 31. Cambridge University Press, Cambridge (1997)

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

    Article  MATH  MathSciNet  Google Scholar 

  66. Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11:659–669 (2005, electronic)

  67. Segal, G.: The definition of conformal field theory. In: Topology, Geometry and Quantum Field Theory. London Mathematical Society Lecture Note Series, vol. 308, pp. 421–577. Cambridge University Press, Cambridge (2004)

  68. Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    Article  ADS  MATH  Google Scholar 

  69. Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421–1451. European Mathematical Society, Zürich (2006)

  70. Stroock, D.W.: Partial Differential Equations for Probabilists. Cambridge Studies in Advanced Mathematics, vol. 112. Cambridge University Press, Cambridge (2012). Paperback edition of the 2008 original

  71. Stroock D.W., Varadhan S.R.S.: Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233. Springer, Berlin (1979)

    Google Scholar 

  72. Vafa C.: Conformal theories and punctured surfaces. Phys. Lett. B 199(2), 195–202 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  73. Werner, W.: Random planar curves and Schramm–Loewner evolutions. In: Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1840, pp. 107–195. Springer, Berlin (2004)

  74. Zhan D.: Stochastic Loewner evolution in doubly connected domains. Probab. Theory Relat. Fields 129(3), 340–380 (2004)

    Article  MATH  Google Scholar 

  75. Zhan D.: Some properties of annulus SLE. Electron. J. Probab. 11(41), 1069–1093 (2006)

    MATH  MathSciNet  Google Scholar 

  76. Zhan, D.:Reversibility of whole-plane SLE. Probab. Theory Relat. Fields. arXiv:1004.1865 (2010, preprint)

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  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Julien Dubédat

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Correspondence to Julien Dubédat.

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Communicated by M. Salmhofer

J. Dubédat was partially supported by NSF Grant DMS-1005749 and the Alfred P. Sloan Foundation.

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Dubédat, J. SLE and Virasoro Representations: Localization. Commun. Math. Phys. 336, 695–760 (2015). https://doi.org/10.1007/s00220-014-2282-8

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