×

SLE and Virasoro representations: fusion. (English) Zbl 1319.81073

The background material for the present paper can be found in [the author, ibid. 336, No. 2, 695–760 (2015; Zbl 1318.82007)]. Algebraic aspects of the present paper may look familiar to readers knowledgeable of the CFT treatment of fusion. However, its main contribution resides in implementing fusion rules for objects and quantitities originating in SLE. Beginning from the seminal BPZ work [A. A. Belavin et al., Nucl. Phys., B 241, No. 2, 333–380 (1984; Zbl 0661.17013)], an implicit idea that certain critical two-dimensional statistical physics models involve weighted null-vectors whose weights are directly related with boundary condtions in the BCFT, is currently interpreted in terms of the dynamics of intefaces in the Schramm-Loewner evolutions (SLE). Starting from \(n\) commuting SLEs, seeded at distinct points, the partition function is found to satusfy null-vector differential equations (at level 2). Higher level null-vector equations are obtained by coalescing seeds one by one. The argument combines the study of Verma modules for the Virasoro algebra with various regularity estimates, those being based on hypoellipticity and stochastic flow arguments.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B68 Virasoro and related algebras
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnold, V.I.: Ordinary Differential Equations. Universitext. Springer, Berlin (2006). Translated from the Russian by Roger Cooke, Second printing of the 1992 edition · Zbl 1126.82011
[2] Bauer M., Bernard D.: \[{{\rm SLE}_{\kappa}}\] SLEκ growth processes and conformal field theories. Phys. Lett. B 543(1-2), 135-138 (2002) · Zbl 0997.60119 · doi:10.1016/S0370-2693(02)02423-1
[3] Bauer M., Bernard D., Kytölä K.: Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5-6), 1125-1163 (2005) · Zbl 1094.82016 · doi:10.1007/s10955-005-7002-5
[4] Bauer M.,Di Francesco P., Itzykson C., Zuber J.-B.: Singular vectors of the Virasoro algebra. Phys. Lett. B 260(3-4), 323-326 (1991) · Zbl 0957.17510 · doi:10.1016/0370-2693(91)91619-7
[5] 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) · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[6] Beliaev D., Johansson Viklund F.: Some remarks on SLE bubbles and Schramm’s two-point observable. Commun. Math. Phys. 320(2), 379-394 (2013) · Zbl 1268.60105 · doi:10.1007/s00220-013-1710-5
[7] Benoit L., Saint-Aubin Y.: Degenerate conformal field theories and explicit expressions for some null vectors. Phys. Lett. B 215(3), 517-522 (1988) · Zbl 0957.17509 · doi:10.1016/0370-2693(88)91352-4
[8] 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) · Zbl 0176.09703
[9] Cardy J.: Corrigendum: “Stochastic Loewner evolution and Dyson’s circular ensembles” [J. Phys. A 36(24), L379-L386 (2003)]. J. Phys. A 36(49), 12343 (2003) · doi:10.1088/0305-4470/36/49/c01
[10] Cardy J.: Stochastic Loewner evolution and Dyson’s circular ensembles. J. Phys. A 36(24), L379-L386 (2003) · Zbl 1038.82074 · doi:10.1088/0305-4470/36/24/101
[11] Cardy J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324(3), 581-596 (1989) · doi:10.1016/0550-3213(89)90521-X
[12] Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515-580 (2012) · Zbl 1257.82020 · doi:10.1007/s00222-011-0371-2
[13] Di Francesco P., Mathieu P., Sénéchal D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York (1997)
[14] Dubédat J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183-1218 (2006) · Zbl 1113.82064 · doi:10.1007/s10955-006-9132-9
[15] Dubédat J.: Commutation relations for SLE. Commun. Pure Appl. Math. 60(12), 1792-1847 (2007) · Zbl 1137.82009 · doi:10.1002/cpa.20191
[16] Dubédat J.: Duality of Schramm-Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 697-724 (2009) · Zbl 1205.60147
[17] Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. (2015). doi:10.1007/s00220-014-2282-8 · Zbl 1318.82007
[18] Feĭgin, B.L., Fuchs, D.B.: Verma modules over the Virasoro algebra. In: Topology (Leningrad, 1982), volume 1060 of Lecture Notes in Math. Springer, Berlin, pp. 230-245 (1984)
[19] Fomin, S.: Loop-erased walks and total positivity. Trans. Am. Math. Soc. 353(9), 3563-3583 (electronic) (2001). doi:10.1090/S0002-9947-01-02824-0 · Zbl 0973.15014
[20] Friedrich R., Kalkkinen J.: On conformal field theory and stochastic Loewner evolution. Nucl. Phys. B 687(3), 279-302 (2004) · Zbl 1149.81352 · doi:10.1016/j.nuclphysb.2004.03.025
[21] Friedrich, R.M.: On connections of conformal field theory and stochastic Loewner evolution. preprint, arXiv:math-ph/0410029 (2004) · Zbl 1113.82064
[22] Gamsa, A., Cardy, J.: The scaling limit of two cluster boundaries in critical lattice models. J. Stat. Mech. Theory Exp. (12): P12009, 26 (electronic) (2005) · Zbl 1109.81049
[23] Goodman R.W.: Nilpotent Lie groups: structure and applications to analysis. Lecture Notes in Mathematics, vol. 562. Springer, Berlin (1976) · Zbl 0347.22001
[24] Grimmett G.: The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006) · Zbl 1122.60087
[25] Hongler C., Kytölä K.: Ising interfaces and free boundary conditions. J. Am. Math. Soc. 26(4), 1107-1189 (2013) · Zbl 1284.82021 · doi:10.1090/S0894-0347-2013-00774-2
[26] Hörmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147-171 (1967) · Zbl 0156.10701 · doi:10.1007/BF02392081
[27] Ince E.L.: Ordinary Differential Equations. Dover Publications, New York (1944) · Zbl 0063.02971
[28] Iohara K., Koga Y.: Representation Theory of the Virasoro algebra. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London (2011) · Zbl 1222.17001 · doi:10.1007/978-0-85729-160-8
[29] Kac V.G., Raina A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, volume 2 of Advanced Series in Mathematical Physics. World Scientific Publishing Co. Inc., Teaneck (1987) · Zbl 0668.17012
[30] Kontsevich, M.: SLE, CFT, and phase boundaries. Arbeitstagung 2003, preprint, MPI 2003 (60) (2003)
[31] Kontsevich M.L.: The Virasoro algebra and Teichmüller spaces. Funct. Anal. Appl. 21(2), 156-157 (1987) · Zbl 0647.58012 · doi:10.1007/BF01078034
[32] Kozdron, M.J., Lawler, G.F.: Estimates of random walk exit probabilities and application to loop-erased random walk. Electron. J. Probab. 10, 1442-1467 (electronic) (2005) · Zbl 1110.60046
[33] Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917-955 (electronic) (2003) · Zbl 1030.60096
[34] Lawler G.F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939-995 (2004) · Zbl 1126.82011 · doi:10.1214/aop/1079021469
[35] Revuz D., Yor M.: Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006 · doi:10.1007/978-3-662-06400-9
[36] Rothschild L.P., Stein E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3-4), 247-320 (1976) · Zbl 0346.35030 · doi:10.1007/BF02392419
[37] Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221-288 (2000) · Zbl 0968.60093 · doi:10.1007/BF02803524
[38] Schramm, O.: A percolation formula. Electron. Commun. Probab. 6, 115-120 (electronic) (2001) · Zbl 1008.60100
[39] Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians, vol. II, pp. 1421-1451. Eur. Math. Soc., Zürich (2006) · Zbl 1112.82014
[40] Stroock, D.W.: Partial Differential Equations for Probabilists, volume 112 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. Paperback edition of the 2008 original (2012) · Zbl 1245.35004
[41] Vostrikova L.: On regularity properties of Bessel flow. Stochastics 81(5), 431-453 (2009) · Zbl 1195.60059
[42] Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the 28th annual ACM symposium on the theory of computing (Philadelphia, PA, 1996), pp. 296-303, New York, ACM (1996) · Zbl 0946.60070
[43] Yoshida, M.: Fuchsian differential equations. Aspects of Mathematics, E11. Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory · Zbl 0618.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.