Abstract
This article is the last of four that completely and rigorously characterize a solution space \({{\mathcal{S}}_N}\) for a homogeneous system of 2N + 3 linear partial differential equations in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE\({_\kappa}\)). The system comprises 2N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2N one-leg boundary operators. In the first two articles (Flores and Kleban in Commun Math Phys, 2012; Flores and Kleban, in Commun Math Phys, 2014), we use methods of analysis and linear algebra to prove that dim \({{\mathcal{S}}_N \leq C_N}\) , with C N the Nth Catalan number. Using these results in the third article (Flores and Kleban, in Commun Math Phys, 2013), we prove that dim \({{\mathcal{S}}_N=C_N}\) and \({{\mathcal{S}}_N}\) is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT.
In this article, we use these results to prove some facts concerning the solution space \({{\mathcal{S}}_N}\) . First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless \({8/\kappa}\) is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion for one-leg boundary operators, assumed in CFT. We also identify particular elements of \({{\mathcal{S}}_N}\) , which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE\({_\kappa}\) process join in a particular connectivity. This leads to new formulas for crossing probabilities of critical lattice models inside polygons with a free/fixed side-alternating boundary condition, which we derive in Flores et al. (Partition functions and crossing probabilities for critical systems inside polygons, in preparation). Finally, we propose a reason for why the exceptional speeds [certain \({\kappa}\) values that appeared in the analysis of the Coulomb gas solutions in Flores and Kleban (Commun Math Phys, 2013)] and the minimal models of CFT are connected.
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References
Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations I. Commun. Math. Phys. Preprint: arXiv:1212.2301 (2012, to appear)
Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations II. Commun. Math. Phys. Preprint: arXiv:1404.0035 (2014, to appear)
Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations III. Commun. Math. Phys. Preprint:arXiv:1303.7182 (2013, to appear)
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)
Francesco P., Mathieu R., Sénéchal D.: Conformal Field Theory. Springer, New York (1997)
Henkel M.: Conformal Invariance and Critical Phenomena. Springer, Berlin (1999)
Bauer M., Bernard D., Kytölä K.: Multiple Schramm-Löwner evolutions and statistical mechanics martingales. J. Stat. Phys. 120, 1125 (2005)
Dubédat J.: Commutation relations for SLE. Comm. Pure Appl. Math. 60, 1792–1847 (2007)
Graham, K.: On multiple Schramm-Löwner evolutions. J. Stat. Mech. P03008 (2007)
Kozdron M.J., Lawler G.: The configurational measure on mutually avoiding SLE paths. Fields Inst. Commun. 50, 199–224 (2007)
Sakai K.: Multiple Schramm-Löwner evolutions for conformal field theories with Lie algebra symmetries. Nucl. Phys. B 867, 429–447 (2013)
Bauer M., Bernard D.: Conformal field theories of stochastic Löwner evolutions. Comm. Math. Phys. 239, 493–521 (2003)
Dotsenko V.S.: Critical behavior and associated conformal algebra of the Z3 Potts model. Nucl. Phys. B 235, 54–74 (1984)
Gruzberg I.A.: Stochastic geometry of critical curves, Schramm-Löwner evolutions, and conformal field theory. J. Phys. A 39, 12601–12656 (2006)
Rushkin I., Bettelheim E., Gruzberg I.A., Wiegmann P.: Critical curves in conformally invariant statistical systems. J. Phys. A 40, 2165–2195 (2007)
Cardy J.: Critical percolation in finite geometries. J. Phys. A Math. Gen. 25, L201–L206 (1992)
Cardy J.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240, 514–532 (1984)
Grimmett G.: Percolation. Springer, New York (1989)
Baxter R.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)
Wu F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)
Fortuin C.M., Kasteleyn P.W.: On the random cluster model I. Introduction and relation to other models. Physica D 57, 536–564 (1972)
Stanley H.E.: Dependence of critical properties on dimensionality of spins. Phys. Rev. Lett. 20, 589–592 (1968)
Ziff R.M., Cummings P.T., Stell G.: Generation of percolation cluster perimeters by a random walk. J. Phys. A Math. Gen. 17, 3009–3017 (1984)
Lawler G.: A self-avoiding walk. Duke Math. J. 47, 655–694 (1980)
Schramm O., Sheffield S.: The harmonic explorer and its convergence to SLE4. Ann. Probab. 33, 2127–2148 (2005)
Weinrib A., Trugman S.A.: A new kinetic walk and percolation perimeters. Phys. Rev. B 31, 2993–2997 (1985)
Madra G., Slade G.: The Self-Avoiding Walk. Birkhäuser, Boston (1996)
Dotsenko V.S., Fateev V.A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys. B 240, 312–348 (1984)
Dotsenko, V.S., Fateev, V.A.: Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge \({c \leq 1}\) . Nucl. Phys. B 251, 691–673 (1985)
Smirnov S.: Towards conformal invariance of 2D lattice models. Proc. Int. Congr. Math. 2, 1421–1451 (2006)
Duminil-Copin, H., Smirnov, S.: Conformal invariance of lattice models. In: Ellwood, D., Newman, C., Sidoravicius, V., Werner, W. (eds). Probability and Statistical Physics in Two and More Dimensions, Clay Mathematics Proceedings, vol. 15, pp. 213–276 (2012)
Dubédat J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123, 1183–1218 (2006)
Francesco P., Golinelli O., Guitter E.: Meanders and the Temperley-Lieb algebra. Comm. Math. Phys. 186, 1–59 (1997)
Francesco P., Guitter E.: Geometrically constrained statistical systems on regular and random lattices: from foldings to meanders. Phys. Reports 415, 1–88 (2005)
Francesco P.: Meander Determinants. Comm. Math. Phys. 191, 543–583 (1998)
Di Francesco, P.: Truncated meanders. In: Jing, N. Misra, K. (eds). Recent Developments in Quantum Affine Algebras and Related Topics, pp. 135–161. American Mathematical Society (1999)
Simmons J.J.H.: Logarithmic operator intervals in the boundary theory of critical percolation. J. Phys. A Math. Theor. 46, 494015 (2013)
Flores, S.M., Simmons, J.J.H., Kleban, P.: Multiple-SLE\({_\kappa }\) connectivity weights for rectangles, hexagons, and octagons (In preparation, 2014)
Flores, S.M., Simmons, J.J.H., Kleban, P., Ziff R.M.: Partition functions and crossing probabilities for critical systems inside polygons (In preparation, 2014)
Bender C.M., Orszag S.A.: Advanced Mathematical Methods for Scientists and Engineers, Asymptotic Methods and Perturbation Theory. Springer, New York (1999)
Gurarie V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993)
Gurarie V.: Logarithmic operators and logarithmic conformal field theories. J. Phys. A Math. Theor. 46, 494003 (2013)
Mathieu P., Ridout D.: From percolation to logarithmic conformal field theory. Phys. Lett. B 657, 120–129 (2007)
Runkel, I., Gaberdiel, M.R., Wood, S.: Logarithmic bulk and boundary conformal field theory and the full centre construction. In: Bai, C., Fuchs, J., Huang, Y.Z., Kong, L., Runkel, I., Schweigert, C. (eds.) Conformal Field Theories and Tensor Categories, pp. 93–168 (2014)
Vasseur, R., Jacobsen, J.L., Saleur, H.: Logarithmic observables in critical percolation. J. Stat. Mech. L07001 (2012)
Bauer M., Bernard D.: 2D growth processes: SLE and Löwner chains. Phys. Rept. 432, 115–221 (2006)
Jokela, N., Järvinen, M., Kytölä, K.: SLE boundary visits. Preprint: arXiv:1311.2297 (2013)
Cardy J.: Boundary conditions, fusion rules, and the Verlinde formula. Nucl. Phys. B 324, 581–596 (1989)
Cardy J.: Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories. Nucl. Phys. B 275, 200–218 (1986)
Sauler H., Bauer M.: On some relations between local height probabilities and conformal invariance. Nucl. Phys. B 320, 591–624 (1989)
Gamsa, A., Cardy, J.: Schramm–Loewner evolution in the three-state Potts model—a numerical study. J. Stat. Mech. P08020 (2007)
Bauer M., Francesco P.: Covariant differential equations and singular vectors in Virasoro representations. Nucl. Phys. B 362, 515–562 (1991)
Lawler G., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004)
Lawler, G., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Lapidus, M.L., Frankenhuysen, M.V. (eds.) Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 2. American Mathematical Society (2004)
Lawler G., Schramm O., Werner W.: Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187, 237–273 (2001)
Smirnov S.: Critical percolation in the plane. C. R. Acad. Sci. Paris Sr. I Math. 333, 239–244 (2001)
Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172, 1435–1467 (2010)
Benoit L., Saint-Aubin Y.: Degenerate conformal field theories and explicit expressions for some null vectors. Phys. Lett. B 215, 517–522 (1988)
Kytölä, K., Peltola, E.: Pure geometries of multiple SLEs (In preparation, 2014)
Kytölä, K., Peltola, E.: Conformally covariant boundary correlation functions with a quantum group. Preprint: arXiv:1408.1384 (2014)
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Flores, S.M., Kleban, P. A Solution Space for a System of Null-State Partial Differential Equations: Part 4. Commun. Math. Phys. 333, 669–715 (2015). https://doi.org/10.1007/s00220-014-2180-0
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DOI: https://doi.org/10.1007/s00220-014-2180-0