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A Solution Space for a System of Null-State Partial Differential Equations: Part 4

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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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Author notes

  1. Steven M. Flores

    Present address: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Steven M. Flores

  2. LASST and Department of Physics and Astronomy, University of Maine, Orono, ME, 04469-5708, USA

    Peter Kleban

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  1. Steven M. Flores
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  2. Peter Kleban
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Correspondence to Steven M. Flores.

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Communicated by M. Salmhofer

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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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