A solution space for a system of null-state partial differential equations 3

Abstract

This article is the third of four that completely characterize a solution space ${\mathcal{S}}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). 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 previous two articles (parts I and II), we use methods of analysis and linear algebra to prove that $\dim {\mathcal{S}}_N\le C_N$, with $C_N$ the $N$th Catalan number. Extending these results, we prove in this article that $\dim {\mathcal{S}}_N=C_N$ and ${\mathcal{S}}_N$ entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of $C_N$ such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of lemma 15 in part I to send each solution of the mentioned set to a vector in ${\mathbb{R}}^{C_N}$, whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric $C_N$ by $C_N$ matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of $C_N$ Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of $C_N$ Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models.

Minor typos from v3 corrected, equation 35 corrected

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Cited by in corpus (22)

• Global and Local Multiple SLEs for $\kappa \le 4$ and Connection Probabilities for Level Lines of GFF
• Pure partition functions of multiple SLEs
• A solution space for a system of null-state partial differential equations 2
• A solution space for a system of null-state partial differential equations 1
• A formula for crossing probabilities of critical systems inside polygons
• Towards a conformal field theory for Schramm-Loewner evolutions
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Multiple SLE type scaling limits: from local to global
• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
• Boundary correlations in planar LERW and UST
• On the Uniqueness of Global Multiple SLEs
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• Connection Probabilities of Multiple FK-Ising Interfaces
• Multiple-SLE connectivity weights for rectangles, hexagons, and octagons
• Standard modules, radicals, and the valenced Temperley-Lieb algebra
• Generators, projectors, and the Jones-Wenzl algebra
• The singularities of Selberg- and Dotsenko-Fateev-like integrals
• Scaling Limits of Crossing Probabilities in Metric Graph GFF
• Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling
• Multiple Ising interfaces in annulus and $2N$-sided radial SLE
• Conformal field theory for annulus SLE: partition functions and martingale-observables
• Multiple backward Schramm--Loewner evolution and coupling with Gaussian free field