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A Solution Space for a System of Null-State Partial Differential Equations: Part 1

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Abstract

This article is the first of four that completely and rigorously 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–Löwner evolution (SLE\({_\kappa}\)). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop-gas model, such as percolation, or more generally the Potts models and O(n) models, at the statistical mechanical critical point. (SLE\({_\kappa}\) partition functions also satisfy these equations.) For such a lattice model in a polygon \({{\mathcal{P}}}\) with its 2N sides exhibiting a free/fixed side-alternating boundary condition \({\vartheta}\), this partition function is proportional to the CFT correlation function

$$\langle\psi_1^c(w_1)\psi_1^c(w_2)\dotsm\psi_1^c(w_{2N-1})\psi_1^c(w_{2N})\rangle^{{\mathcal{P}}}_\vartheta,$$

where the w i are the vertices of \({{\mathcal{P}}}\) and where \({\psi_1^c}\) is a one-leg corner operator. (Partition functions for “crossing events” in which clusters join the fixed sides of \({{\mathcal{P}}}\) in some specified connectivity are linear combinations of such correlation functions.) When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider.

In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than C N , the Nth Catalan number. While our motivations are based in CFT, our proofs are completely rigorous. This proof is contained entirely within this article, except for the proof of Lemma 14, which constitutes the second article (Flores and Kleban, in Commun Math Phys, arXiv:1404.0035, 2014). In the third article (Flores and Kleban, in Commun Math Phys, arXiv:1303.7182, 2013), we use the results of this article to prove that the solution space of this system of PDEs has dimension C N and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article (Flores and Kleban, in Commun Math Phys, arXiv:1405.2747, 2014), we prove further CFT-related properties about these solutions, some useful for calculating cluster-crossing probabilities of critical lattice models in polygons.

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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, University of Michigan, Ann Arbor, MI, 48109-2136, USA

    Steven M. Flores

  2. LASST, 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 1. Commun. Math. Phys. 333, 389–434 (2015). https://doi.org/10.1007/s00220-014-2189-4

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