2024/07/12 by Guillaume Baverez, Antoine Jego, Antoine Jégo +2 · 3 citations
Mathematics · Physics and Astronomy · #17B68. Secondary: 47L55 #30C62 #81T40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Primary: 60J67 #Probability (math.PR) #Representation Theory (math.RT) #math-ph #math.CV #math.MP #math.PR #math.RT #msc:17B68. #msc:30C62 #msc:47L55 #msc:60J67 #msc:81T40
paper · pdf · doi:10.48550/arxiv.2407.09080
60 pages, 1 figure. Substantial revision with improved presentation
openalex publication_date 2024/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
This paper initiates the study of the conformal field theory of the SLEκ loop measure ν for κ∈(0,4], the range where the loop is almost surely simple. First, we construct two commuting representations (Ln,Ln)n∈ℤ of the Virasoro algebra with central charge cM=1-6((2)/(√κ)-(√κ)/(2))2≤1 as (unbounded) first order differential operators on L2(ν). Second, we introduce highest-weight representations and characterise their structure: in particular, we prove the existence of vanishing singular vectors at arbitrary levels on the Kac table. Third, we prove an integration by parts formula for the SLE loop measure, and use it to define the Shapovalov form of the representation, a non degenerate (but not positive definite) Hermitian form Q on L2(ν) with a remarkably simple geometric expression. The fact that Q differs from the L2(ν)-inner product is a manifestation of non-unitarity. Finally, we write down a spectral resolution of Q using the joint diagonalisation of L0 and L0. As an application of these results, we provide the first proof of the uniqueness of restriction measures, as conjectured by Kontsevich and Suhov. Our results lay the groundwork for an in-depth study of the CFT of SLE: in forthcoming works, we will define correlation functions on Riemann surfaces, and prove conformal Ward identities, BPZ equations, and conformal bootstrap formulas.