Conformal covariance of the Liouville quantum gravity metric for γ∈ (0,2)

2019/05/01 by Ewain Gwynne, Gwynne, Ewain, Jason Miller +1
Physics and Astronomy · #Black Holes and Theoretical Physics #Complex Variables (math.CV) #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1905.00384

openalex publication_date 2019/05/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

For γ∈ (0,2), U⊂ \mathbb C, and an instance h of the Gaussian free field (GFF) on U, the γ-Liouville quantum gravity (LQG) surface associated with (U,h) is formally described by the Riemannian metric tensor e^γh (dx² + dy²) on U. Previous work by the authors showed that one can define a canonical metric (distance function) D_h on U associated with a γ-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for γ-LQG surfaces. That is, if U,\widetildeU are domains, ϕ\colon U → \widetildeU is a conformal transformation, Q=2/γ+γ/2, and \widetilde h = h∘ϕ^-1 + Qlog|(ϕ^-1)'|, then D_h(z,w) = D_\widetildeh(ϕ(z),ϕ(w)) for all z,w ∈ U. This proves that D_h is intrinsic to the quantum surface structure of (U,h), i.e., it does not depend on the particular choice of parameterization.

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