Duality and the Knizhnik-Polyakov-Zamolodchikov Relation in Liouville Quantum Gravity

Abstract

We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dμ[subscript γ]=ε[superscript γ[superscript 2]/2]e[superscript γh [subscript ε](z)]dz, where dz is the Lebesgue measure on D, γ is a real parameter, 0≤γ<2, and h[subscript ε](z) denotes the mean value on the circle of radius ε centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case γ>2 is shown to be related to the quantum measure dμγ′, γ′<2, by the fundamental duality γγ′=4.