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

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≤γ2 is shown to be related to the quantum measure dμγ′, γ′