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 dmicro_{gamma}=epsilon;{gamma;{2}/2}e;{gammah_{epsilon}(z)}dz, where dz is the Lebesgue measure on D, gamma is a real parameter, 0<or=gamma<2, and h_{epsilon}(z) denotes the mean value on the circle of radius epsilon centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case gamma>2 is shown to be related to the quantum measure dmu_{gamma;{'}}, gamma;{'}<2, by the fundamental duality gammagamma;{'}=4.
Duplantier
Bertrand
B
Institut de Physique Théorique, CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, France.
Sheffield
Scott
S