Gaussian Free Fields and KPZ Relation in R^4

Abstract: This work aims to extend part of the two dimensional results of Duplantier and Sheffield on Liouville quantum gravity to four dimensions, and indicate possible extensions to other even-dimensional spaces R^2n as well as Riemannian manifolds. Let "\Theta" be the Gaussian free field on R^4 with the underlying Hilbert space being the Sobolev space H^2 witb the inner product determined by the operator (I-\Delta)^2. Assume "\theta" is a generic element from \Theta. We consider a sequence of random Borel measures on R^4, each of which is absolutely continuous with respect to the Lebesgue measure dx and the density function is given by the exponential of a centered Gaussian family parametrized by x in R^4. We show that with probability 1, this sequence of measures weakly converges to a limit random measure which can be "formally" written as "exp(2\gamma\theta(x)dx". In this setting, we also prove a KPZ relation, which is the quadratic relation between the scaling exponent of a bounded Borel set on R^4 under the Lebesgue measure and its counterpart under the random measure obtained above. Our approach is similar to the one used by Duplantier and Sheffield in 2D but with adaptations to R^4.