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

摘要

For gamma is an element of(0, 2), U subset of C, and an instance h of the Gaussian free field (GFF) on U, the gamma -Liouville quantum gravity (LQG) surface associated with (U, h) is formally described by the Riemannian metric tensor e(gamma h)(dx(2) + dy(2)) on U. Previous work by the authors showed that one can define a canonical metric (distance function) D-h on U associated with a gamma-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for gamma-LQG surfaces. That is, if U, - U are domains, phi : U - U is a conformal transformation, Q = 2/gamma+gamma/2, and (h) over tilde = h omicron phi(-1) + Qlog vertical bar(phi(-1))'vertical bar, then Dh(z, w) = D (h) over tilde (phi(z), phi(w)) for all z, w is an element of 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.