Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension

Abstract

For large classes of even-dimensional Riemannian manifolds $(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels $k$ which exhibit a precise logarithmic divergence: $|k(x,y)-\log \frac{1}{d(x,y)}|\le C$. They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field $h$, we define the quantum Liouville measure, a random measure on $M$, heuristically given as $$ dμ_g^{h}(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)$$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_{\ell },\ell \in \mathbb{N}$. In terms on the quantum Liouville measure, we define the Liouville Brownian motion on $M$ and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorous meaning to the Polyakov-Liouville measure $$d\hat{\mathbf{I}}{½}_g^{\ast }(h)=\frac{1}{Z_g^{\ast }}\exp (-\int \hat{I}˜Q_{g}h+m{e}^{{\hat{I}}^{3}h}d{\text{vol}}_g)\exp (-\frac{a_n}{2}{\mathfrak{p}}_g(h,h))dh,$$ and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary compact manifolds.

New notion of `singular' h, now Sobolev function rather than distribution. Detailed discussion on plain vs. adjusted LQG measure. Additional results on Polyakov-Liouville measure