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: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. 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 $$ d\mu_g^{h}(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,d \text{vol}_g(x)$$ 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 $$ d\boldsymbol{\nu}^*_g(h) =\frac1{Z^*_g} \exp\Big(- \int \Theta\,Q_g h + m e^{\gamma h} d \text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ 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.