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

For large classes of even-dimensional Riemannian manifolds (M,g)$(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields h=hg$h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: |k(x,y)-log1d(x,y)|<= C$\big\vert k(x,y)-\log\frac1{d(x,y)}\big\vert \le C$. They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field h$h$, we define the Liouville Quantum Gravity measure, a random measure on M$M$, heuristically given as d mu g gamma h(x)& colone;e gamma h(x)-gamma 22k(x,x)dvolg(x),$$\begin{equation*} d\mu_g<^>{\gamma h}(x)\coloneqq e<^>{\gamma h(x)-\frac{\gamma<^>2}2k(x,x)}\,d \mbox{vol}_g(x)\,, \end{equation*}$$and rigorously obtained as almost sure weak limit of the right-hand side with h$h$ replaced by suitable regular approximations h & ell;,& ell;is an element of N$h_\ell, \ell\in{\mathbb N}$. In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on M$M$ and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's Q$Q$-curvature: we give a rigorous meaning to the Polyakov-Liouville measure d nu g & lowast;(h)=1Zg & lowast;exp-integral Theta Qgh+me gamma hdvolgxexp-an2pg(h,h)dh$$\begin{eqnarray*} \hspace*{13.5pc} d\boldsymbol{\nu}<^>*_g(h) &=& \frac1{Z<^>*_g} \exp\left(- \int \Theta\,Q_g h + m e<^>{\gamma h} d \mbox{vol}_g\right)\\ &&\times \exp\left(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\right) dh \end{eqnarray*}$$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 concerning the logarithmic divergence of the kernel k$k$ rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds.