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For large classes of even-dimensional Riemannian manifolds $(M,g)$, weconstruct and analyze conformally invariant random fields. These centeredGaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, arecharacterized by their covariance kernels $k$ which exhibit a preciselogarithmic divergence: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. They share afundamental quasi-invariance property under conformal transformations.  In terms of the co-polyharmonic Gaussian field $h$, we define the quantumLiouville measure, a random measure on $M$, heuristically given as $$d\mu_g^{h}(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,d \text{vol}_g(x)$$ andrigorously obtained as almost sure weak limit of the right-hand side with $h$replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. Interms on the quantum Liouville measure, we define the Liouville Brownian motionon $M$ and the random GJMS operators.  Finally, we present an approach to a conformal field theory in arbitrary evendimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorousmeaning to the Polyakov-Liouville measure $$ d\boldsymbol{\nu}^*_g(h)=\frac1{Z^*_g} \exp\Big(- \int \Theta\,Q_g h + m e^{\gamma h} d\text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ andwe derive the corresponding conformal anomaly.  The set of admissible manifolds is conformally invariant. It includes allcompact 2-dimensional Riemannian manifolds, all compact non-negatively curvedEinstein manifolds of even dimension, and large classes of compact hyperbolicmanifolds of even dimension. However, not every compact even-dimensionalRiemannian manifold is admissible.  Our results rely on new sharp estimates for heat kernels and higher orderGreen kernels on arbitrary compact manifolds.

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