Abstract: On large classes of closed even-dimensional Riemannian manifolds M, we construct and study the Copolyharmonic Gaussian Field, i.e. a conformally invariant log-correlated Gaussian field of distributions on M. This random field is defined as the unique centered Gaussian field with covariance kernel given as the resolvent kernel of Graham—Jenne—Mason—Sparling (GJMS) operators of maximal order. The corresponding Gaussian Multiplicative Chaos is a generalization to the 2m-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. We study the associated Liouville Brownian motion and random GJMS operator, the higher-dimensional analogues of the 2d Liouville Brownian Motion and of the random Laplacian. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.

(arXiv:2105.13925, joint work with Ronan Herry, Eva Kopfer, Karl-Theodor Sturm)