Conformally Invariant Random Geometry on Riemannian Manifolds of Even Dimension

We construct and study conformally invariant, log-correlated Gaussian random fields on compact Riemannian manifolds of general even dimension uniquely defined through its covariance kernel given as inverse of the Graham-Jenne-Mason-Sparling (GJMS) operator. The corresponding Gaussian Multiplicative Chaos is a generalization to the n-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. 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.

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