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

2021/05/28 by Lorenzo Dello Schiavo, Schiavo, Lorenzo Dello, Ronan Herry +5
Environmental Science · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hydrology and Drought Analysis #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2105.13925

openalex publication_date 2021/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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\frac1d(x,y)|≤ 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 dμ_g^h(x):= e^γh(x)-\fracγ²2k(x,x) d vol_g(x) and rigorously obtained as almost sure weak limit of the right-hand side with h replaced by suitable regular approximations h_ℓ, ℓ∈\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\boldsymbolν^*_g(h) =\frac1Z^*_g exp(- ∫ Θ Q_g h + m e^γh d vol_g) exp(-(a_n)/(2) \mathfrak p_g(h,h)) 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.

Related

• Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension
• Random Riemannian Geometry in 4 Dimensions
• Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously
• Around the semi-classical limit of boundary Liouville conformal field theory
• Existence and uniqueness of the Liouville quantum gravity metric for γ∈ (0,2)