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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2105.13925 |
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| _version_ | 1866913941219704832 |
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| author | Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor |
| author_facet | Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor |
| contents | 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: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. 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 $$ dμ_g^{h}(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)$$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. 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 $$ d\boldsymbolν^*_g(h) =\frac1{Z^*_g} \exp\Big(- \int Θ\,Q_g h + m e^{γh} d \text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_13925 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension Schiavo, Lorenzo Dello Herry, Ronan Kopfer, Eva Sturm, Karl-Theodor Probability 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: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. 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 $$ dμ_g^{h}(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)$$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. 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 $$ d\boldsymbolν^*_g(h) =\frac1{Z^*_g} \exp\Big(- \int Θ\,Q_g h + m e^{γh} d \text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ 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. |
| title | Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension |
| topic | Probability |
| url | https://arxiv.org/abs/2105.13925 |