Liouville quantum gravity on the Riemann sphere. (English) Zbl 1336.83042
Summary: In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by A. M. Polyakov [“Quantum geometry of bosonic strings”, Phys. Lett., B 103, No. 3, 207–210 (1981; doi:10.1016/0370-2693(81)90743-7)]. We establish some of its fundamental properties like conformal covariance under \(\mathrm{PSL}_2(\mathbb C)\)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.
MSC:
| 83E30 | String and superstring theories in gravitational theory |
| 83C45 | Quantization of the gravitational field |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 53Z05 | Applications of differential geometry to physics |
| 37K65 | Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 60G60 | Random fields |
Keywords:
Liouville quantum field theory; Riemann sphere; conformal covariance; Seiberg bounds; KPZ scaling laws; Weyl anomaly; random planar mapsReferences:
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