Local Conformal Structure of Liouville Quantum Gravity
| Autor: | Antti Kupiainen, Rémi Rhodes, Vincent Vargas |
|---|---|
| Přispěvatelé: | Department of Mathematics and Statistics [Helsinki], Falculty of Science [Helsinki], University of Helsinki-University of Helsinki, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0013,Liouville,Géométrie quantique de Liouville et flots turbulents(2015), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), ANR: Liouville,ANR-15-CE40-0013, Université Paris-Est (UPE), Department of Mathematics and Statistics, Mathematical physics, Mind and Matter, Helsingin yliopisto = Helsingfors universitet = University of Helsinki-Helsingin yliopisto = Helsingfors universitet = University of Helsinki, École normale supérieure - Paris (ENS-PSL) |
| Rok vydání: | 2018 |
| Předmět: |
Structure constants
FREE-FIELD [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] FOS: Physical sciences Conformal map String theory integrability field theory Liouville 60D05 81T40 81T20 114 Physical sciences 01 natural sciences operator product expansion Theoretical physics Conformal symmetry 0103 physical sciences FOS: Mathematics 111 Mathematics Operator product expansion structure correlation function 0101 mathematics Quantum field theory string model Ward identity: conformal 2D Mathematical Physics Mathematics cohomology: quantum field theory: conformal Conformal field theory Probability (math.PR) 010102 general mathematics differential equations Statistical and Nonlinear Physics Mathematical Physics (math-ph) 16. Peace & justice INVARIANCE [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] UNIQUENESS quantum gravity gravitation symmetry: conformal [PHYS.GRQC]Physics [physics]/General Relativity and Quantum Cosmology [gr-qc] Quantum gravity 010307 mathematical physics Mathematics - Probability |
| Zdroj: | Communications in Mathematical Physics Communications in Mathematical Physics, Springer Verlag, 2018 Commun.Math.Phys. Commun.Math.Phys., 2019, 371 (3), pp.1005-1069. ⟨10.1007/s00220-018-3260-3⟩ |
| ISSN: | 0010-3616 1432-0916 |
| DOI: | 10.1007/s00220-018-3260-3 |
| Popis: | Liouville Conformal Field Theory (LCFT) is an essential building block of Polyakov's formulation of non critical string theory. Moreover, scaling limits of statistical mechanics models on planar maps are believed by physicists to be described by LCFT. A rigorous probabilistic formulation of LCFT based on a path integral formulation was recently given by the present authors and F. David in \cite{DKRV}. In the present work, we prove the validity of the conformal Ward identities and the Belavin-Polyakov-Zamolodchikov (BPZ) differential equations (of order $2$) for the correlation functions of LCFT. This initiates the program started in the seminal work of Belavin-Polyakov-Zamolodchikov \cite{BPZ} in a probabilistic setup for a non-trivial Conformal Field Theory. We also prove several celebrated results on LCFT, in particular an explicit formula for the 4 point correlation functions (with insertion of a second order degenerate field) leading to a rigorous proof of a non trivial functional relation on the 3 point structure constants derived earlier in the physics literature by Teschner \cite{Tesc}. The proofs are based on exact identities which rely on the underlying Gaussian structure of LCFT combined with estimates from the theory of critical Gaussian Multiplicative Chaos and a careful analysis of singular integrals (Beurling transforms and generalizations). As a by-product, we give bounds on the correlation functions when two points collide making rigorous certain predictions from physics on the so-called "operator product expansion" of LCFT. Major revision |
| Databáze: | OpenAIRE |
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