Smoothness of correlation functions in Liouville conformal field theory. (English) Zbl 1459.81100
Liouville conformal field theory (LCFT) was introduced by Polyakov in 1981 and has attracted extensive interest in physics and mathematics communities. It is an important example of two-dimensional conformal field theory which provides invaluable insight on the role of conformal metrics in two dimensions.
While the original work utilizes the path integral formulation, it is also possible to study LCFT with the conformal bootstrap method. However, for LCFT, the equivalence of the path integral and conformal bootstrap approaches is far from obvious, and how to unify them remains an open problem. Along this direction, the present paper provides a rigorous proof of the smoothness of the correlation functions in LCFT. This result serves as the first step towards proving that the correlation functions of LCFT satisfy the higher Ward identities and the higher Belavin-Polyakov-Zamolodchikov equations, which is a prediction of the conformal bootstrap approach.
While the original work utilizes the path integral formulation, it is also possible to study LCFT with the conformal bootstrap method. However, for LCFT, the equivalence of the path integral and conformal bootstrap approaches is far from obvious, and how to unify them remains an open problem. Along this direction, the present paper provides a rigorous proof of the smoothness of the correlation functions in LCFT. This result serves as the first step towards proving that the correlation functions of LCFT satisfy the higher Ward identities and the higher Belavin-Polyakov-Zamolodchikov equations, which is a prediction of the conformal bootstrap approach.
Reviewer: Hong-Hao Tu (Dresden)
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 60D05 | Geometric probability and stochastic geometry |
| 81T08 | Constructive quantum field theory |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 81V17 | Gravitational interaction in quantum theory |
| 83C45 | Quantization of the gravitational field |
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