Conformal field theory on the plane

Abstract

We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the $H_3^{+}$ model, and the $S{U}_2$ and ${\widetilde{SL}}_2(\mathbb{R})$ WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.

145 pages. Public domain. Suggest modifications at https://github.com/ribault/CFT-Review . Version 6: Many small improvements

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• Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
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• The DOZZ Formula from the Path Integral
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• Integrability of Liouville theory: proof of the DOZZ Formula
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• Constructive Liouville Conformal Field Theory
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• Mabuchi spectrum from the minisuperspace
• Wilson Loop Invariants from $W_N$ Conformal Blocks
• Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus
• On the CFT describing the spin clusters in 2d Potts model
• Level Set Percolation in Two-Dimensional Gaussian Free Field
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• Probabilistic conformal blocks for Liouville CFT on the torus
• Path integral approach to analytic continuation of Liouville theory: the pencil region
• Remarks on BMS${}_3$ invariant field theories: correlation functions and nonunitary CFTs
• Standard modules, radicals, and the valenced Temperley-Lieb algebra
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