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The aim of this course is to give an introduction to the Liouville CFT. This is the simplest non-trivial CFT with continuum spectrum of the central charge and conformal dimensions. The Liouville CFT describes the measure in the space of Riemann metrics and as such is an essential component of the world-sheet description of the theories of two-dimensional quantum gravity, also known as Liouville gravity.

1. Integration measure over Riemann metrics of a surface with and without boundary. Conformal anomaly and balance of the central charges. Liouville action for 2D quantum gravity. KPZ scaling rule. 2. Classical Liouville theory. Classical solutions in Minkowski and Euclidean spaces. Canonical formalism and quantisation. Wheeler-De Witt equation. Wave functions and Hilbert space. 3. Liouville theory on a sphere: Lagrangian density, energy-momentum tensor, central charge, primary fields and spectrum of anomalous dimensions. Reflection symmetry and reflection coefficient. Two- and three-point correlation functions. 4. Boundary Liouville theory. Correlation functions. Boundary reflection amplitude. Difference equation for the three-point function.

[1] A. M. Polyakov. Fine Structure of Strings. Nucl. Phys., B268:406–412, 1986. [2] O. Alvarez. Theory of Strings with Boundaries: Fluctuations, Topology, and Quantum Geometry. Nucl. Phys. B, 216:125–184, 1983. [3] F. David. Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge. Mod. Phys. Lett., A3:1651, 1988. [4] N. Seiberg, Notes on Quantum Liouville Theory and Quantum Gravity, https://academic.oup.com/ptps/article/doi/10.1143/ PTP.102.319/1904500 [5] A. B. Z. Zamolodchikov and A. Zamolodchikov. Lectures on liouville theory and matrix models, http://qft.itp.ac.ru/ZZ.pdf. [6] A. B. Zamolodchikov and A. B. Zamolodchikov. Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys., B477:577–605, 1996. [7] V. Fateev, A. B. Zamolodchikov, and A. B. Zamolodchikov. Boundary Liouville field theory. I: Boundary state and boundary two-point function. 2000. [8] I. Kostov, B. Ponsot, and D. Serban. Boundary liouville theory and 2d quantum gravity. Nucl.Phys., B683:309–362, 2004.