Two and three-point functions in Liouville theory

Abstract

Based on our generalization of the Goulian-Li continuation in the power of the 2D cosmological term we construct the two and three-point correlation functions for Liouville exponentials with generic real coefficients. As a strong argument in favour of the procedure we prove the Liouville equation of motion on the level of three-point functions. The analytical structure of the correlation functions as well as some of its consequences for string theory are discussed. This includes a conjecture on the mass shell condition for excitations of noncritical strings. We also make a comment concerning the correlation functions of the Liouville field itself.

15 pages, Latex, Revised version: A sign error in formula (50) is corrected

References in corpus (8)

• World Sheet and Space Time Physics in Two Dimensional (Super) String Theory
• On Correlation Functions for Non-critical Strings with c<1 but d>1
• Gravity-Matter Couplings from Liouville Theory
• The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity
• On the Exact Operator Formalism of Two-Dimensional Liouville Quantum Gravity in Minkowski Spacetime
• On the Liouville Approach to Correlation Functions for 2-D Quantum Gravity
• Analysis of All Dimensionful Parameters Relevant in Gravitational Dressing of Conformal Theories
• Conformally invariant off-shell string physics

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