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Quantum Liouville theory in the background field formalism. I: Compact Riemann surfaces
"... Abstract. Using Polyakov’s functional integral approach with the Liouville action functional defined in [ZT87c] and [TT03a], we formulate quantum Liouville theory on a compact Riemann surface X of genus g> 1. For the partition function 〈X 〉 and for the correlation functions with the stressenergy ..."
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Abstract. Using Polyakov’s functional integral approach with the Liouville action functional defined in [ZT87c] and [TT03a], we formulate quantum Liouville theory on a compact Riemann surface X of genus g> 1. For the partition function 〈X 〉 and for the correlation functions with the stressenergy tensor components 〈 ∏ n i=1 T(zi) ∏ l k=1 ¯ T ( ¯wk)X〉, we describe Feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution — the hyperbolic metric on X. Extending analysis in [Tak93, Tak94, Tak96a, Tak96b], we define the regularization scheme for any choice of global coordinate on X, and for Schottky and quasiFuchsian global coordinates we rigorously prove that one and twopoint correlation functions satisfy conformal Ward identities in all orders of the perturbation theory. Obtained results are interpreted in terms of complex geometry
Zamolodchikov relations and Liouville hierarchy in SL(2,R) k WZNW model
, 2004
"... We study the connection between Zamolodchikov operatorvalued relations in Liouville field theory and in the SL(2, R)k WZNW model. In particular, the classical relations in SL(2, R)k can be formulated as a classical Liouville hierarchy in terms of the isotopic coordinates, and their covariance is ea ..."
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We study the connection between Zamolodchikov operatorvalued relations in Liouville field theory and in the SL(2, R)k WZNW model. In particular, the classical relations in SL(2, R)k can be formulated as a classical Liouville hierarchy in terms of the isotopic coordinates, and their covariance is easily understood in the framework of the AdS3/CFT2 correspondence. Conversely, we find a closed expression for the classical Liouville decoupling operators in terms of the so called uniformizing Schwarzian operators and show that the associated uniformizing parameter plays the same role as the isotopic coordinates in SL(2, R)k. The solutions of the jth classical decoupling equation in the WZNW model span a spin j reducible representation of SL(2, R). Likewise, we show that in Liouville theory solutions of the classical decoupling equations span spin j representations of SL(2, R), which is interpreted as the isometry group of the hyperbolic upper halfplane. We also discuss the connection with
unknown title
, 2003
"... hepth/0308131 Polyakov conjecture for hyperbolic singularities ..."
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Classical geometry from the quantum Liouville theory
, 2005
"... Zamolodchikov’s recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are si ..."
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Zamolodchikov’s recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are simply related to the geodesic length functions of the hyperbolic geometry on the 4punctured Riemann sphere. Such relation provides new powerful methods for both numerical and analytical calculations of these functions. The consistency conditions for the factorization of the 4point classical Liouville action in different channels are numerically verified. The factorization yields efficient numerical methods to calculate the 4point classical action and, by the Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of the 4punctured sphere.