Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces

Details Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces

2005-08-25

arxiv.org/abs/hep-th/0508188v2

Commun.Math.Phys. 268 (2006) 135-197

Physics

High Energy Physics

High Energy Physics - Theory

67 pages, 4 figures (Typos corrected as in the published version)

Scientific paper

10.1007/s00220-006-0091-4

Using Polyakov's functional integral approach with the Liouville action functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function and for the correlation functions with the stress-energy tensor components $<\prod_{i=1}^{n}T(z_{i})\prod_{k=1}^{l}\bar{T}(\w_{k})X>$, 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 \cite{LT1,LT2,LT-Varenna,LT3}, we define the regularization scheme for any choice of global coordinate on X, and for Schottky and quasi-Fuchsian global coordinates we rigorously prove that one- and two-point correlation functions satisfy conformal Ward identities in all orders of the perturbation theory. Obtained results are interpreted in terms of complex geometry of the projective line bundle $\cE_{c}=\lambda_{H}^{c/2}$ over the moduli space $\mathfrak{M}_{g}$, where c is the central charge and $\lambda_{H}$ is the Hodge line bundle, and provide Friedan-Shenker \cite{FS} complex geometry approach to CFT with the first non-trivial example besides rational models.

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