Affine Jordan cells, logarithmic correlators, and hamiltonian reduction

Abstract

We study a particular type of logarithmic extension of SL(2,R) Wess-Zumino-Witten models. It is based on the introduction of affine Jordan cells constructed as multiplets of quasi-primary fields organized in indecomposable representations of the Lie algebra sl(2). We solve the simultaneously imposed set of conformal and SL(2,R) Ward identities for two- and three-point chiral blocks. These correlators will in general involve logarithmic terms and may be represented compactly by considering spins with nilpotent parts. The chiral blocks are found to exhibit hierarchical structures revealed by computing derivatives with respect to the spins. We modify the Knizhnik-Zamolodchikov equations to cover affine Jordan cells and show that our chiral blocks satisfy these equations. It is also demonstrated that a simple and well-established prescription for hamiltonian reduction at the level of ordinary correlators extends straightforwardly to the logarithmic correlators as the latter then reduce to the known results for two- and three-point conformal blocks in logarithmic conformal field theory.

arXiv comment31 pages, v2: reference corrected, v3: version to be published

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• On logarithmic solutions to the conformal Ward identities
• Towards the Construction of Local Logarithmic Conformal Field Theories
• Coset construction of logarithmic minimal models: branching rules and branching functions
• Staggered modules of $N=2$ superconformal minimal models
• Staggered and affine Kac modules over $A_1^{(1)}$
• Indecomposable Representations in Z_n Symmetric b,c Ghost Systems via Deformations of the Virasoro Field
• Special function models of indecomposable $sl(2)$ representations: The Laguerre Case