Toward logarithmic extensions of \widehatsℓ (2)_k conformal field models

2007/01/31 by AM Semikhatov, A. M. Semikhatov · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Central charge #Conformal field theory #Conformal map #Dimension (graph theory) #Indecomposable module #Integrable system #Kernel (algebra) #Logarithm #Modular group #Torus #hep-th #math-ph #math.MP #math.QA

paper · pdf · doi:10.1007/s11232-007-0135-8

published as Theor.Math.Phys.153:1597-1642,2007 · 54 pages, amsart++, xy, graphicx, times&mathptmx. V2: references added, some changes in the presentation

arxiv created 2007/04/15 · openalex publication_date 2007/12/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

For positive integer p=k+2, we construct a logarithmic extension of the ^sl(2)_k conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a three-boson realization of ^sl(2)_k. The currents W^-(z) and W⁺(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p-2 and charge 2p-1, and a (theta=1)-twisted highest-weight state of the same dimension 4p-2 and charge -2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p-1 integrable representation characters they generate a modular group representation whose structure is described as a deformation of the (9p-3)-dimensional representation R_p-1 ⊕ C² \tensor R_p-1 ⊕ R_p-1 ⊕ C² \tensor R_p-1 ⊕ C³ \tensor R_p-1, where R_p-1 is the SL(2,Z)-representation on integrable representation characters and R_p-1 is a (p+1)-dimensional SL(2,Z)-representation known from the logarithmic (p,1) model. The dimension 9p-3 is conjecturally the dimension of the space of torus amplitudes, and the Cⁿ with n=2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. Under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p,1) model.

Citations

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• TFT construction of RCFT correlators I: partition functions
• The logarithmic triplet theory with boundary

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• Kazhdan–Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models