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 TMF, 2007, Volume 153, Number 3, Pages 291–346 (Mi tmf6139)

Toward logarithmic extensions of $\widehat{s\ell}(2)_k$ conformal field models

A. M. Semikhatov

P. N. Lebedev Physical Institute, Russian Academy of Sciences

Abstract: For positive integers $p=k+2$, we construct a logarithmic extension of the $\widehat{s\ell}_k$ conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of $\widehat{s\ell}_k$. 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 by a $(\theta=1)$-twisted highest-weight state of the same dimension $4p-2$ and opposite 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 $\mathscr{R}_{p+1}\oplus\mathbb{C}^2{\otimes}\mathscr{R}_{p+1}\oplus \mathscr{R}_{p-1}\oplus\mathbb{C}^2\otimes \mathscr{R}_{p-1}\oplus\mathbb{C}^3\otimes\mathscr{R}_{p-1}$, where $\mathscr{R}_{p-1}$ is the $SL(2,\mathbb{Z})$-representation on $\widehat{s\ell}_k$ integrable-representation characters and $\mathscr{R}_{p+1}$ is a $(p+1)$-dimensional $SL(2,\mathbb{Z})$-representation known from the logarithmic $(p,1)$ model. The dimension $9p-3$ is conjecturally the dimension of the space of torus amplitudes, and the $\mathbb{C}^n$ with $n=2$ and $3$ suggest the Jordan cell sizes in indecomposable $W$-algebra modules. We show that under Hamiltonian reduction, the $W$-algebra currents map into the currents of the triplet $W$-algebra of the logarithmic $(p,1)$ model.

Keywords: logarithmic conformal field theory, $W$-algebra, fermionic screening, butterfly resolution, characters, modular transformation

DOI: https://doi.org/10.4213/tmf6139

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English version:
Theoretical and Mathematical Physics, 2007, 153:3, 1597–1642

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Revised: 15.04.2007

Citation: A. M. Semikhatov, “Toward logarithmic extensions of $\widehat{s\ell}(2)_k$ conformal field models”, TMF, 153:3 (2007), 291–346; Theoret. and Math. Phys., 153:3 (2007), 1597–1642

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf6139
• https://doi.org/10.4213/tmf6139
• http://mi.mathnet.ru/eng/tmf/v153/i3/p291

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. M. Semikhatov, “Factorizable ribbon quantum groups in logarithmic conformal field theories”, Theoret. and Math. Phys., 154:3 (2008), 433–453
2. A. M. Gainutdinov, “A generalization of the Verlinde formula in logarithmic conformal field theory”, Theoret. and Math. Phys., 159:2 (2009), 575–586
3. Semikhatov A.M., “Higher string functions, higher-level Appell functions, and the logarithmic $\widehat{\mathrm{sl}}(2)_k/\mathrm{u}(1)$ CFT model”, Comm. Math. Phys., 286:2 (2009), 559–592
4. Gainutdinov A.M., Tipunin I.Yu., “Radford, Drinfeld and Cardy boundary states in the $(1,p)$ logarithmic conformal field models”, J. Phys. A, 42:31 (2009), 315207, 30 pp.
5. Semikhatov A.M., “A Heisenberg Double Addition to the Logarithmic Kazhdan-Lusztig Duality”, Letters in Mathematical Physics, 92:1 (2010), 81–98
6. Bushlanov P.V. Gainutdinov A.M. Tipunin I.Yu., “Kazhdan-Lusztig Equivalence and Fusion of Kac Modules in Virasoro Logarithmic Models”, Nucl. Phys. B, 862:1 (2012), 232–269
7. Semikhatov A.M. Tipunin I.Yu., “Logarithmic (Sl)Over-Cap(2) CFT Models From Nichols Algebras: I”, J. Phys. A-Math. Theor., 46:49, SI (2013), 494011
8. Hadjiivanov L. Furlan P., “Quantum Groups as Generalized Gauge Symmetries in WZNW Models. Part i. the Classical Model”, Phys. Part. Nuclei, 48:4 (2017), 509–563
9. Hadjiivanov L. Furlan P., “Quantum Groups as Generalized Gauge Symmetries in WZNW Models. Part II. the Quantized Model”, Phys. Part. Nuclei, 48:4 (2017), 564–621
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