Higher String Functions, Higher-Level Appell Functions, and the Logarithmic $${{{\widehat{s\ell}2_k/u(1)}}}$$ CFT Model

Semikhatov, A. M.

doi:10.1007/s00220-008-0677-0

Cited by 7 publications

(8 citation statements)

References 52 publications

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“…The word "parafermionic" is somewhat overloaded here (and, in particular, is not related to the parafermions discussed in the context of logarithmic conformal field theories in [32]); although it is motivated by the discussion in [33], "anyonic" might be a better choice.…”

Section: First-order "Parafermionic" Systemsmentioning

confidence: 97%

A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

Semikhatov

2009

Theor Math Phys

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We show that the full matrix algebra Matp(C) is a U-module algebra for U = Uq s (2), a quantum s (2) group at the 2pth root of unity. The algebra Matp(C) decomposes into a direct sum of projective U-modules P + n with all odd n, 1 ≤ n ≤ p. In terms of generators and relations, this U-module algebra is described as the algebra of q-differential operators "in one variable" with the relations ∂z = q − q −1 + q −2 z∂ and z p = ∂ p = 0. These relations define a "parafermionic" statistics that generalizes the fermionic commutation relations. By the Kazhdan-Lusztig duality, it is to be realized in a manifestly quantumgroup-symmetric description of (p, 1) logarithmic conformal field models. We extend the Kazhdan-Lusztig duality between U and the (p, 1) logarithmic models by constructing a quantum de Rham complex of the new U-module algebra and discussing its field theory counterpart.

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Section: First-order "Parafermionic" Systemsmentioning

confidence: 97%

A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

Semikhatov

2009

Theor Math Phys

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“…As can be seen in [52], a more natural object from the "screening/quantum-group" standpoint may be given by the coset b s (2)/u(1), but we leave its "logarithmization" for the future. (Added in proof: this was considered in [53].) generators of the triplet W -algebra of the logarithmic (p, 1) models.…”

Section: Introductionmentioning

confidence: 97%

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

Semikhatov

2007

Theor Math Phys

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For positive integers p = k + 2, we construct a logarithmic extension of the b s (2) 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 b s (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 by a (θ=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 ofs (2) k integrable-representation characters and Rp+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 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.

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“…), and can be quite interesting at the level of characters (cf. [27]), but the meaning of "algebras" generated by mutually nonlocal fields would then have to be worked out first, before the "three-boson" results in this paper can be restated in that language. 2 From p sℓp2q k p sℓp2q k p sℓp2q k to W algebras.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic $\widehat{s\ell }(2)$ CFT models from Nichols algebras: I

Semikhatov¹,

Tipunin²

2013

J. Phys. A: Math. Theor.

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We construct chiral algebras that centralize rank-two Nichols algebras with at least one fermionic generator. This gives "logarithmic" W -algebra extensions of a fractional-level p sℓp2q algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic CFT models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization UpXq of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the UpXq simple modules are proposed, as a first step toward a functorial relation between UpXq and W -algebra representation categories.

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Higher String Functions, Higher-Level Appell Functions, and the Logarithmic $${{{\widehat{s\ell}2_k/u(1)}}}$$ CFT Model

Cited by 7 publications

References 52 publications

A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

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

Logarithmic $\widehat{s\ell }(2)$ CFT models from Nichols algebras: I

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