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Continued fractions and fermionic representations for characters of M(p,p′) minimal models

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Abstract

We present fermionic sum representations of the characters χ (p, p′)τ, s of the minimal M(p,p′) models for all relatively prime integers p′>p for some allowed values of r and s. Our starting point is biomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1/2 chain of anisotropy −Δ=−cos(π(p/p′)). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of {p/p′} (and p/p′), where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the Hermitian irreducible representations of SU q- (2) (and SU q+ (2)) with q−=exp(iπ{p/p}) (and q+=exp(iπ(p/p′))). We also establish the duality relation M(p,p′) ↔ M(p′−p,p′) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.

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Authors and Affiliations

  1. Physikalisches Institut der Rheinischen, Friedrich-Wilhelms Universität Bonn, Nussallee 12, D-53115, Bonn, Germany

    Alexander Berkovich

  2. Institute for Theoretical Physics, State University of New York, 11794-3840, Stony Brook, NY, USA

    Barry M. McCoy

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  1. Alexander Berkovich
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  2. Barry M. McCoy
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Dedicated to Prof. Vladimir Rittenberg on his 60th birthday

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Berkovich, A., McCoy, B.M. Continued fractions and fermionic representations for characters of M(p,p′) minimal models. Lett Math Phys 37, 49–66 (1996). https://doi.org/10.1007/BF00400138

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  • DOI: https://doi.org/10.1007/BF00400138

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