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

Abstract

We present fermionic sum representations of the characters χ τ, s (p, p′) 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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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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Mathematics Subject Classification (1991)

  • 81T10

Key words

  • conformal field theory
  • fermionic sum representations
  • continued fractions
  • Andrews-Bailey transformations
  • Rogers-Ramanujan identities