New Pentagon Identities Revisited

Shahriyar Jafarzade

doi:10.1088/1742-6596/1194/1/012054

Shahriyar Jafarzade

Open Access

https://doi.org/10.1088/1742-6596/1194/1/012054

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Abstract

We present a new solution to the pentagon identity in terms of gamma function. We obtain this solution by taking the gamma function limit from the pentagon identity related to the three-dimesional index. This limit corresponds to the identification of the sphere partition function of dual theories and being equivalent to the star-triangle relation in statistical mechanics which corresponds to the “strongly coupled” regime of the Faddeev-Volkov model.

Highlights

Following identity is called the pentagon identity [1] for non-commutative operators xy = qyx l(y)l(x) = l(x)l(−xy)l(y), (1)where l(x) = ∞ i=1(1 − xqi) and |q| < 1 for complex variable x
Pentagon identity for the quantum dilogarithm has been interpreted as the restricted star-triangle relation in statistical mechanics [1] and Pachner-moves [3]
Considering the same duality for three dimensional index leads to the following pentagon identity [14] 4

Summary

Introduction

3. Considering the same duality for three dimensional index leads to the following pentagon identity [14] 4. 2. Gamma function limit we consider the gamma function limit of the pentagon identity related to the three dimensional index. Gamma function limit we consider the gamma function limit of the pentagon identity related to the three dimensional index This limit corresponds to the new pentagon identities in terms of gamma function. This pentagon identity have interpretation as the identification of the sphere partition function for the dual theories. It is interesting that this identity corresponds to the identification of the sphere partition functions [13] for the three dimensional duality which we discussed below the formula (8). Statistical mechanics interpretation of this pentagon identity is equivalent to the star-triangle relation with continuous and discrete spin parameters.

Other Gamma Function Limits

Conclusion