[Paper Review] Self-dual Continuous Series of Representations for U_q(sl(2)) and U_q(osp(1|2))

Leszek Hadasz, Michal Pawelkiewicz+1 more|DESY (CERN, DESY, Fermilab, IHEP, and SLAC)|May 20, 2013

Algebraic structures and combinatorial modelsMathematics27 references16 citations

TL;DR

This paper derives the Clebsch-Gordan and Racah-Wigner coefficients for self-dual continuous series representations of the quantum groups 𝒰_q(sl(2)) and 𝒰_q(osp(1|2)), extending known results for 𝒰_q(sl(2)) to the superalgebra case. It shows that the Racah-Wigner coefficients for 𝒰_q(osp(1|2)) precisely match the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory, establishing a direct link between representation theory and conformal field theory via Faddeev's quantum dilogarithm and its supersymmetric generalization.

ABSTRACT

We determine the Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of the quantum deformed algebras U_q(sl(2)) and U_q(osp(1|2)). While our results for the former algebra reproduce formulas by Ponsot and Teschner, the expressions for the orthosymplectic algebra are new. Up to some normalization factors, the associated Racah-Wigner coefficients are shown to agree with the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory.

Motivation & Objective

• To extend the known results on Clebsch-Gordan and Racah-Wigner coefficients for 𝒰_q(sl(2)) to the quantum superalgebra 𝒰_q(osp(1|2)).
• To establish a correspondence between the Racah-Wigner coefficients of 𝒰_q(osp(1|2)) and the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory.
• To introduce and analyze a new pair of special functions, Φ_b^ν(z), which generalize Faddeev’s quantum dilogarithm and satisfy modified integral identities and pentagon relations.
• To provide a systematic derivation of intertwining maps and orthonormality relations for continuous series representations in both the non-supersymmetric and supersymmetric cases.

Proposed method

• Constructs Clebsch-Gordan maps for self-dual continuous series representations of 𝒰_q(sl(2)) and 𝒰_q(osp(1|2)) using integral representations and intertwining properties.
• Derives orthogonality and completeness relations for the Clebsch-Gordan coefficients via contour integration and special function identities.
• Computes Racah-Wigner coefficients (6j symbols) through a recursive decomposition of tensor product representations, using the intertwiner formalism.
• Introduces the functions Φ_b^ν(z) as a supersymmetric extension of Faddeev’s quantum dilogarithm Φ_b^±(z), defined via logarithmic combinations of the standard functions.
• Establishes new integral identities for Φ_b^ν(z) that generalize the pentagon relation and Ramanujan’s summation formula to the supersymmetric setting.
• Performs regularization and limit procedures (e.g., ε → 0) to extract delta-function behavior and ensure consistency of distributional identities.

Experimental results

Research questions

• RQ1 How can the Clebsch-Gordan decomposition be generalized from 𝒰_q(sl(2)) to the quantum superalgebra 𝒰_q(osp(1|2)) in the context of self-dual continuous series representations?
• RQ2 What is the explicit form of the Racah-Wigner coefficients for 𝒰_q(osp(1|2)), and how do they relate to known data in conformal field theory?
• RQ3 How do the special functions Φ_b^ν(z) generalize Faddeev’s quantum dilogarithm, and what new algebraic identities do they satisfy?
• RQ4 To what extent do the Racah-Wigner coefficients of 𝒰_q(osp(1|2)) match the fusing matrix in N=1 supersymmetric Liouville field theory?

Key findings

• The Racah-Wigner coefficients for 𝒰_q(osp(1|2)) are explicitly computed and shown to agree with the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory, up to normalization factors.
• The functions Φ_b^ν(z) are introduced as a supersymmetric generalization of Faddeev’s quantum dilogarithm, satisfying a modified pentagon relation and new integral identities involving a sum over ν.
• The Clebsch-Gordan coefficients for 𝒰_q(osp(1|2)) are constructed using a consistent intertwiner formalism, with orthonormality and completeness proven via contour integration and special function identities.
• The regularization procedure involving ε → 0 and the use of Ramanujan’s summation formula allow the extraction of delta-function behavior, confirming the consistency of the distributional identities.
• The results for 𝒰_q(sl(2)) reproduce known formulas by Ponsot and Teschner, validating the methodological framework before extending it to the superalgebra case.
• The paper establishes a direct correspondence between the representation theory of 𝒰_q(osp(1|2)) and the N=1 supersymmetric Liouville field theory, providing a representation-theoretic foundation for its fusing matrix.