The universal Racah-Wigner symbol for Uq(osp(1|2))

Michal Pawelkiewicz; Paulina Suchanek; Volker Schomerus

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The universal Racah-Wigner symbol for Uq(osp(1|2))

Michal Pawelkiewicz, Volker Schomerus, Paulina Suchanek

High Energy Physics - Theory · Submitted 25 Jul 2013

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The authors' main claim We propose a new and elegant formula for the Racah-Wigner symbol of self-dual continuous series of representations of Uq(osp(1|2)).

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We propose a new and elegant formula for the Racah-Wigner symbol of self-dual continuous series of representations of Uq(osp(1|2)). It describes the entire fusing matrix for both NS and R sector of N=1 supersymmetric Liouville field theory. In the NS sector, our formula is related to an expression derived in [1]. Through analytic continuation in the spin variables, our universal expression reproduces known formulas for the Racah-Wigner coefficients of finite dimensional representations.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a new and elegant formula for the Racah-Wigner symbol of self-dual continuous series of representations of Uq(osp(1\|2)).	contribution 0.98	0.95	0.81	asserted 1.00
s1	It describes the entire fusing matrix for both NS and R sector of N=1 supersymmetric Liouville field theory.	interpretation 0.55	0.74	0.35	asserted 1.00
s2	In the NS sector, our formula is related to an expression derived in [1].	result 0.43	0.27	0.05	asserted 1.00
s3	Through analytic continuation in the spin variables, our universal expression reproduces known formulas for the Racah-Wigner coefficients of finite dimensional representations.	result 0.84	0.85	0.15	asserted 1.00

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Original abstract

We propose a new and elegant formula for the Racah-Wigner symbol of self-dual continuous series of representations of Uq(osp(1|2)). It describes the entire fusing matrix for both NS and R sector of N=1 supersymmetric Liouville field theory. In the NS sector, our formula is related to an expression derived in [1]. Through analytic continuation in the spin variables, our universal expression reproduces known formulas for the Racah-Wigner coefficients of finite dimensional representations.

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References

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
hep-th 2026-10 conditional novelty 7.0 of 10

The torus one-point amplitudes of the four super Virasoro minimal strings are computed from the RNS worldsheet and match random matrix predictions.
Solving the Schwarzian via the Conformal Bootstrap
hep-th 2017-05 conditional novelty 7.0 of 10

Exact correlation functions of the Schwarzian theory, including the out-of-time-ordered four-point function, are obtained as a large-central-charge limit of 2D Virasoro conformal field theory, with the crossing kernel...
Crossing Symmetry in Alpha Space
hep-th 2017-02 accept novelty 7.0 of 10

Decomposing 1d CFT four-point functions in eigenfunctions of the conformal Casimir turns crossing symmetry into an integral eigenvalue problem whose kernel is computed in closed form as a Wilson function.
Solving 3d Gravity with Virasoro TQFT
hep-th 2023-04 conditional novelty 6.0 of 10

3d quantum gravity with negative cosmological constant is reformulated as two copies of a Virasoro TQFT, making gravity partition functions on hyperbolic 3-manifolds algorithmically computable.
Lens partition function, pentagon identity and star-triangle relation
hep-th 2020-09 conditional novelty 4.0 of 10

The authors obtain a hyperbolic hypergeometric identity from gauge symmetry breaking of an S^3_b/Z_r partition function duality and express it as a star-triangle relation and a pentagon identity.

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13 arXiv papers cite this in 22 sentences

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Papers citing it · 13 sentences without a confident label

Builds on it 1 paper

And, finally, we give general expression for the supersymmetric Racah-Wigner symbols, considered in [ref] , in all available parametrizations for both the Neveu-Schwarz and Ramond sectors.

A parafermionic hypergeometric function and supersymmetric 6j-symbols · Introduction · 0.81

Compares with it 2 papers

Then we check that the restriction to the Neveu-Schwarz sector of the general supersymmetric Racah-Wigner symbol, suggested in [ref] , is in agreement with the Neveu-Schwarz sector Racah-Wigner symbols derived in [ref] .

A parafermionic hypergeometric function and supersymmetric 6j-symbols · Introduction · 0.91
Let us mention that in [ref] a similar restriction has been implemented for Racah-Wigner 6j-symbols.

Matrix factorization for solutions of the Yang-Baxter equation · Trigonometric factorization · 0.93

Cites it as background 5 papers

In this last appendix we discuss the fusion kernel of (spacelike) $𝒩=1$ Liouville theory in the NS-sector as obtained in [ref] .

On the three-point functions in timelike N=1 Liouville CFT · Relation to the $𝒩=1$ fusion kernel · 0.80
Some attempts to find the fusion matrix also in the Ramond sector can be found in [ref] .

Comments on fusion matrix in N=1 super Liouville field theory · Introduction · 0.95
While several results are known on this object, the fusion matrix with this specific configuration is not yet available [ref] .

Solving the Schwarzian via the Conformal Bootstrap · $𝒩=1$ Schwarzian theory · 0.87

What it cites · 8 sentences without a confident label

Compares with 1 paper

Thus the Racah-Wigner symbol [ref] coincides with the fusion matrix of Liouville theory [ref] .

6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories · The Racah-Wigner symbol of · 0.89

Cites as background 3 papers

As we recalled in the introduction, the fusing matrix of $N=1$ Liouville field theory should be a central ingredient in the construction of a new 3-dimensional topological quantum field theory, just as Faddeev's quantum dilogarithm [ref] , i.e. the building block of the fusing matrix on Liouville field theory, is used to construct SL(2) Chern-Simons or quantum Teichmueller theory, see e.g. [ref] .

Chern-Simons Theory and S-duality · The Racah-Wigner symbol of · 0.85
As we recalled in the introduction, the fusing matrix of $N=1$ Liouville field theory should be a central ingredient in the construction of a new 3-dimensional topological quantum field theory, just as Faddeev's quantum dilogarithm [ref] , i.e. the building block of the fusing matrix on Liouville field theory, is used to construct SL(2) Chern-Simons or quantum Teichmueller theory, see e.g. [ref] .

A TQFT from quantum Teichm\"uller theory · The Racah-Wigner symbol of · 0.85
As we recalled in the introduction, the fusing matrix of $N=1$ Liouville field theory should be a central ingredient in the construction of a new 3-dimensional topological quantum field theory, just as Faddeev's quantum dilogarithm [ref] , i.e. the building block of the fusing matrix on Liouville field theory, is used to construct SL(2) Chern-Simons or quantum Teichmueller theory, see e.g. [ref] .

On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory · The Racah-Wigner symbol of · 0.85

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