Abstract
We propose a new and elegant formula for the Racah-Wigner symbol of selfdual continuous series of representations of U q (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 RacahWigner coefficients of finite dimensional representations.
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L. Hadasz, M. Pawelkiewicz and V. Schomerus, Self-dual Continuous Series of Representations for \( \mathcal{U} \) q (sl(2)) and \( \mathcal{U} \) q (osp(1|2)), arXiv:1305.4596 [INSPIRE].
L. Hadasz, On the fusion matrix of the N = 1 Neveu-Schwarz blocks, JHEP 12 (2007) 071 [arXiv:0707.3384] [INSPIRE].
D. Chorazkiewicz and L. Hadasz, Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks, JHEP 01 (2009) 007 [arXiv:0811.1226] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
B. Ponsot and J. Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U(q)(SL(2, \( \mathbb{R} \))), Commun. Math. Phys. 224 (2001) 613 [math/0007097] [INSPIRE].
J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry and supersymmetric gauge theories, arXiv:1202.4698 [INSPIRE].
A.N. Kirillov and N.Y. Reshetikhin, Representations of the algebra U q (sl(2)), q orthogonal polynomials and invariants of links, in New developments in the theory of knots, T. Kohno eds., World Scientific, Singapore, pg. 202.
V.G. Kac, Infinite dimensional Lie algebras and groups, Proceedings of the Conference Held at Cirm, Luminy, Marseille France (1988), Advanced Series in Mathematical Physics. Vol. 7, World Scientific, Singapore (1989).
P. Minnaert and M. Mozrzymas, Racah coefficients and 6j symbols for the quantum superalgebra U q (osp(1/2)), J. Math. Phys. 36 (1995) 907 [INSPIRE].
P. Minnaert and M. Mozrzymas, Analytical formulae for Racah coefficients and 6 − j symbols of the quantum superalgebra U q (osp(1|2)), J. Math. Phys. 38 (1997) 2676 [INSPIRE].
H. Saleur, Quantum Osp(1|2) and Solutions of the Graded Yang-Baxter Equation, Nucl. Phys. B 336 (1990) 363 [INSPIRE].
I. Ennes, P. Ramadevi, A. Ramallo and J. Sanchez de Santos, Duality in osp(1|2) conformal field theory and link invariants, Int. J. Mod. Phys. A 13 (1998) 2931 [hep-th/9709068] [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
L. Hadasz, Z. Jaskolski and M. Piatek, Analytic continuation formulae for the BPZ conformal block, Acta Phys. Polon. B 36 (2005) 845 [hep-th/0409258] [INSPIRE].
L. Álvarez-Gaumé, C. Gomez and G. Sierra, Quantum Group Interpretation of Some Conformal Field Theories, Phys. Lett. B 220 (1989) 142 [INSPIRE].
V. Dotsenko and V. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].
P. Furlan, A.C. Ganchev and V. Petkova, Fusion Matrices and C < 1 (Quasi)local Conformal Theories, Int. J. Mod. Phys. A 5 (1990) 2721 [Erratum ibid. A 5 (1990) 3641] [INSPIRE].
G. Felder, J. Fröhlich and G. Keller, Braid Matrices and Structure Constants for Minimal Conformal Models, Commun. Math. Phys. 124 (1989) 647 [INSPIRE].
E. Witten, private communication.
D. Chorazkiewicz, L. Hadasz and Z. Jaskolski, Braiding properties of the N = 1 super-conformal blocks (Ramond sector), JHEP 11 (2011) 060 [arXiv:1108.2355] [INSPIRE].
A. Recknagel and V. Schomerus, Boundary Conformal Field Theory and the Worldsheet Approach to D-branes, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2014).
N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [INSPIRE].
A. Belavin, M. Bershtein, B. Feigin, A. Litvinov and G. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Comm. Math. Phys. 319 (2013) 269 [arXiv:1111.2803] [INSPIRE].
V. Schomerus and P. Suchanek, Liouville’s imaginary shadow, JHEP 12 (2012) 020 [arXiv:1210.1856] [INSPIRE].
L. Hadasz and Z. Jaskolski, Super-Liouville — Double-Liouville correspondence, arXiv:1312.4520 [INSPIRE].
V. Schomerus, Worldsheet duality for spacetime fermions, talk presented at the Gauge theory angle at integrability workshop, Simons Center, Stony Brook (2012), http://media.scgp.stonybrook.edu/video/video.php?f=20121113 1 qtp.mp4
L. Faddeev and R. Kashaev, Quantum Dilogarithm, Mod. Phys. Lett. A 9 (1994) 427 [hep-th/9310070] [INSPIRE].
A.Y. Volkov, Noncommutative hypergeometry, Commun. Math. Phys. 258 (2005) 257 [math/0312084] [INSPIRE].
R. Kashaev, The Hyperbolic volume of knots from quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 [INSPIRE].
R. Kashaev, Quantization of Teichmueller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105 [INSPIRE].
R. Kashaev, The quantum dilogarithm and Dehn twists in quantum Teichmller theory, in NATO Science Series II. Vol. 35: Integrable structures of exactly solvable two-dimensional models of quantum field theory, Kluwer Academic Publishers, Dordrecht The Netherlands (2001).
J. Teschner, On the relation between quantum Liouville theory and the quantized Teichmüller spaces, Int. J. Mod. Phys. A 19S2 (2004) 459 [hep-th/0303149] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
J.E. Andersen and R. Kashaev, A TQFT from quantum Teichmüller theory, arXiv:1109.6295 [INSPIRE].
I. Nidaiev and J. Teschner, On the relation between the modular double of U q (SL(2, \( \mathbb{R} \))) and the quantum Teichmueller theory, arXiv:1302.3454 [INSPIRE].
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Pawelkiewicz, M., Schomerus, V. & Suchanek, P. The universal Racah-Wigner symbol for U q (osp(1|2)). J. High Energ. Phys. 2014, 79 (2014). https://doi.org/10.1007/JHEP04(2014)079
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DOI: https://doi.org/10.1007/JHEP04(2014)079