×

zbMATH — the first resource for mathematics

On moduli space of symmetric orthogonal matrices and exclusive Racah matrix \(\overline{S}\) for representation \(R=[3,1]\) with multiplicities. (English) Zbl 1397.81091
Summary: Racah matrices and higher \(j\)-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of the author [Mod. Phys. Lett. A 33, No. 12, Article ID 1850062, 19 p. (2018; Zbl 1386.81097)], we obtain the exclusive Racah matrix \(\overline{S}\) for the currently-front-line case of representation \(R = [3, 1]\) with non-trivial multiplicities, where it is actually operator-valued, i.e., depends on the choice of bases in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of \(\overline{S}\) needs to be reconstructed from orthogonality conditions. Therefore, we discuss the abundance of symmetric orthogonal matrices to which \(\overline{S}\) belongs, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus, the knowledge approximately matches the freedom, and this explains why the method can work – with some limited addition of educated guesses. A similar calculation for \(R = [r, 1]\) with \(r > 3\) should also be doable.

MSC:
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Morozov, A.
[2] Landau, L. D.; Lifshitz, E. M., Quantum mechanics: non-relativistic theory, (1977), Pergamon Press · Zbl 0178.57901
[3] Chern, S.-S.; Simons, J.; Witten, E.; Reshetikhin, N.; Turaev, V.; Guadagnini, E.; Martellini, M.; Mintchev, M.; Guadagnini, E.; Martellini, M.; Mintchev, M.; Turaev, V. G.; Viro, O. Y.; Kaul, R. K.; Govindarajan, T. R.; Kaul, R. K.; Govindarajan, T. R.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Ramadevi, P.; Govindarajan, T. R.; Kaul, R. K.; Morozov, A.; Smirnov, A.; Smirnov, A.; Mironov, A.; Morozov, A.; Morozov, An.; Anokhina, A.; Mironov, A.; Morozov, A.; Morozov, An., (Rebhan, A.; Katzarkov, L.; Knapp, J.; Rashkov, R.; Scheidegger, E., Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, (2013), World Scientific Publishing Co. Pte. Ltd.), Nucl. Phys. B, 868, 271-313, (2013)
[4] Alexander, J. W.; Jones, V. F.R.; Jones, V. F.R.; Jones, V. F.R.; Kauffman, L.; Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B.R.; Millet, K.; Ocneanu, A.; Przytycki, J. H.; Traczyk, K. P.; Morozov, A., Trans. Am. Math. Soc., Invent. Math., Bull. Am. Math. Soc., Ann. Math., Topology, Bull. Am. Math. Soc., Kobe J. Math., Theor. Math. Phys., 187, 2, 447-454, (2016)
[5] Gu, J.; Jockers, H., Commun. Math. Phys., 338, 393-456, (2015)
[6] Mironov, A.; Morozov, A.; Sleptsov, A.; Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Singh, V. K.; Nawata, S.; Ramadevi, P.; Kumar Singh, Vivek; Mironov, A.; Morozov, A.; Mironov, A.; Morozov, A.; Morozov, An.; Ramadevi, P.; Kumar Singh, Vivek; Sleptsov, A., J. High Energy Phys., J. High Energy Phys., Nucl. Phys. B, 899, 395-413, (2015)
[7] Caudron, A.; Bonahon, F.; Siebenmann, L. C.; Gabai, D., Genera of arborescent links, vol. 339, Publ. Math. Orsay, vol. 82-4, (1986), AMS
[9] Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 03, (2012)
[10] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An.; Mironov, A.; Morozov, A.; Mironov, A.; Morozov, A., Int. J. Mod. Phys. A, Phys. Lett. B, 755, 47-57, (2016)
[11] Tuba, I.; Wenzl, H.
[12] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.; Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., J. High Energy Phys., JETP Lett., 104, 56-61, (2016)
[13] Shakirov, Sh.; Sleptsov, A., in press
[14] Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A.; Galakhov, D.; Melnikov, D.; Mironov, A.; Morozov, A.; Sleptsov, A., Phys. Lett. B, Nucl. Phys. B, 899, 194-228, (2015)
[15] Mironov, A.; Morozov, A.; Shakirov, Sh.; Mironov, A.; Morozov, A.; Shakirov, Sh.; Mironov, A.; Morozov, A.; Shakirov, Sh.; Smirnov, A., J. High Energy Phys., Int. J. Mod. Phys. A, Nucl. Phys. B, 855, 128-151, (2012) · Zbl 1247.81397
[16] Zamolodchikov, Al., Commun. Math. Phys., 96, 419, (1984)
[17] Mironov, A.; Morozov, A., Phys. Lett. B, 682, 118-124, (2009)
[18] Sh. Shakirov, in press.
[19] Mironov, A.; Morozov, A.; Morozov, An., AIP Conf. Proc., 1562, 123, (2013)
[20] Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A., Phys. Lett. B, 760, 45-58, (2016)
[21] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An., J. High Energy Phys., 2012, (2012)
[22] Itoyama, H.; Mironov, A.; Morozov, A.; Morozov, An.; Nawata, S.; Ramadevi, P.; Zodinmawia; Sun, X.; Fuji, H.; Gukov, S.; Stosic, M.; Sulkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A., Int. J. Mod. Phys. A, J. High Energy Phys., J. High Energy Phys., 07, (2015)
[23] Morozov, A.; Morozov, A.; Kononov, Ya.; Morozov, A.; Kononov, Ya.; Morozov, A., Nucl. Phys. B, J. High Energy Phys., Mod. Phys. Lett. A, 31, 38, 582-605, (2016)
[24] Dunin-Barkowski, P.; Mironov, A.; Morozov, A.; Sleptsov, A.; Smirnov, A., J. High Energy Phys., 03, (2013)
[25] Dolotin, V.; Morozov, A., Introduction to non-linear algebra, WS 2007 · Zbl 1134.15001
[26] Arthamonov, S.; Mironov, A.; Morozov, A.; Arthamonov, S.; Mironov, A.; Morozov, A.; Morozov, An.; Kononov, Ya.; Morozov, A., Theor. Math. Phys., J. High Energy Phys., Pis’ma Zh. Eksp. Teor. Fiz., 101, 931-934, (2015)
[27] Rosso, M.; Jones, V. F.R.; Lin, X.-S.; Zheng, H., J. Knot Theory Ramif., Trans. Am. Math. Soc., 362, 1-18, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.