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Interplay between symmetries of quantum 6j-symbols and the eigenvalue hypothesis

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Abstract

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of \(U_q(sl_N)\) is uniquely determined by eigenvalues of the corresponding quantum \(\mathcal {R}\)-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also, due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6j-symbols, about which almost nothing is known for \(N>2\), with the exception of the tetrahedral symmetries, complex conjugation and transformation \(q \longleftrightarrow q^{-1}\). In this paper, we prove the eigenvalue hypothesis in \(U_q(sl_2)\) case and show that it is equivalent to 6j-symbol symmetries (the Regge symmetry and two argument permutations). Then, we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of \(U_q(sl_N)\) and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries.

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Notes

  1. All parameters, corresponding to the second matrix we will label by \(\ \tilde{ }\).

  2. \(R_i=[r_i], \ \widetilde{R}_i=[\widetilde{r}_i]\)

References

  1. Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246 (1985). https://doi.org/10.1090/S0273-0979-1985-15361-3

    Article  MathSciNet  MATH  Google Scholar 

  2. Przytycki, J.H., Traczyk, P.: Invariants of links of Conway type. J. Knot Theor. 4, 115–139 (1987). https://doi.org/10.1142/S0218216513500788. arXiv:1610.06679

    Article  MathSciNet  MATH  Google Scholar 

  3. Satoshi Nawata, Ramadevi, P., Zodinmawia: Colored HOMFLY polynomials from Chern–Simons theory. J. Knot Theor. 22, 1350078 (2013). https://doi.org/10.1142/S0218216513500788. arXiv:1302.5144

    Article  MathSciNet  MATH  Google Scholar 

  4. Yu, N., Turaev, V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127, 1–26 (1990). https://doi.org/10.1007/BF02096491

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Kirillov, A.N., Reshetikhin, N.Y.: Representations of the algebra \(U_q(sl(2))\), \(q\)-orthogonal polynomials and invariants of links. In: New Developments in the Theory of Knots, pp. 202–256. World Scientific, 1990. https://doi.org/10.1142/9789812798329_0012

  6. Rosengren, H.: An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic). Ramanujan J. 13, 133–168 (2007). arXiv:math/0312310

    Article  MathSciNet  Google Scholar 

  7. Freidel, L., Louapre, D.: Asymptotics of 6j and 10j symbols. Class. Quant. Grav. 20, 1267–1294 (2003). https://doi.org/10.1088/0264-9381/20/7/303. arXiv:hep-th/0209134

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Teschner, J., Vartanov, G.: 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories. Lett. Math. Phys. 104, 527–551 (2014). https://doi.org/10.1007/s11005-014-0684-3. arXiv:1202.4698

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Liu, J., Perlmutter, E., Rosenhaus, V., Simmons-Duffin, D.: \(d\)-Dimensional SYK, adS loops, and \(6j\) symbols. JHEP 03, 052 (2019). https://doi.org/10.1007/JHEP03(2019)052. arXiv:1808.00612

    Article  MathSciNet  MATH  Google Scholar 

  10. Saswati, D., Mironov, A., Morozov, A., Morozov, A., Ramadevi, P., Vivek Kumar, S., Sleptsov, A.: Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations. Ann. Henri Poincare, 2019. arXiv:1805.03916https://doi.org/10.1007/s00023-019-00841-z

  11. Alekseev, V., Morozov, A., Sleptsov, A.: Multiplicity-free \(U_q(sl_N)\) 6-j symbols: relations, asymptotics, symmetries. Nucl. Phys. B 960, 115164 (2020)

    Article  Google Scholar 

  12. Mironov, A., Morozov, A., Morozov, A.: Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid. JHEP 03, 034 (2012). https://doi.org/10.1007/JHEP03(2012)034. arXiv:1112.2654

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Itoyama, H., Mironov, A., Morozov, A., Morozov, A.: Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations. Int. J. Mod. Phys. A28, 1340009 (2013). https://doi.org/10.1142/S0217751X13400095. arXiv:1209.6304

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Mironov, A., Morozov, A.: Universal Racah matrices and adjoint knot polynomials: arborescent knots. Phys. Lett. B 755, 47–57 (2016). https://doi.org/10.1016/j.physletb.2016.01.063. arXiv:1511.09077

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Anokhina, A., Morozov, A.: Cabling procedure for the colored HOMFLY polynomials. Teor. Mat. Fiz. 178, 3–68 (2014). https://doi.org/10.1007/s11232-014-0129-2. arXiv:1307.2216

    Article  MathSciNet  MATH  Google Scholar 

  16. Saswati Dhara, Mironov, A., Morozov, A., Morozov, A., Ramadevi, P., Vivek Kumar Singh, Sleptsov A: Eigenvalue hypothesis for multistrand braids. Phys. Rev D97(12), 126015 (2018). https://doi.org/10.1103/PhysRevD.97.126015. arXiv:1711.10952

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Bishler, L., Morozov, A., Sleptsov, A., Shakirov, S.: On the block structure of the quantum \(\cal{R}\)-matrix in the three-strand braids. Int. J. Mod. Phys A33(17), 1850105 (2018). https://doi.org/10.1142/S0217751X18501051. arXiv:1712.07034

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Mironov, A., Morozov, A.: Eigenvalue conjecture and colored Alexander polynomials. Eur. Phys. J. C 78(4), 284 (2018). https://doi.org/10.1140/epjc/s10052-018-5765-5. arXiv:1610.03043

    Article  Google Scholar 

  19. Mishnyakov, V., Sleptsov, A.: Perturbative analysis of the colored Alexander polynomial and KP soliton \(\tau \)-functions. Nucl. Phys. B 965, 115334 (2021). https://doi.org/10.1016/j.nuclphysb.2021.115334

    Article  MathSciNet  Google Scholar 

  20. Mishnyakov, V., Sleptsov, A., Tselousov, N.: A new symmetry of the colored Alexander polynomial. In: Annales Henri Poincaré, pp. 1–31. Springer, (2021)

  21. Mishnyakov, V., Sleptsov, A., Tselousov, N.: A novel symmetry of colored HOMFLY polynomials coming from \(\mathfrak{sl}(N|M)\) superalgebras. arXiv:2005.01188, (2020)

  22. Morozov, A., Sleptsov, A.: New symmetries for the \(U_q(sl_N)\)\(6\)-\(j\) symbols from the Eigenvalue conjecture. JETP Lett. 108(10), 697–704 (2018). https://doi.org/10.1134/S0021364018220058. arXiv:1905.01876

    Article  ADS  Google Scholar 

  23. Klimyk, A., Schmudgen, K.: Quantum Groups and Their Representations. Springer, Berlin (1997)

    Book  Google Scholar 

  24. Jie, G., Jockers, H.: A note on colored HOMFLY polynomials for hyperbolic knots from WZW models. Commun. Math. Phys. 338(1), 393–456 (2015). https://doi.org/10.1007/s00220-015-2322-z. arXiv:1407.5643

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Mironov, A., Morozov, A., Andrey, M.: Character expansion for HOMFLY polynomials. I. Integrability and difference equations. In Anton, R., Ludmil, K., Johanna, K., Radoslav, R., Emanuel, S (eds.), Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer, pp. 101–118. 2011. arXiv:1112.5754, https://doi.org/10.1142/9789814412551_0003

  26. Biedenharn, L., Schwinger, J., Van Dam, H.: Quantum theory of angular momentum (1965)

  27. Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, Berlin (1991)

    MATH  Google Scholar 

  28. Nawata, S., Ramadevi, P.Z.: Multiplicity-free quantum 6\(j\)-symbols for \(U_{q}(\mathfrak{sl}{_N})\). Lett. Math. Phys 103, 1389–1398 (2013). https://doi.org/10.1007/s11005-013-0651-4

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Our work was partly supported by the grant of the Foundation for the Advancement of Theoretical Physics “BASIS” (A.M., A.S. and A.V.), 17-01-00585 (A.M.), 18-31-20046 (A.S.), 19-51-50008-Yaf-a (A.M.), 18-51-05015-Arm-a (A.M, A.S.), 18-51-45010-Ind-a (A.M, A.S.), 19-51-53014-GFEN-a (A.M, A.S.), 20-01-00644 (A.M., A.S. and A.V.), by President of Russian Federation grant MK-2038.2019.1 (A.M.). The work was also partly funded by RFBR and NSFB according to the research project 19-51-18006 (A.M.). On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Authors and Affiliations

  1. ITEP, Moscow, Russia, 117218

    Victor Alekseev, Andrey Morozov & Alexey Sleptsov

  2. Institute for Information Transmission Problems, Moscow, Russia, 127994

    Victor Alekseev, Andrey Morozov & Alexey Sleptsov

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141701

    Victor Alekseev, Andrey Morozov & Alexey Sleptsov

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  1. Victor Alekseev
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  2. Andrey Morozov
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  3. Alexey Sleptsov
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Correspondence to Alexey Sleptsov.

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Alekseev, V., Morozov, A. & Sleptsov, A. Interplay between symmetries of quantum 6j-symbols and the eigenvalue hypothesis. Lett Math Phys 111, 50 (2021). https://doi.org/10.1007/s11005-021-01386-1

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