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A note on colored HOMFLY polynomials for hyperbolic knots from WZW models. (English) Zbl 1328.81193
Summary: Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of \(\mathrm{SU}(N)\). The crossing matrices directly relate to 6j-symbols of the quantum group \(\mathcal{U}_q\mathfrak{su}(N)\). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation \(\{2, 1\}\) for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or – in the limit to classical groups – to determine color factors for Yang Mills amplitudes.

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
58J28 Eta-invariants, Chern-Simons invariants
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
81T45 Topological field theories in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Knot Atlas
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[1] Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121, 351, (1989) · Zbl 0667.57005
[2] Labastida, J.: Chern-Simons gauge theory: ten years after. AIP Conf. Proc 484, 1 (1999). arXiv:hep-th/9905057 · Zbl 1162.81414
[3] Mariño M.: Chern-Simons Theory, Matrix Models and Topological Strings, International Series of Monographs on Physics, vol. 131. The Clarendon Press Oxford University Press, Oxford (2005) · Zbl 1093.81002
[4] Mariño, M.: Chern-Simons theory and topological strings. Rev. Mod. Phys. 77, 675-720 (2005). arXiv:hep-th/0406005 · Zbl 1205.81013
[5] Gukov, S., Saberi, I.: Lectures on knot homology and quantum curves. AMS Contemp. Math. 613, (2014). arXiv:1211.6075 [hep-th] · Zbl 1320.81084
[6] Witten, E.: Chern-Simons gauge theory as a string theory. Prog. Math. 133, 637-678 (1995). arXiv:hep-th/9207094 · Zbl 0844.58018
[7] Gopakumar, R., Vafa, C.: On the gauge theory / geometry correspondence. Adv. Theor. Math. Phys. 3, 1415-1443 (1999). arXiv:hep-th/9811131 · Zbl 0972.81135
[8] Labastida, J., Mariño, M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217, 423-449, (2001). arXiv:hep-th/0004196 · Zbl 1018.81049
[9] Labastida, J., Mariño, M., Vafa, C.: Knots, links and branes at large N. JHEP 0011, 007 (2000). arXiv:hep-th/0010102 · Zbl 0286.20054
[10] Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B577, 419-438 (2000). arXiv:hep-th/9912123 · Zbl 1036.81515
[11] Mariño, M., Vafa, C.: Framed knots at large N. Contemp. Math. 310, 185-204 (2002). arXiv:hep-th/0108064 · Zbl 1042.81071
[12] Brini, A., Eynard, B., Mariño, M.: Torus knots and mirror symmetry. Ann. Henri Poincare 13, 1873-1910 (2012). arXiv:1105.2012 [hep-th]
[13] Diaconescu, D., Shende, V., Vafa, C.: Large N duality, lagrangian cycles, and algebraic knots. Commun. Math. Phys. 319, 813-863 (2013). arXiv:1111.6533 [hep-th] · Zbl 1309.57011
[14] Jockers, H., Klemm, A., Soroush, M.: Torus knots and the topological vertex. Lett. Math. Phys. 104, 953-989 (2014). arXiv:1212.0321 [hep-th] · Zbl 1302.57021
[15] Ng, L.: Framed knot contact homology. Duke Math. J. 141, 365-406 (2008). arXiv:math/0407071 · Zbl 1145.57010
[16] Ng, L.: A topological introduction to knot contact homology. In: Contact and Symplectic Topology, Bolyai Society Mathematical Studies vol. 26, pp 485-530, (2014). arXiv:1210.4803 [math.GT] · Zbl 1351.53094
[17] Aganagic, M., Vafa, C.: Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, (2012). arXiv:1204.4709 [hep-th]
[18] Aganagic, M., Ekholm, T., Ng, L., Vafa, C.: Topological strings, D-model, and knot contact homology. Adv. Theor. Math. Phys. 18(4), 827-956 (2014). arXiv:1304.5778 [hep-th] · Zbl 1315.81076
[19] Gu, J., Jockers, H., Klemm, A., Soroush, M.: Knot nvariants from topological recursion on augmentation varieties, (2014). arXiv:1401.5095 [hep-th]
[20] Rama Devi, P., Govindarajan, T., Kaul, R.: Three-dimensional Chern-Simons theory as a theory of knots and links. 3. Compact semisimple group. Nucl. Phys. B402, 548-566 (1993). arXiv:hep-th/9212110 · Zbl 0941.57500
[21] Zodinmawia, Ramadevi, P.: SU(N) quantum Racah coefficients and non-torus links. Nucl. Phys. B870, 205-242 (2013). arXiv:1107.3918 [hep-th] · Zbl 1262.81168
[22] Nawata, S., Ramadevi, P., Zodinmawia, Sun, X.: Super-A-polynomials for Twist knots. JHEP 1211, 157 (2012). arXiv:1209.1409 [hep-th] · Zbl 1397.57029
[23] 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). arXiv:1209.6304 [math-ph] · Zbl 1259.81082
[24] Itoyama, H., Mironov, A., Morozov, A., Morozov, A.: HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations. JHEP 1207, 131 (2012). arXiv:1203.5978 [hep-th] · Zbl 1397.57012
[25] Kawagoe, K.: On the formulae for the colored HOMFLY polynomials (2012). arXiv:1210.7574 [math.GT] · Zbl 1341.57007
[26] Nawata, S., Ramadevi, P., Zodinmawia: Colored HOMFLY polynomials from Chern-Simons theory. J. Knot Theor. 22, 1350078 (2013). arXiv:1302.5144 [hep-th] · Zbl 1296.57015
[27] Anokhina, A., Mironov, A., Morozov, A., Morozov, A.: Knot polynomials in the first non-symmetric representation. Nucl. Phys. B882, 171-194 (2014). arXiv:1211.6375 [hep-th] · Zbl 1285.81035
[28] Anokhina, A., Morozov, A.: Cabling procedure for the colored HOMFLY polynomials. Teor. Mat. Fiz. 178, 3-68 (2014). arXiv:1307.2216 [hep-th] · Zbl 1318.81055
[29] Ramadevi, P., Sarkar, T.: On link invariants and topological string amplitudes. Nucl. Phys. B600, 487-511 (2001). arXiv:hep-th/0009188 · Zbl 1097.81742
[30] Borhade, P., Ramadevi, P., Sarkar, T.: U(N) framed links, three manifold invariants, and topological strings. Nucl. Phys. B678, 656-681 (2004). arXiv:hep-th/0306283 · Zbl 1097.81709
[31] Butler P.: Point Group Symmetry Applications: Methods and Tables. Plenum Press, New York (1981) · Zbl 0587.20031
[32] Haase, R.W.; Dirl, R., The symmetric group: algebraic formulas for some \(S\)_{\(f\)} 6\(j\) symbols and \({S_f⊃ S_{f_1}× S_{f_2} 3jm}\) symbols, J. Math. Phys., 27, 900-913, (1986) · Zbl 0598.22012
[33] Cvitanović P.: Group Theory. Princeton University Press, Princeton, NJ (2008) · Zbl 1152.22001
[34] Elvang, H., Cvitanović, P., Kennedy, A.D.: Diagrammatic young projection operators for U(n), (2003). arXiv:hep-th/0307186 · Zbl 1067.81054
[35] Behrend, R.E., Pearce, P.A., Petkova, V.B., Zuber, J.-B.: Boundary conditions in rational conformal field theories. Nucl. Phys. B570, 525-589 (2000). arXiv:hep-th/9908036 · Zbl 1028.81520
[36] Felder, G., Frohlich, J., Fuchs, J., Schweigert, C.: The geometry of WZW branes. J. Geom. Phys. 34, 162-190 (2000). arXiv:hep-th/9909030 · Zbl 1002.81042
[37] Felder, G., Frohlich, J., Fuchs, J., Schweigert, C.: Correlation functions and boundary conditions in RCFT and three-dimensional topology. Compos. Math. 131, 189-237 (2002). arXiv:hep-th/9912239 · Zbl 1002.81045
[38] Wigner E.P.: Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra. Academic Press, New York (1959) · Zbl 0085.37905
[39] Gorshkov, A.V., Hermele, M., Gurarie, V., Xu, C., Julienne, P.S., Ye, J., Zoller, P., Demler, E., Lukin, M.D., Rey, A.M.: Two-orbital SU(N) magnetism with ultracold alkaline-earth atoms. Nat. Phys. 6, 289-295 (2010). arXiv:0905.2610 [cond-mat]
[40] Moore, G.W.; Seiberg, N., Classical and quantum conformal field theory, Commun. Math. Phys., 123, 177, (1989) · Zbl 0694.53074
[41] Bar-Natan, D., Morrison, S., et al.: The knot atlas · Zbl 1130.57012
[42] Nawata, S., Ramadevi, P., Zodinmawia: Multiplicity-free quantum 6j-symbols for \({U_q(\mathfrak{sl}_N)}\). Lett. Math. Phys. 103, 1389-1398 (2013). arXiv:1302.5143 [hep-th] · Zbl 1330.17020
[43] Moore G.W., Seiberg, N.: Lectures on RCFT. In: Green, M.B., et al. (eds.) Superstrings ’89: Proceedings of the Trieste Spring School, pp. 1-129. World Scientific Publishing Co. Pte. Ltd., (1990) · Zbl 0985.81739
[44] Alvarez-Gaume, L.; Gomez, C.; Sierra, G., Duality and quantum groups, Nucl. Phys. B, 330, 347, (1990) · Zbl 0764.17021
[45] Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras. I, II. J. Am. Math. Soc. 6, 905-947, (1993, 949-1011) · Zbl 0786.17017
[46] Kazhdan, D.; Lusztig, G., Tensor structures arising from affine Lie algebras. III, J. Am. Math. Soc., 7, 335-381, (1994) · Zbl 0802.17007
[47] Kazhdan, D.; Lusztig, G., Tensor structures arising from affine Lie algebras. IV, J. Am. Math. Soc., 7, 383-453, (1994) · Zbl 0802.17008
[48] Finkelberg, M., An equivalence of fusion categories, Geom. Funct. Anal., 6, 249-267, (1996) · Zbl 0860.17040
[49] Lienert, C.R.; Butler, P.H., Racah-Wigner algebra for q-deformed algebras, J. Phys. A Math. Gen., 25, 1223, (1992) · Zbl 0758.17009
[50] Pan, F., Racah coefficients of quantum group uq (n), J. Phys. A Math. Gen., 26, 4621, (1993) · Zbl 0829.33013
[51] Haase, R.W.: The symmetric group and the unitary group: an application of group-subgroup transformation theory. Ph.D. thesis, University of Canterbury. Physics (1983)
[52] Butler, P.; King, R., Symmetrized Kronecker products of group representations, Can. J. Math, 26, 328-339, (1974) · Zbl 0286.20054
[53] Searle, B.: Calculation of 6j symbols. Ph.D. thesis, University of Canterbury. Physics (1988)
[54] Kaul, R., Govindarajan, T.: Three-dimensional Chern-Simons theory as a theory of knots and links. Nucl. Phys. B380, 293-336 (1992). arXiv:hep-th/9111063 · Zbl 0938.81553
[55] Kaul, R.; Govindarajan, T., Three-dimensional Chern-Simons theory as a theory of knots and links. 2. multicolored links, Nucl. Phys., B393, 392-412, (1993) · Zbl 0938.57504
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