Abstract
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full \({{\rm Sl}(2, \mathbb {Z})}\) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated with torus knots in the large N Gopakumar–Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.
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Aganagic, M., Cheng, M.C.N., Dijkgraaf, R., Krefl, D., Vafa, C.: Quantum geometry of refined topological strings. arXiv:1105.0630 [hep-th]
Aganagic M., Klemm A., Mariño M., Vafa C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005) hep-th/0305132
Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1–28 (2002) hep-th/0105045
Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. hep-th/0012041
Axelrod S., Della Pietra S., Witten E.: Geometric quantization of Chern– Simons gauge theory. J. Diff. Geom. 33, 787–902 (1991)
Beasley, C.: Localization for Wilson loops in Chern–Simons theory. arXiv: 0911.2687 [hep-th]
Beasley, C., Witten, E.: Non-abelian localization for Chern–Simons theory. arXiv:hep-th/0503126
Bouchard V., Klemm A, . Mariño M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009) arXiv:0709.1453 [hep-th]
Bos M., Nair V.P.: U(1) Chern–Simons theory and c = 1 conformal blocks. Phys. Lett. B 223, 61 (1989)
Brini, A., Cavalieri, R.: Open orbifold Gromov–Witten invariants of [C 3/Z n ]: localization and mirror symmetry. arXiv:1007.0934 [math.AG]
Brini, A.: Open topological strings and integrable hierarchies: remodeling the A-model. arXiv:1102.0281 [hep-th]
Chekhov, L., Eynard, B., Marchal, O.: Topological expansion of the Bethe ansatz, and quantum algebraic geometry. arXiv:0911.1664 [math-ph]
Correale R., Guadagnini E.: Large N Chern–Simons field theory. Phys. Lett. B 337, 80–85 (1994)
Rama Devi P., Govindarajan T.R., Kaul R.K.: Three-dimensional Chern–Simons theory as a theory of knots and links. 3. Compact semisimple group. Nucl. Phys. B 402, 548–566 (1993) hep-th/9212110
Diaconescu D.-E., Florea B.: Large N duality for compact Calabi–Yau threefolds. Adv. Theor. Math. Phys. 9, 31–128 (2005) hep-th/0302076
Dolivet Y., Tierz M.: Chern–Simons matrix models and Stieltjes–Wigert polynomials. J. Math. Phys. 48, 023507 (2007) hep-th/0609167
Dunfield, N.M., Gukov, S., Rasmussen, J.: The superpolynomial for knot homologies. math/0505662 [math.GT]
Elitzur S., Moore G.W., Schwimmer A., Seiberg N.: Remarks on the canonical quantization of the Chern–Simons–Witten theory. Nucl. Phys. B 326, 108 (1989)
Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. math-ph/0702045
Eynard, B., Orantin, N.: Topological expansion of mixed correlations in the hermitian 2 matrix model and x-y symmetry of the F(g) invariants. arXiv:0705.0958 [math-ph]
Gopakumar R., Vafa C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999) hep-th/9811131
Gorsky, E.: q, t-Catalan numbers and knot homology. arXiv:1003.0916 [math.AG]
Gukov S., Schwarz A.S., Vafa C.: Khovanov–Rozansky homology and topological strings. Lett. Math. Phys. 74, 53–74 (2005) hep-th/0412243
Iqbal A., Kashani-Poor A.-K.: The vertex on a strip. Adv. Theor. Math. Phys. 10, 317–343 (2006) hep-th/0410174
Iqbal A., Kozcaz C., Vafa C.: The refined topological vertex. JHEP 0910, 069 (2009) hep-th/0701156
Isidro J.M., Labastida J.M.F., Ramallo A.V.: Polynomials for torus links from Chern–Simons gauge theories. Nucl. Phys. B 398, 187–236 (1993) hep-th/ 9210124
Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 121, 335 (1987)
Kallen, J.: Cohomological localization of Chern–Simons theory. arXiv:1104.5353 [hep-th]
Katz S.H., Liu C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5, 1–49 (2002)
Koshkin S.: Conormal bundles to knots and the Gopakumar–Vafa conjecture. Adv. Theor. Math. Phys. 11, 591–634 (2007) math/0503248
Labastida J.M.F., Llatas P.M., Ramallo A.V.: Knot operators in Chern–Simons gauge theory. Nucl. Phys. B 348, 651–692 (1991)
LabastidaJ. M. F., Mariño M.: The HOMFLY polynomial for torus links from Chern–Simons gauge theory. Int. J. Mod. Phys. A 10, 1045–1089 (1995) hep-th/9402093
Labastida J.M.F., Mariño M., Vafa C.: Knots, links and branes at large N. JHEP 0011, 007 (2000) hep-th/0010102
Labastida J.M.F., Ramallo A.V.: Operator formalism for Chern–Simons theories. Phys. Lett. B 227, 92 (1989)
Lawrence R., Rozansky L.: Witten–Reshetikhin–Turaev invariants of Seifert manifolds. Commun. Math. Phys. 205, 287 (1999)
Lickorish W.B.R.: An introduction to knot theory. Springer, Heidelberg (1997)
Lin X.-S., Zheng H.: On the Hecke algebras and the colored HOMFLY polynomial. Trans. Am. Math. Soc. 362, 1–18 (2010) arXiv:math.QA/0601267
Mariño M.: Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants. Commun. Math. Phys. 253, 25 (2004) hep-th/0207096
Mariño, M.: Knot invariants, matrix models, and open strings (2002, unpublished)
Mariño M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008) hep-th/0612127
Mariño, M.: Chern–Simons theory, the 1/N expansion, and string theory. arXiv:1001.2542 [hep-th]
Mariño, M., Vafa, C.: Framed knots at large N. hep-th/0108064
Morton H.R., Manchón P.M.G.: Geometrical relations and plethysms in the Homfly skein of the annulus. J. Lond. Math. Soc. 78, 305–328 (2008)
Oblomkov, A., Shende, V.: The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link. arXiv:1003.1568 [math.AG]
Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419–438 (2000) hep-th/9912123
Rosso M., Jones V.F.R.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramif. 2, 97–112 (1993)
Stevan S.: Chern–Simons invariants of torus links. Ann. Henri Poincaré 11, 1201–1224 (2010) arXiv:1003.2861 [hep-th]
Taubes C. H.: Lagrangians for the Gopakumar–Vafa conjecture Adv. Theor. Math. Phys. 5, 139–163 (2001) math/0201219 [math-dg]
Tierz M.: Soft matrix models and Chern–Simons partition functions. Mod. Phys. Lett. A 19, 1365–1378 (2004) hep-th/0212128
Tierz M.: Schur polynomials and biorthogonal random matrix ensembles. J. Math. Phys. 51, 063509 (2010)
Traczyk P.: Periodic knots and the skein polynomial. Invent. Math. 106, 73–84 (1991)
Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)
Witten E.: Gauge theories and integrable lattice models. Nucl. Phys. B 322, 629 (1989)
Zhou J.: A proof of the full Mariño–Vafa conjecture. Math. Res. Lett. 17, 1091–1099 (2010) arXiv:1001.2092 [math.AG]
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Communicated by Krzysztof Gawe¸dzki.
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Brini, A., Mariño, M. & Eynard, B. Torus Knots and Mirror Symmetry. Ann. Henri Poincaré 13, 1873–1910 (2012). https://doi.org/10.1007/s00023-012-0171-2
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DOI: https://doi.org/10.1007/s00023-012-0171-2