Abstract
We formulate a refinement of SU(N) Chern–Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern–Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern–Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern–Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern–Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S 3 colored by fundamental representation, the index equals the Poincaré polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S 3 has a large-N dual which is the refined topological string on \({X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}\) ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SL N knot homology. We also provide a matrix model description of some amplitudes of the refined Chern–Simons theory on S 3.
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References
Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)
Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Jones V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–112 (1985)
Jones V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126(2), 335–388 (1987)
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(2), 239246 (1985). doi:10.1090/S0273-0979-1985-15361-3
Khovanov M.: A categorification of the Jones polynomial. Duke. Math. J. 101, 359–426 (2000)
Kronheimer P.B., Mrowka T.S.: Khovanov homology is an unknot-detector. Publ. Math. IHES 113(1), 97–208 (2011)
Khovanov M., Rozansky L.: Matrix factorizations and link homology. Fundam. Math. 199, 1–91 (2008) arXiv:math/0401268
Khovanov M., Rozansky L.: Matrix factorizations and link homology II. Geom. Topol. 12, 1387–1425 (2008) arXiv:math/0505056
Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419 (2000) arXiv:hep-th/9912123
Gopakumar R., Vafa C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 3 1415 (1999)arXiv:hep-th/981113
Gopakumar, R., Vafa, C.: M theory and topological strings. 1. hep-th/9809187
Gopakumar, R., Vafa, C.: M theory and topological strings. 2. hep-th/9812127
Gukov S., Schwarz A.S., Vafa C.: Khovanov–Rozansky homology and topological strings. Lett. Math. Phys. 74, 53–74 (2005) hep-th/0412243
Dunfield N.M., Gukov S., Rasmussen J.: The superpotential for knot homologies. Exp. Math. 15, 129 (2006) math/0505662
Rasmussen J.: Khovanov–Rozansky homology of two-bridge knots and links. Duke Math. J. 136(3), 551–583 (2007) arXiv:math.GT/0508510
Rasmussen, J.: Some differentials on Khovanov–Rozansky homology. arXiv:math/0607544
Witten, E.: Fivebranes and knot. arXiv:1101.3216
Witten E.: Chern–Simons gauge theory as a string theory. Prog. Math. 133, 637–678 (1995) hep-th/9207094
Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004) arXiv:hep-th/0206161
Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. In: Ethigof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics Progress in Mathematics, vol. 244, pp. 525–596 (2006)
Marino, M.: Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants. Commun. Math. Phys. 253, 25–49 (2004). hep-th/0207096
Aganagic M., Mariño M., Vafa C.: All loop topological string amplitudes from Chern–Simons theory. Commun. Math. Phys. 247, 467–512 (2004)
Dijkgraaf, R.: Vafa, C.: Toda theories, matrix models, topological strings, and N = 2 gauge systems. arXiv:0909.2453
Verlinde E.P.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988)
Moore, G., Seiberg, N.: Lectures on RCFT. In: Lee, H.C. (ed.) Physics, Geometry, And Topology (Banff, AB, 1989). NATO Adv. Sci. Inst. Ser. B Phys., vol. 238, pp. 263-361. Plenum, New York (1990)
Iqbal, A., Kozcaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). hep-th/0701156
Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). hep-th/0310272
Aganagic M., Cheng M.C.N., Dijkgraaf R., Krefl D., Vafa C.: Quantum geometry of refined topological strings. JHEP 1211, 19 (2012)
Vafa C.: Black holes and Calabi–Yau threefolds. Adv. Theor. Math. Phys. 2, 207 (1998) hep-th/9711067
Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS states. arXiv:1006.0146
Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. hep-th/0012041
Taubes C.: Lagrangians for the Gopakumar–Vafa conjecture. Geom. Topol. Monogr. 8, 73–95 (2006)
Aganagic M., Klemm A., Marino M., Vafa C.: The Topological vertex. Commun. Math. Phys. 254, 425–478 (2005) hep-th/0305132
Labastida, J.M.F., Marino, M., Vafa, C.: Knots, links, and branes at large N. JHEP 0011, 007 (2000). hep-th/0010102
Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. hepth/0602087
Aganagic M., Yamazaki M.: Open BPS wall crossing and M-theory. Nucl. Phys. B 834, 258–272 (2010) arXiv:0911.5342
Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences. arXiv:1006.3435
Cheng M.C.N., Dijkgraaf R., Vafa C.: Non-perturbative topological strings and conformal blocks. JHEP 1109, 022 (2011)
Candelas P., De La Ossa X.C., Green P.S., Parkes L.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74 (1991)
Gaiotto D., Witten E.: Supersymmetric boundary conditions in N = 4 super Yang–Mills theory. J. Stat. Phys. 135, 789–855 (2009) arXiv:0804.2902
Gaiotto, D., Witten, E.: Janus configurations, Chern–Simons couplings, and the theta-angle in N = 4 super Yang–Mills theory. JHEP 1006, 097 (2010). arXiv:0804.2907
Hollowood, T.J., Iqbal, A., Vafa C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). hep-th/0310272
Beasley C., Witten E.: Non-abelian localization for Chern–Simons theory. J. Differ. Geom. 70, 183–323 (2005) hep-th/0503126
Beasley C.: Localization for Wilson loops in Chern–Simons theory. Adv. Theor. Math. Phys. 17, 1–240 (2013)
Beasley, C.: Remarks on Wilson loops and Seifert loops in Chern–Simons theory, AMS/IP Atud. Adv. Math. 50, 1–17 (2011). (AMS)
Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in N = 2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010). arXiv:0909.0945
Dimofte T., Gukov S., Hollands L.: Vortex counting and Lagrangian 3-manifolds. Cett. Math. Phys. 98, 255–287 (2011)
Aganagic M., Ooguri H., Saulina N., Vafa C.: Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304–348 (2005) hep-th/0411280
Aganagic, M., Vafa, C.: G(2) manifolds, mirror symmetry and geometric engineering. hep-th/0110171
Aganagic M., Klemm A., MarinoM. Vafa C.: Matrix model as a mirror of Chern–Simons theory. JHEP 0402, 010 (2004) hep-th/0211098
Gukov S., Iqbal A., Kozcaz C., Vafa C.: Link homologies and the refined topological vertex. Commun. Math. Phys. 298, 757–785 (2010) arXiv:0705.1368
Macdonald I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1995)
Macdonald, I.G.: A new class of symmetric functions. Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Seminaire Lotharingien, pp 131–171 (1988)
Macdonald, I.G.: Orthogonal polynomials associated with root systems (1988). (Preprint)
Etingof P., Kirillov A. Jr: On Cherednik–Macdonald–Mehta identities, q-alg 9712051. Electron. Res. Announc. 4, 43–47 (1998)
Iqbal, A., Kozcaz, C., Refined Hopf link revisited. JHEP 04, 046 (2012). arXiv:1111.0525
Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304–348 (2005). hep-th/0411280
Cherednik I.: Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122(1), 119–145 (1995)
Cherednik I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. Second Ser. 141(1), 191–216 (1995)
Kirillov A. Jr: On inner product in modular tensor categories. I. J. Am. Math. soc. 9, 1135–1169 (1996)
Cherednik I., Ostrik V.: From double affine Hecke algebra to Fourier transform. Sel. Math. (N.S.) 9(2), 161249 (2003)
Hansen S.K.: Reshetikhin–Turaev invariants of Seifert 3-manifolds and a rational surgery formula. Algebr. Geom. Topol. 1, 627–686 (2001) math.GT/0111057
Brini A., Eynard B., Marino M.: Torus knots and mirror symmetry. Ann. Henri Poincare 13, 1873–1910 (2012) arXiv:1105.2012
t’Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. 72, 461 (1974)
Dunin-Barkowski, P., Mironov, A., Morozov, A., Sleptsov A., Smirnov, A.: Superpolynomials for toric knots from evolution induced by cut-and-join operators. JHEP 03, 021 (2013)
Cherednik I.: Jones polynomials of torus knots via DAHA. Int. Math. Res. Not. 2013(23), 5366–5425 (2013)
Gorsky, E., Oblomkov, A., Rasmussen, J., Shende, V.: Torus knots and the rational DAHA. arXiv:1207.4523
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Communicated by N. A. Nekrasov
The paper is a version of arXiv:1105.5117 prepared for CMP.
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Aganagic, M., Shakirov, S. Knot Homology and Refined Chern–Simons Index. Commun. Math. Phys. 333, 187–228 (2015). https://doi.org/10.1007/s00220-014-2197-4
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DOI: https://doi.org/10.1007/s00220-014-2197-4