Refined Chern-Simons Theory and Knot Homology

The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural deformation of the geometric background. Analogously with the unrefined case, the solution of refined Chern-Simons theory is given in terms of S and T matrices, which are the proper Macdonald deformations of the usual ones. This provides a direct way to compute refined Chern-Simons invariants of a wide class of three-manifolds and knots. The knot invariants of refined Chern-Simons theory are conjectured to coincide with the knot superpolynomials -- Poincare polynomials of the triply graded knot homology theory. This conjecture is checked for a large number of torus knots in S^3, colored by the fundamental representation. This is a short, expository version of arXiv:1105.5117, with some new results included.

[1]  Miranda C. N. Cheng,et al.  UvA-DARE ( Digital Academic Repository ) Quantum Geometry of Refined Topological Strings , 2012 .

[2]  M. Yamazaki,et al.  Open BPS wall crossing and M-theory , 2009, 0911.5342.

[3]  M. Taki.  Surface operator, bubbling Calabi-Yau and AGT relation , 2010, 1007.2524.

[4]  L. Alday,et al.  Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry , 2009, 0909.0945.

[5]  Mikhail Khovanov,et al.  Matrix factorizations and link homology , 2008 .

[6]  All Loop Topological String Amplitudes from Chern-Simons Theory , 2002, hep-th/0206164.

[7]  Miranda C. N. Cheng,et al.  Non-perturbative topological strings and conformal blocks , 2010, 1010.4573.

[8]  C. Vafa,et al.  Disk Instantons, Mirror Symmetry and the Duality Web , 2001, hep-th/0105045.

[9]  Alexander Kirillov.  On inner product in modular tensor categories. I , 1995 .

[10]  Nathan Seiberg,et al.  Remarks on the canonical quantization of the Chern-Simons-Witten theory , 1989 .

[11]  Jacob Rasmussen,et al.  The Superpolynomial for Knot Homologies , 2005, Exp. Math..

[12]  M. Aganagic,et al.  Knot Homology from Refined Chern-Simons Theory , 2011, 1105.5117.

[13]  Non-Abelian localization for Chern-Simons theory , 2005, hep-th/0503126.

[14]  Khovanov-Rozansky homology of two-bridge knots and links , 2005, math/0508510.

[15]  C. Vafa,et al.  Matrix models, geometric engineering and elliptic genera , 2003, hep-th/0310272.

[16]  E. Witten,et al.  Rigid Surface Operators , 2008, 0804.1561.

[17]  C. Vafa,et al.  R-Twisting and 4d/2d Correspondences , 2010, 1006.3435.

[18]  Chern-Simons gauge theory as a string theory , 1992, hep-th/9207094.

[19]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[20]  H. Awata,et al.  Five-dimensional AGT conjecture and the deformed Virasoro algebra , 2009, 0910.4431.

[21]  H. Awata,et al.  Five-Dimensional AGT Relation and the Deformed β-Ensemble , 2010, 1004.5122.

[22]  S. Pasquetti,et al.  A& B model approaches to surface operators and Toda thoeries , 2010, 1004.2025.

[23]  C. Vafa,et al.  Khovanov-Rozansky Homology and Topological Strings , 2004, hep-th/0412243.

[24]  V. Ostrik,et al.  From double Hecke algebra to Fourier transform , 2001, math/0111130.

[25]  Lagrangians for the Gopakumar-Vafa conjecture , 2002, math/0201219.

[26]  E. Witten,et al.  Knot Invariants from Four-Dimensional Gauge Theory , 2011, 1106.4789.

[27]  E. Verlinde,et al.  Fusion Rules and Modular Transformations in 2D Conformal Field Theory , 1988 .

[28]  Kenneth C. Millett,et al.  A new polynomial invariant of knots and links , 1985 .

[29]  Jacob Rasmussen,et al.  Some differentials on Khovanov–Rozansky homology , 2006, math/0607544.

[30]  C. Vafa,et al.  M-theory and a topological string duality , 2006, hep-th/0602087.

[31]  L. Rozansky,et al.  Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds , 1999 .

[32]  Knot Invariants and Topological Strings , 1999, hep-th/9912123.

[33]  H. Awata,et al.  Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String , 2010, 1008.0574.

[34]  C. Vafa,et al.  Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems , 2009, 0909.2453.

[35]  Vaughan F. R. Jones.  Index for subfactors , 1983 .

[36]  L. Alday,et al.  Affine SL(2) Conformal Blocks from 4d Gauge Theories , 2010, Letters in Mathematical Physics.

[37]  Reshetikhin{Turaev invariants of Seifert 3{manifolds and a rational surgery formula , 2001, math/0111057.

[38]  B. Eynard,et al.  Torus Knots and Mirror Symmetry , 2011, 1105.2012.

[39]  Macdonald's evaluation conjectures and difference Fourier transform , 1994, q-alg/9412016.

[40]  E. Witten.  Khovanov homology and gauge theory , 2011, 1108.3103.

[41]  J. Schwarz,et al.  World volume action of the M theory five-brane , 1997, hep-th/9701166.

[42]  C. Vafa,et al.  Link Homologies and the Refined Topological Vertex , 2007, 0705.1368.

[43]  D. Gaiotto.  Surface operators in $ \mathcal{N} $ = 2 4d gauge theories , 2009, 0911.1316.

[44]  C. Beasley.  Localization for Wilson Loops in Chern-Simons Theory , 2009, 0911.2687.

[45]  I. Cherednik.  Double affine Hecke algebras and Macdonald's conjectures , 1995 .

[46]  Mikhail Khovanov.  A categorification of the Jones polynomial , 1999 .

[47]  M. Aganagic,et al.  Orientifolds and the refined topological string , 2012, 1202.4456.

[48]  J. R. Neil.  COMBINATORIAL CALCULATION OF THE VARIOUS NORMALIZATIONS OF THE WITTEN INVARIANTS FOR 3-MANIFOLDS , 1992 .

[49]  Mikhail Khovanov,et al.  Matrix factorizations and link homology II , 2008 .

[50]  Lagrangians for the Gopakumar-Vafa conjecture , 2001 .

[51]  G. Hooft.  A Planar Diagram Theory for Strong Interactions , 1974 .

[52]  A. Iqbal,et al.  Refined Hopf link revisited , 2011, 1111.0525.

[53]  Orthogonal polynomials associated with root systems , 2000, math/0011046.

[54]  Nikita A. Nekrasov.  Seiberg-Witten prepotential from instanton counting , 2002 .

[55]  C. Vafa,et al.  The Topological Vertex , 2003, hep-th/0305132.

[56]  The refined topological vertex , 2007, hep-th/0701156.

[57]  R. Gopakumar,et al.  On the Gauge Theory/Geometry Correspondence , 1998, hep-th/9811131.

[58]  I. G. MacDonald,et al.  A new class of symmetric functions. , 1988 .

[59]  E. Witten.  Fivebranes and Knots , 2011, 1101.3216.

[60]  S. Gukov,et al.  Vortex Counting and Lagrangian 3-Manifolds , 2010, 1006.0977.

[61]  L. Rozansky.  A largek asymptotics of Witten's invariant of Seifert manifolds , 1993, hep-th/9303099.

[62]  L. Jeffrey.  Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation , 1992 .