1202.2489 (Mina Aganagic et al.)

Refined Chern-Simons Theory and Knot Homology    [PDF]

Mina Aganagic, Shamil Shakirov

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.

View original: http://arxiv.org/abs/1202.2489