Proceedings of Symposia in Pure Mathematics
Volume 0, XXXX
Refined Chern-Simons Theory and Knot Homology
Mina Aganagic and Shamil Shakirov
Abstract. 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. Analo-
gously 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 de-
formations 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 Poincar´ e polynomials of the triply graded knot ho-
mology theory. This conjecture is checked for a large number of torus knots
in S3, colored by the fundamental representation. This is a short, expository
version of arXiv:1105.5117, with some new results included.
1. Introduction
One of the beautiful stories in the marriage of mathematics and physics de-
veloped from Witten’s realization [1] that three dimensional Chern-Simons theory
on S3 computes the polynomial invariant of knots constructed by Jones in [2] .
While Jones constructed an invariant J(K, q) of knots in three dimensions, his
construction relied on projections of knots to two dimensions. This obscured the
three dimensional origin of the Jones polynomial. The fact that Chern-Simons
theory is a topological quantum field theory in three dimensions made it manifest
that the Jones polynomial is an invariant of the knot, and independent of the two
dimensional projection. Moreover, it also gave rise to new topological invariants of
three-manifolds and knots in them. For any three-manifold M and a knot in it, the
Chern-Simons path integral, with a Wilson loop observable inserted along the knot,
gives a topological invariant that depends only on M, K and the representation of
the gauge group. Moreover, Chern-Simons theory gives a whole family of invariants
associated to M and K, by changing the gauge group G and the representation R
on the Wilson line. The Jones polynomial J(K, q) corresponds to G = SU(2), and
R the fundamental, two dimensional representation of G. Taking G = SU(n) in-
stead, one computes the HOMFLY polynomial H(K, q, a) [3] evaluated at a =
qn.
The work in [1] was made even more remarkable by the fact that it explained how
to solve Chern-Simons theory for any M and collection of knots in it.
Based on talks presented by M.A. at several conferences and workshops, including the String-
Math 2011 Conference at the University of Pennsylvania.
c XXXX American Mathematical Society
1
Proceedings of Symposia in Pure Mathematics
Volume 85 , 2012
3
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