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

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Proceedings of Symposia in Pure Mathematics

Volume 85 , 2012

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