In [Quantum Field Theory and the Jones Polynomial][1], Witten showed how to get the Jones polyomial as a Wilson Loop in Chern-Simons theory.  The Chern-Simons Lagrangian is 
$$ \mathcal{L} = \frac{k}{4\pi} \int_M \mathrm{Tr}(A \wedge dA +  \frac{2}{3} A \wedge A \wedge A )$$
Here you're integrating over a 3-manifold (e.g.  $M= S^3)$, but you're also integrating over the moduli space of connections $A$ on $M$, so $A$ takes values in some lie algebra, e.g.  $\mathfrak{g} = \mathfrak{su}(2)$.

Based on this information they can calculate the partition function for $M = S^3, \mathfrak{g}=\mathfrak{su}(2)$ to be $$ Z(S^3) = \sqrt{\frac{2}{k+2}}\sin \frac{\pi}{k+2} $$

In this theory, one can also define ``Wilson loops" over closed curves in your 3-manifold, i.e. knots.
$$ W_R(C) = \mathrm{Tr}_R\left[ P \exp \int_C A \cdot dx \right]$$ 
Remember if we exponentiate an element of the Lie algebra $A \in \mathfrak{g}$ then $e^A$ is going to be an element of the Lie group $G$. So $e^{\int_C A dx} \in G$.  Proving the Wilson loops give you Jones polynomials involves the Atiyah-Singer index theorem and some surgery theory of manifolds.  Wilson loops can be used to derive Khovanov Homology.

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Lately, in the physics literature, there is a tendency to derive things from 6-dimensional gauge theory and "dimensionally" reduce down to lower dimensions. Unfortunately I am in a hurry, and I refer you to **Section 6, pp 120-123** for the definition of "monodromy defect" which I can fill in later

> In gauge theory with gauge group G on
> any manifold X, let U be a submanifold
> of codimension
> 2. Let C be a conjugacy class in G. Then one considers gauge theory on X\U
> with the condition that the gauge
> fields have a monodromy around U that
> is in the conjugacy class C. A surface
> operator supported on U is defined by
> asking in addition that the fields
> should have the mildest type of
> singularity consistent with this
> monodromy or (depending on the
> context) by imposing additional
> conditions on the singular behavior
> along U. **We will call codimension two
> operators of this sort monodromy
> defects**.

So in gauge theory, there are line operators and sometimes surface operators.  Since Chern-Simons theory is 3-dimensional, co-dimension 2 is 3-2 = 1-dimensional.   Witten wants to re-derive some properties of knots using these operators instead.

  [1]: http://projecteuclid.org/euclid.cmp/1104178138