In Quantum Field Theory and the Jones Polynomial, 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.

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.

This post imported from StackExchange MathOverflow at 2015-04-04 12:37 (UTC), posted by SE-user john mangual