# monodromy defects and Chern-Simons

In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to descriptions based on codimension two defects in various dimensions, as we explain briefly in section 6. This matter certainly merits much closer attention."

Can somebody explain what monodromy defects are?

[Edit: The quoted sentence appears in the introduction of this arxiv preprint of Ed Witten. --PLC]

• Please include a full citation (with link to PDF, if available) for the quote. Also, please provide some indication of what you do and don't already know --- your question currently is rather vague, and seems to ask for someone to write a long expository article just for you, but you could focus your question with a bit of background. – Theo Johnson-Freyd Feb 4 '11 at 5:22

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.