monodromy defects and Chern-Simons

Question

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]

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

Comments

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.

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

Answer 1

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.

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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.

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