# Universality of Abelian 2+1 Chern-Simons Theory

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Consider the following abelian Chern-Simons theory in 2+1 dimensions:

$\mathcal{L}_{CS} = \frac{\kappa}{2} \epsilon^{\mu\nu\rho} A_{\mu}\partial_{\nu}A_{\rho}$

Are there unique Universality classes for each value of $\kappa$, or all Lagrangian densities of the above form but with different non-zero $\kappa$values belong to the same universality class? In other words, is it justified to absorb $\kappa$by rescaling the gauge field variables? Obviously there is a good reason that people keep $\kappa$ explicit, but I don't see the problem with such a seemingly innocuous rescaling. Does the rescaling meddle with the gauge transformation on the edge?

retagged Nov 23, 2014
As discussed in this question http://www.physicsoverflow.org/24333/list-of-known-universality-classes?show=24333#q24333 from a general RG point of view, universality classes are roughly speaking defined by fixed points of the RG flow in theory space.

Assuming that the theory is given by a certain Lagrangian the coupling constants are the parameters of the theory. The RG flow in theory space, spanned by the (in principle infinitely many) coupling constants, then describe how the theory evolves when changing scale by RG transformations.

So to me it seems not to make sense to absorb the coupling constant into the gauge field, or to talk about about universality classes for each value of the (or several) coupling constant which corresponds just to a single point in theory space. As I understand it, one considers rather the whole basin of attraction corresponding to a fixed point (universality class) which consists of all the points in theory space that would flow into the fixed point when starting the RG flow from them.

I am not 100% sure if those general remarks hold for topological field theories, such as the one in the question, too or if I am completely missing the point and you are up to something different and more subtleties should be discussed.

Thanks for the comment, Dilaton. RG flows, as far as I know, are not unique for a given theory. The exact RG flow depends on many things such as the details of the regularization scheme, etc. In contrast, the fixed-point theory should be unique. What matters is the observables of a given theory as opposed to the explicit form of the Lagranigian density. In this case, unless there is a reason that such a rescaling is not possible or is problematic (e.g., in a simple non-linear sigma-model) it is hard to imagine that such a rescaling causes any difficulty.

Two theories could belong to the same universality class but may not be connected by any RG flow line in the infinite-dimensional space of possible theories. However, there should be RG flows that take them to the same fixed-point theory that characterizes that universality class.

I don't know much about topological field theories, but in the case of a CS theory, the theory has a gap. This means under the RG transformation the gap grows to infinity and the fixed-point theory becomes seemingly trivial as it has no dynamics. Nonetheless, we know from concrete examples in Condensed Matter physics (topological insulators and more broadly, Symmetry-Protected Topological states and long-range entangled states) that not all gapped theories belong to the same universality class. Actually, this is the question that I am asking here in the specific case of abelian CS theories in 2+1D.

Chern-Simons theory does not flow. It's already topological, and you can distinguish Chern-Simons theory at different levels eg. by computing the ground state degeneracy on a torus.

Thanks for your comment, Ryan. Do you mean that a given CS theory is already a fixed-point theory or there is no point in carrying out the RG transformation on it? If the former is the case, shouldn't its gap grow under RG (i.e., as we probe its low-energy observables)?

As for determining its spectrum, in analogy with topological insulators, it seems to me that we may need more than its spectrum to judge to what universality class it belongs. I suppose, one way out would be to consider non-compact manifolds instead of a compact ones such as a torus or higher genus manifolds. I mean, using boundary properties to uniquely determine the universality class (using the bulk-boundary correspondence). In any case, my question here is more focused on CS theories on the same manifold but with different $\kappa$ values.

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