Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

157 submissions , 130 unreviewed
4,092 questions , 1,495 unanswered
4,954 answers , 21,129 comments
1,470 users with positive rep
560 active unimported users
More ...

  Universality of Abelian 2+1 Chern-Simons Theory

+ 4 like - 0 dislike
565 views

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?

asked Nov 23, 2014 in Theoretical Physics by SG (20 points) [ no revision ]
retagged Nov 23, 2014 by Dilaton
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.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysi$\varnothing$sOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...