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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Relation between conformal and topological field theories

+ 3 like - 0 dislike
703 views

The Chern-Simons (CS) theory is a topological quantum field theory (TQFT).

  1. The question is, is a conformal field theory (CFT) a topological quantum theory?

  2. Or the reverse, topological quantum field theory is a CFT?

  3. What is a conformal field theory (CFT)?

This post imported from StackExchange Physics at 2015-03-04 16:11 (UTC), posted by SE-user Gilmar
asked Oct 15, 2014 in Theoretical Physics by Gilmar (15 points) [ no revision ]
A CFT is simply a field theory with conformal symmetry, which you can probably easily check for...

This post imported from StackExchange Physics at 2015-03-04 16:11 (UTC), posted by SE-user Danu

2 Answers

+ 6 like - 0 dislike

A conformal transformation is one which alters the metric up to a factor, i.e.

$$g_{\mu\nu}(x)\to\Omega^2(x)g_{\mu\nu}(x)$$

A field theory described by a Lagrangian invariant up to a total derivative under a conformal transformation is said to be a conformal field theory. These transformations include

  • Scaling or dilations $x^\mu \to \lambda x^\mu$
  • Rotations $x^\mu \to M^\mu_\nu x^\nu$
  • Translations $x^\mu \to x^\mu + c^\mu$

In addition to these, the conformal group includes a set of special conformal transformations given by,

$$x^\mu \to \frac{x^\mu-b^\mu x^2}{1-2b \cdot x + b^2 x^2}$$

If you compute the generators of the conformal transformations, and the algebra they satisfy, with some manipulation it may be shown there is an isomorphism between the conformal group in $d$ dimensions and the group $SO(d+1,1)$. In two dimensions, the conformal group is rather special; it is simply the group of all analytic maps; this set is infinite-dimensional since one requires an infinite number of parameters to specify all functions analytic in some neighborhood. The global variety of conformal transformations, i.e. those which are not functions of the coordinates but constants, in $d=2$ are equivalent to $SL(2,\mathbb{C})$.


On the other hand, a topological field theory is one which is invariant under all transformations which do not alter the topology of spacetime, e.g. they may not puncture it and increase the genus. The correlation functions do not depend on the metric, and are in fact topological invariants.


Hence, a topological field theory is invariant under conformal transformations by the fact that it does not even depend on the metric. However, not all conformal field theories are topological field theories.

This post imported from StackExchange Physics at 2015-03-04 16:11 (UTC), posted by SE-user JamalS
answered Oct 15, 2014 by JamalS (895 points) [ no revision ]
+ 2 like - 0 dislike

Roughly, every chiral part of a rational CFT gives a TFT theory. For example for WZW models the chiral parts are current algebras. The corresponding TFT is Chern-Simons theory.

The point is the representation of a chiral rational CFT is a modular tensor category. From a modular tensor category one can construct a 3D TFT via the Reshetikhin-Turaev construction.

Conversely, from a given rational chiral CFT $V$, Fuchs-Runkel-Schweigert used the associated TFT to construct (all) 2D CFTs having $V$ as chiral parts. They give a construction of all correlation functions on an arbitrary Riemann surface using the TFT.

This post imported from StackExchange Physics at 2015-03-04 16:11 (UTC), posted by SE-user Marcel
answered Oct 19, 2014 by Marcel (300 points) [ no revision ]

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$\hbar$ysicsO$\varnothing$erflow
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).
Please complete the anti-spam verification




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

Your rights
...