# Relation between conformal and topological field theories

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)?

• A CFT is simply a field theory with conformal symmetry, which you can probably easily check for...
– Danu
Oct 15 '14 at 5:59

## 2 Answers

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.

• Conformal field Theory in 1D is a Topological Field theory Jul 10 '18 at 12:54

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.