# Are Schwarz-type TQFTs also conformally invariant?

Let's consider a 3D manifold $Y$ for brevity, and a topological invariant $Z(Y)$ which does not depend on the geometry of $Y$.

A Schwarz-type topological quantum field theory (TQFT) has the defining feature that the action $S$ is independent of the geometry of $Y$. More explicitly, we may write $${\delta S\over \delta g_{\mu\nu}}=0,$$ where $g_{\mu\nu}$ is a metric defined on $Y$. However, it is also well known that the expression on the left hand side is nothing more than the stress-energy tensor $T^{\mu\nu}={\delta S\over \delta g_{\mu\nu}}$. So equivalently, we may rewrite this as $$T^{\mu\nu}=0.$$

This tells us a couple of things:

• We have a vanishing Hamiltonian $H=T^{00}=0$ -- there are no dynamics involved in such a TQFT. This fact also holds true for a Witten-type TQFT.
• Having vanishing diagonal terms also mean that we have $\text{Tr}\,T=T^\mu_{\phantom{\mu}\mu}=0$.

The first point is easy to understand. Intuitively a topological invariant should be unaffected when we stretch $Y$ without tearing or gluing. So if we identify one of these dimensions as being time-like (so that we really have a 2+1D manifold, rather than a 3D one), then translations in time should leave $Z(Y)$ unchanged.

The second point needs clarifying. The condition $\text{Tr}\,T=0$ also implies conformal invariance. Does this necessarily mean that the TQFT on $Y$ is also conformally invariant?

In Witten's work on the Jones polynomial for example, a 2D conformal field theory (CFT) is indeed defined on a cross-section of $Y$, upon choosing the Chern-Simons action. However, there is no explicit mention of a 3D CFT being defined on $Y$. Is the Chern-Simons theory also conformally invariant, and if so, does it hold true for higher dimensional Schwarz-type TQFTs?

I really appreciate any help that can point me in the right direction. Thanks in advance!

• Thank you for the useful notes, and sorry for the delayed reply. So all TQFTs are also conformally invariant -- in particular for the case of the Chern-Simons theory, all interesting quantities (such as $Z(Y)$) transform trivially under the conformal group and hence we get nothing new. On the other hand, introducing Wilson loops (or Wilson lines), we obtain static charges on a cross section of $Y$. Consequently, interesting quantities no longer transform trivially under the conformal group, and it is now useful to talk about the corresponding 2D CFT. I hope I'm getting this right. – KSP Jan 7 '18 at 7:42