Putting same CFT on different surfaces

Question

What does it mean to put a CFT on different surfaces? I am initially motivated by the question of comparing same CFT on plane and cylinder but I believe this question can be generalized. Can a CFT put on every two dimensional surface? What about higher dimensions?

Answer 1

Easiest case

Conformal transformations are locally composed of a rotation and rescaling. So if the space you're interested in is related to flat space by a Weyl rescaling factor $\Omega(x)^2$, the theory will have its $n$-point functions change according to \begin{equation} \left < \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_g = \prod_{i = 1}^n \Omega(x_i)^{\Delta_i} \left < \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_{\Omega^2 g} \quad (91) \end{equation} where I've used the numbering from TASI lectures by David Simmons-Duffin.

Harder case

Perturbations to the metric are governed by insertions of the stress tensor. So if you can go from flat space to the new space by a diffeomorphism, you can use \begin{equation} \left < \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_g = \left < \exp\left ( \int (g_{\mu\nu} - \eta_{\mu\nu}) T^{\mu\nu}(x) dx \right ) \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_\eta. \quad (8) \end{equation} The Taylor expansion of this involves infinitely many correlation functions. So in $d > 2$, all we can say is that solving a theory in flat space allows you to solve it in a deformed space. In $d = 2$, things are nicer because stress tensor insertion obey the Virasoro Ward identity. Therefore, solving for a correlator in flat space allows you to solve it in a deformed space.

Hardest case

If the curved manifold you're looking at has a non-trivial topology, the CFT might fail to be consistent there. One necessary condition called modular invariance reads \begin{equation} Z(\tau) = Z(\tau + 1) = Z(-1 / \tau) \end{equation} and is discussed in most standard references. Here, $Z$ is the torus partition function defined by \begin{equation} Z = \mathrm{Tr} \left [ e^{2\pi i \tau (L_0 - c/24)} e^{2\pi i \bar{\tau} (\bar{L}_0 - \bar{c} / 24)} \right ] \end{equation} while $\tau$ is the complex structure of the torus. To make this sufficient, one also needs a modular covariance condition to be obeyed by the one-point functions. Moreover, Moore and Seiberg showed that 2d CFTs consistent on the sphere and the torus are automatically consistent on all Riemann surfaces. In higher dimensions, the analogous problem is wide open.

Update

For a small diffeomorphism $\delta g_{\mu\nu}$, you can take the derivative of (8) at $\eta_{\mu\nu}$ to get \begin{equation} \delta \left < \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_\eta = \int \delta g_{\mu\nu} \left < T^{\mu\nu}(x) \mathcal{O}_1(x_1) \dots \mathcal{O}_n(x_n) \right >_\eta dx. \end{equation} This also makes the connection with (91) easy to see because in the case of a Weyl transformation $\delta g_{\mu\nu} = \omega(x) \eta_{\mu\nu}$ and the whole thing becomes an integral involving the trace of the stress tensor. Using \begin{equation} T^\mu_\mu(x) \mathcal{O}_i(x_i) = \Delta_i \delta(x - x_i) \mathcal{O}_i(x_i), \quad (92) \end{equation} we end up with an overall factor of $\sum_i \omega_i(x_i) \Delta_i$ which is exactly the result of taking $\Omega(x) = e^{\omega(x)}$ to first order.