Conformal fields on compactified manifolds? An apparent paradox!

I would appreciate it if someone tells me how a cft on a compactified manifold (e.g. by means of periodic boundary conditions) can be meaningful? The global conformal invariance is broken due to the scale over which the manifold is compactified (e.g. the period). The local conformal invariance of course still classically exists, but for the simple case of a field on a cylinder for instance, the trace of the stress-energy tensor becomes non-zero (in proportion with the central charge); hence apparently compactification of the space-time manifold is in contrast with having conformal symmetry.

-

If you mean compactifications of the world volumes at which CFTs such as those from the AdS/CFT are defined, then it is true of course that they break the conformal symmetry in general. The conformal symmetry is only broken by the global effects, not locally, however. In particular, some compactifications such as your cylinder preserve the conformal symmetry completely. This is particularly well-known for the $N=4$ gauge theory on $S^3\times {\mathbb R}$, a multi-dimensional cylinder, which preserves all the conformal symmetry including the $SO(4,2)$ conformal group.
 Is your last example similar to the global breaking of conformal symmetry when going from the complex plane to the cylinder (compactifying and so introducing a length scale)? It gives a "vaccum Casimir energy" $E \propto \frac{c}{L^2}$ but still we have $T^{z\bar{z}}=0$, thus tracelessness and local conformal invariance? – Learning is a mess May 22 at 22:09