Conformal fields on compactified manifolds? An apparent paradox!

Question

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.

Answer 1

First, if you talk about world sheet CFTs describing compactifications of string theory, as your focus on the "central charge" and "compactifications" suggests, for those we must realize that the world sheet conformal symmetry has nothing to do with the would-be spacetime conformal symmetry. Spacetime transformations are manifested as internal symmetries of the world sheet CFT so they are completely independent from the world sheet conformal transformations that act nonlocally on the world sheet. String theory introduces a preferred distance scale in the spacetime, the string scale (and derived scales dependent on the string coupling, such as the spacetime Planck scale as well), so the spacetime effective theories produced by string theory are never conformal (because they're not scale-invariant).

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.

No, the stress-energy tensor of such a theory still exactly vanishes. The theory is locally the same regardless of the compactification.