By computing the Noether charges and their algebra, we show that in Liouville classical theory there is a genuine center of the Virasoro algebra, and that it is related to the trace of the energy-momentum tensor, in curved space. The calculations in the curved spacetime context were far from trivial, and we had to invent novel techniques to circumvent the difficulties encountered by Jackiw and other workers.
Eventually, we proved Jackiw’s conjecture, hence, with these results in our hands, we are now fully entitled to call this phenomenon a classical gravitational anomaly, as we are able to see that the phenomenon, behind the obstruction to the symmetry noticed by Jackiw, is just the same one that we know from the quantum case: the occurrence of infinite degrees of freedom. Those of a quantum field, in the standard cases, and those of the curved spacetime, in this classical instance of the phenomenon.
A noticeable byproduct here, in the curved context, is that if we trade the Weyl anomaly for the diffeomorphic anomaly, we are able to provide a compact formula for the violation of the diffeomorphic invariance. This formula allows us to find the improvements necessary to remedy Weyl, while losing diffeo invariance.
The importance of conformal field theories is difficult to overestimate, from the early days of string theory, to the present intense days of the AdS/CFT correspondence. Liouville has a prominent role among them, and so it has in a variety of other fields of theoretical investigation. Therefore, on the one hand, it is crucial to learn about the symmetries and anomalies of Liouville field theory already at the classical level. On the other hand, the research presented here opens a debate about phenomena that have been always thought to be due to quantum features of the theory.
Among the latter, first and foremost is the Hawking phenomenon, that in two dimensions is known to be in one-to-one correspondence with the Weyl/trace anomaly. Notice also that, the phenomenon we are disclosing here may even be more general as diffeo anomalies have been seen to be related to Hawking in any dimension.