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To fully clarify the invariance of the classical Liouville field theory underthe Virasoro algebra, we first elucidate in detail the concept of classicalanomaly, discuss the occurrence of two symmetry algebras associated to thisproblem, and provide some new formulae to compute the classical center in ageneral fashion. We apply this to the study of the symmetries of the free bosonin two dimensions. Moving to Liouville, we see how this gives rise to anenergy-momentum tensor with non-tensorial conformal transformations, in flatspace, and a non-vanishing trace, in curved space. We provide a variety ofimprovements of the (local) theory, that restore Weyl invariance. With explicitcomputations, we show that the covariant conservation of theWeyl-invariance-improved energy-momentum tensor is lost, in general, and relatethe chosen improvement with the corresponding subset of preserveddiffeomorphisms. The non-tensorial transformation rule of theWeyl-invariance-improved energy-momentum tensor in curved space is explicitlyback-traced to the Virasoro center.

Trace and diffeomorphism anomalies of the classical Liouville theory, Virasoro algebras, Weyl-gauging and all that