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By a representation theoretic picture I mean viewing the central charge as something that arises as an obstruction to lifting a projective representation of the Witt algebra of the generators of conformal transformations into a linear representation, while by a physical picture I mean viewing it as something to do with the Casimir energy and the trace of the stress tensor.

One can think about more general anomalies and see the two perspectives there. The first picture has something to do with the symmetry group (in this case the conformal group) failing to commute with certain observables (eg. the L operators). The second picture is when we compute expectation values of these operators (eg. the stress tensor) they transform under the symmetries.

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