Diff Anomalies in Four Dimensions

An extension of the diffeomorphism algebra is a kind of diff anomaly, but according to standard wisdom there are no diff anomalies at all in four dimensions. This is correct, within the framework of QFT without horizontal fuzziness. Hence a quantum theory of gravity must go beyond QFT.

To elaborate on this point, let us consider an analogous but simpler system, namely Yang-Mills theory in four dimensions. In the Hamiltonian formulation, the constraint algebra is the algebra of maps from 3-space to some finite-dimensional Lie algebra. As has been well known to experts since the 1980s [Pressley-Segal, Loop groups, chapters 4 and 9], this algebra of maps admits two qualitatively different extensions: the central extension and the Mickelsson-Faddeev extension. Let us contrast the two.

The Mickelsson-Faddeev extension
1. is proportional to the third Casimir invariant.
2. describes gauge anomalies in QFT.
3. treats all space points on an equal footing.
4. has no non-trivial unitary representations, at least not of lowest-energy type [Pickrell 1989].
5. is therefore a bad anomaly, which must not arise in nature.
6. and indeed, cancels in the standard model.

In contrast, the central extension
1. is proportional to the second Casimir invariant.
2. does not arise within the framework of QFT without horizontal fuzziness.
3. does not treat space points equally, but depends on a privileged one-dimensional curve, "the observer's trajectory".
4. has many non-trivial unitary representations.
5. is therefore a good anomaly, which cannot be ruled out on the basis of unitarity.

The multi-dimensional Virasoro algebra is the diffeomorphism analogue of this central extension, which we may also call the multi-dimensional affine algebra.

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