Claud Lovelace: membranes in 28 dimensions

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Claud Lovelace was the first Gentleman who encountered the critical dimension, D=26, in string theory more than 35 years ago. As you know, the supersymmetric string critical dimension is D=10.

Membranes with one temporal dimension, d=2+1, lead to D=10+1 and it could perhaps be natural to think about a bosonic M-theory in D=26+1 dimensions. Also, one may imagine worldvolume theories with two times, i.e. d=2+2, in D=10+2 or D=26+2 dimensions. Note that in all these cases, you are left with 8 (super case) or 24 (bosonic case) transverse dimensions.

Is this story going beyond this simple numerology? Most string theorists would answer Probably not. However, a few brave souls including Lovelace have been trying to construct a new mathematical formalism with symmetries and ghosts that would be as powerful as conformal field theory of 2D string worldsheets - but it would be different, higher-dimensional, and perhaps more fundamental.




Self-duality is needed for the 2+2-dimensional worldvolumes to replace the conformal structure on the world sheet. In the past, the corresponding theory needed diffeomorphism ghosts. The "central charge" cancellation led to the critical dimension D=28-1566=-1538, well below zero.

See: Calculable membrane theory

Lovelace now claims to have gotten rid of the diffeomorphism ghosts, returning him to the D=28 game for the 2+2-dimensional membrane.

The Riemann tensor is decomposed into the Ricci scalar; the de-traced Ricci tensor; and the self-dual and anti-self-dual components of the Weyl tensor. Instead of one conformal anomaly in 4D, there are four analogous anomalies in 4D: the squares of the two Weyl tensors; the Euler density; the Hirzebruch signature density.

The 4D topological gravity is shown to be not-quite-topological: one component of g_{mu nu}, out of ten, survives - a fact that should be obvious to the people who have used N=(2,1) strings to describe such self-dual systems. This surviving scalar plays the role of the conformal mode. An anomaly turns out to be (D-28)/360. I don't quite understand the formalism yet.