Symmetries of field configurations on compact spaces, spinors and no-go theorems

Most modern research into theories with more than four spacetime dimensions has taken place within the context of string theory and/or modern presentations of quantum field theories. They often involve making an ansatz for a Lagrangian, composed of fields described by fibre bundles and assumed to transform as multiplets of SU(N) gauge symmetries. Often, supersymmetry and/or quantisation of the theory are explicitly imposed. It is assumed that the Coleman–Mandula no-go theorem prevents non-trivial mixing of Lorentz symmetries and other symmetries.

By contrast, this work has been developed in a context in which higher-dimensional theories are rethought from first principles. The only fundamentals are the geometry of spacetime and its symmetry properties under transformations.

We consider diffeomorphisms on closed spacelike hypersurfaces, and the induced ‘dragging action’ on fields on these compact spaces. We use the action of translations and rotations on vector and scalar fields as examples.

Any fields on such a space can be decomposed into harmonics. These form multiplets under rotational diffeomorphisms.

We find that even classical fields or fields describing geometry can be decomposed into such harmonics, distinguished by their ‘quantum numbers’, which describe their transformation properties.

We show that if the three familiar spatial dimensions form a compact space, and there are six extra dimensions also forming a compact space, there are harmonics carrying the quantum numbers of a single generation of fermions in the Standard Model.

In such a theory, the relevant no-go theorem is not that of Coleman and Mandula, but that of O’Raifeartaigh. We find that the theory exploits a loophole in this theorem. The Poincaré group and the isometry group for the extra dimensions are embedded in an SO(10) group. However, this is only manifest in a non-physical ‘decompactification limit’. Away from this limit, the extra symmetries are non-linearly realised.