The Einstein-Hilbert action with cosmological constant as a functional of generic form
The geometrical underpinnings of a specific class of Dirac operators is discussed. It is demonstrated how this class of Dirac operators allow to relate various geometrical functionals like, for example, the Yang-Mills action and the functional of non-linear $\sigma-$models (i.e. of (Dirac) harmonic maps). These functionals are shown to be similar to the Einstein-Hilbert action with cosmological constant (EHC). The EHC may thus be regarded as a "generic functional". As a byproduct, the geometrical setup presented also allows to avoid the issue of "fermion doubling" as usually encountered, for instance, in the geometrical discussion of the Standard Model in terms of Dirac operators. Furthermore, it is demonstrated how the geometrical setup presented allows to derive the cosmological constant term of the EHC from the Einstein-Hilbert functional and the action of a purely gauge coupling Higgs field.