Celebrating the possible unification of Gravity and Gauge interactions the day after Mardi Gras

... and the day before the expected announcement of detection of gravitational waves, is it reasonable?

In General Relativity the Lorentz group is realized as a local symmetry of the tangent manifold. There exists no spinor representations of the diffeomorphisms and this dictates the use of this local symmetry in curved space-time. Usually the dimension of the tangent space is taken to be equal to the dimension of the curved manifold and the Lorentz symmetry is then simply a manifestation of the equivalence principle for spaces without torsion. Considering the group of local Lorentz transformations in tangent space, we can reformulate General Relativity as a gauge theory where the gauge fields are the spin-connections. If the dimensions of space time and tangent space are the same, the gauge fields (spin-connections) simply encode the same amount of information about dynamics of the gravitational field as the affine connections and nothing more. However, the dimension of the tangent group must not necessarily be the same as the dimension of the manifold [1] 

... one can unify gauge interactions with gravity by considering higher dimensional tangent spaces in a four dimensional space-time. The gauged tangent space Lorentz group describes simultaneously the symmetry groups of gravity and gauge interactions, provided a metricity condition is satisfied. The spin-connections of the higher dimensional tangent space fully incorporate information on the affine connection of space-time as well as the gauge fields. Those connections which are responsible for gravity are “composite” because they satisfy extra constraints which allow to express them in terms of the derivatives of the vielbeins. On the other hand the spin-connections responsible for gauge interactions do not obey any constraints and hence are independent. The complete geometric unification of gravity and gauge interactions is realized by writing the action of the theory just in terms of curvature invariants of the tangent group which contains the Yang-Mills action for gauge fields. The realistic group which unifies the gravity with gauge interactions and contains the Standard Model is SO (1, 13) in a fourteen dimensional tangent space. It corresponds to SO(10) grand unified theory concerning the gauge fields content, however, it has double the number of fermions, half of which can be made very massive via Brout-Englert-Higgs mechanism. The SO(1,13) is broken first {to SO(1,3)×SO(4)×SO(10) then} to SO(1,3)×U(1)×SU(2)×SU(3) and then to SO(1,3)×U(1)_em×SU(3) by using Brout-Englert-Higgs mechanism. Since the Dirac operator plays a fundamental role in this setting, it is natural to look for connections between this construction and that of noncommutative geometry. In addition, the need to add Higgs scalar fields suggests that a total unification of gravity, gauge and Higgs fields within one geometrical setting, should be possible by replacing the continuous four dimensional manifold by a noncommutative space which has both discrete and continuous structures [5]. This possibility and others will be the subject of future investigations.

//update 

Notes added: ... Michel Dubois-Violette, communicated to us the following. In 1970, R. Greene has proved that a 4-dimensional Lorentzian manifold admits locally an isometric smooth free embedding in Minkowski space M(1, 13) [9]. There is a similar result proved the same year for the Euclidean signature in M.L. Gromov and V.A. Rokhlin [10]. This means that one can include an arbitrary deformation of the four-manifold in the same flat space and eventually expect to quantize space-time in the fixed Minkowski space M(1, 13). 

On Unification of Gravity and Gauge Interactions

Ali H. Chamseddine, Viatcheslav Mukhanov

(Submitted on 6 Feb 2016 (v1), last revised 14 Feb 2016 (this version, v2))