Gravity and Unification: Insights from SL(2N,C) Gauge Symmetries

Abstract: The perspective that gravity may govern the unification of all known elementary forces calls for an extension of the gauge gravity symmetry group $SL(2,C)$ to the broader local symmetry $SL(2N,C)$, where $N$ reflects the internal $SU(N)$ symmetry subgroup. This extension is shown to lead to a consistent hyperunification framework, provided that the tetrad fields of $% SL(2,C)$ retain their invertibility condition in the extended theory, thus maintaining their connection to gravity. As a result -- while the full gauge multiplet of $SL(2N,C)$ typically comprises vector, axial-vector and tensor field submultiplets of $SU(N)$ -- only the vector submultiplet and singlet tensor field may manifest in an observed particle spectrum. The axial-vector submultiplet remains decoupled from ordinary matter, while the tensor submultiplet acquires the Planck scale order mass. Consequently, the effective symmetry of the theory reduces to $SL(2,C)\times SU(N)$, bringing together $SL(2,C)$ gauge gravity and $SU(N)$ grand unification. As all states in $SL(2N,C)$ are also classified by their spin magnitudes, some $% SU(N)$ grand unified models, including the standard $SU(5)$, appear unsuitable for the standard spin-$1/2$ quarks and leptons. However, applying $SL(2N,C)$ symmetry to a model of composite quarks and leptons, where constituent chiral preons form the fundamental representations, identifies $% SL(16,C)$ with its effective $SL(2,C)\times SU(8)$ symmetry accommodating all three quark-lepton families, as the most compelling candidate for hyperunification of the existing fundamental forces.