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

Abstract

The perspective that gravity may govern the unification of all elementary forces calls for extending the gauge-gravity symmetry $SL(2,C)$ to the broader local symmetry $SL(2N,C)$, where $N$ reflects the internal $SU(N)$ subgroup. This extension yields a consistent hyperunification framework in which -- aside from the linear gravity Lagrangian, to which only tensor fields contribute -- the quadratic curvature sector is fully unified across all gauge submultiplets. Tetrad fields play a central role: once dynamical, their invertibility -- treated as a nonlinear sigma-model type length constraint -- naturally implies condensation and thereby triggers spontaneous breaking of $SL(2N,C)$. As a result, while the full gauge multiplet contains vector, axial-vector, and tensor submultiplets, only the vector submultiplet remains in the observed spectrum; the axial-vector and tensor submultiplets acquire large masses at the symmetry-breaking scale. The effective symmetry reduces to $SL(2,C)\times SU(N)$, collecting together $SL(2,C)$ gauge gravity and the $SU(N)$ grand-unified sector. Since states in $SL(2N,C)$ are also classified by spin magnitudes, many $SU(N)$ GUT models -- such as standard $SU(5)$% -- appear ill-suited for fundamental spin-$1/2$ quarks and leptons. By contrast, applying $SL(2N,C)$ to a composite framework with chiral preons in fundamental representations points to $SL(16,C)$, with effective accommodating all three quark-lepton families, as a compelling candidate for hyperunification of all fundamental forces.

Comments: Published version, 30 pages, to appear in Nuclear Physics B (2025)