SL(2N,C) Yang-Mills Theories: Direct Internal Forces and Emerging Gravity

Abstract

A four-dimensional gauge-gravity unification based on local $SL(2N,C)$ symmetry is developed in a universal Yang--Mills-type setting, which, however, appears dynamically consistent only in the symmetry-broken phase. In the exact symmetry limit the theory may only be formulated in a premetric framework, where the accompanying tetrad multiplets, though promoted to dynamical fields, do not yet satisfy the conventional invertibility conditions. An ordinary Einstein--Cartan spacetime geometry emerges only in the broken post-soldering phase, in which the $SL(2N,C)$ tetrad multiplets are treated as constrained dynamical fields selecting a neutral internal symmetry branch. This realizes the breaking $SL(2N,C)\to SL(2,C)\times SU(N)$, thereby lifting all noncompact internal directions, while the surviving neutral tetrad is, as usual, associated with the gravitational field. A special ghost-free curvature-squared Lagrangian provides a consistent quadratic sector for the spin connection, propagating only admissible connection modes: the massless $SU(N)$ vector fields together with massive axial-vector and pseudoscalar multiplets. The Einstein--Cartan linear curvature term is argued to arise radiatively from fermion loops, thereby relating the gravitational scale to the same $SL(2N,C)$-covariant matter sector that defines the unified gauge coupling. Finally, the matter sector points to a deeper elementarity of $SL(2N,C)$ spinors, identified with preon constituents whose bound states form the observed quarks and leptons. Anomaly matching between preons and composites singles out $N=8$. The chain $SL(16,C)\to SL(2,C)\times SU(8)$ then naturally yields three composite quark--lepton families, while filtering out extraneous heavy states.

30 pages, published version, to appear in Phys. Rev. D