Unification of elementary forces in gauge SL(2N,C) theories

TL;DR

This paper proposes that gauge theories based on the group SL(2N,C) could unify all elementary forces, including gravity, by naturally reducing to local SL(2,C) and SU(N) symmetries, with implications for grand unified theories.

Contribution

It introduces a framework where SL(2N,C) gauge theories unify gravity and gauge interactions, and explores their implications for grand unified theories, especially highlighting the SU(8) unification for three families.

Findings

The theory naturally reduces to SL(2,C) gravity and SU(N) GUTs.

Many traditional GUTs like SU(5) are not compatible with standard fermions.

SU(8) GUT from SL(16,C) may unify all three families of quarks and leptons.

Abstract

We argue that the gauge $SL(2N,C)$ theories may point to a possible way where the known elementary forces, including gravity, could be consistently unified. Remarkably, while all related gauge fields are presented in the same adjoint multiplet of the $SL(2N,C)$ symmetry group, the tensor field submultiplet providing gravity can be naturally suppressed in the weak-field approach developed for accompanying tetrad fields. As a result, the whole theory turns out to effectively possess the local $SL(2,C)\times SU(N)$ symmetry so as to naturally lead to the $SL(2,C)$ gauge gravity, on the one hand, and the $SU(N)$ grand unified theory, on the other. Since all states involved in the $SL(2N,C)$ theories are additionally classified according to their spin values, many possible $SU(N)$ GUTs - including the conventional one-family $SU(5)$ theory - appear not to be relevant for the standard $1/2$ spin quarks and leptons. Meanwhile, the $SU(8)$ grand unification for all three families of composite quarks and leptons that stems from the $SL(16,C)$ theory seems to be of special interest that is studied in some detail.