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

J. L. Chkareuli

TL;DR

This work proposes a four-dimensional gauge–gravity unification based on the non-compact group $SL(2N,\mathbb{C})$, where dynamical tetrads obey a nonlinear length constraint that triggers spontaneous breaking to $SL(2,\mathbb{C})\times SU(N)$ and lifts noncompact directions. It constructs a ghost-free, curvature-squared sector with a single universal gauge coupling, and shows that an Einstein–Cartan linear curvature term can be radiatively generated by fermion loops after tetrad condensation. The matter sector is treated via a composite preon model, with anomaly matching uniquely selecting $N=8$ and yielding an $SL(16,\mathbb{C})$ metaflavor that produces three composite quark–lepton families under $SL(2,\mathbb{C})\times SU(8)$. The framework thus provides a fully four-dimensional, gauge-based route to unify gravity with internal interactions, with gravity emerging from quantum effects and a rich preon/composite structure predicting a distinctive heavy spectrum and family pattern.

Abstract

We develop a four-dimensional gauge-gravity unification based on the \(SL(2N,\mathbb C)\) gauge theory taken in a universal Yang--Mills type setting. The accompanying tetrads are promoted to dynamical fields whose contracted invertibility condition is interpreted as a nonlinear sigma-model type length constraint. This triggers tetrad condensation and spontaneously breaks $\mathrm{SL}(2N, \mathbb{C})\to \mathrm{SL}(2,\mathbb{C})\times\mathrm{SU}(N)$, lifting all noncompact directions. A special ghost-free curvature-squared Lagrangian provides a consistent quadratic sector, while an Einstein--Cartan linear curvature term is induced radiatively from fermion loops. Below the breaking scale, only a neutral tetrad associated with graviton and $\mathrm{SU}(N)$ vector fields remain massless, whereas axial-vector and tensor fields of the entire gauge multiplet acquire heavy masses. The matter sector clearly points to a deeper elementarity of \(SL(2N,\mathbb C)\) spinors, which can be 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 $\mathrm{SL}(16,\mathbb{C})\to \mathrm{SL}(2,\mathbb{C})\times\mathrm{SU}(8)$ then naturally yields three composite quark--lepton families, while filtering out extraneous heavy states.

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

TL;DR

This work proposes a four-dimensional gauge–gravity unification based on the non-compact group , where dynamical tetrads obey a nonlinear length constraint that triggers spontaneous breaking to and lifts noncompact directions. It constructs a ghost-free, curvature-squared sector with a single universal gauge coupling, and shows that an Einstein–Cartan linear curvature term can be radiatively generated by fermion loops after tetrad condensation. The matter sector is treated via a composite preon model, with anomaly matching uniquely selecting and yielding an metaflavor that produces three composite quark–lepton families under . The framework thus provides a fully four-dimensional, gauge-based route to unify gravity with internal interactions, with gravity emerging from quantum effects and a rich preon/composite structure predicting a distinctive heavy spectrum and family pattern.

Abstract

We develop a four-dimensional gauge-gravity unification based on the \(SL(2N,\mathbb C)\) gauge theory taken in a universal Yang--Mills type setting. The accompanying tetrads are promoted to dynamical fields whose contracted invertibility condition is interpreted as a nonlinear sigma-model type length constraint. This triggers tetrad condensation and spontaneously breaks , lifting all noncompact directions. A special ghost-free curvature-squared Lagrangian provides a consistent quadratic sector, while an Einstein--Cartan linear curvature term is induced radiatively from fermion loops. Below the breaking scale, only a neutral tetrad associated with graviton and vector fields remain massless, whereas axial-vector and tensor fields of the entire gauge multiplet acquire heavy masses. The matter sector clearly points to a deeper elementarity of \(SL(2N,\mathbb C)\) spinors, which can be identified with preon constituents whose bound states form the observed quarks and leptons. Anomaly matching between preons and composites singles out . The chain then naturally yields three composite quark--lepton families, while filtering out extraneous heavy states.

Paper Structure

This paper contains 22 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\ SL(2N,C)$ gauge theories
  3. Neutral spinors -- $SL(2,C)$ symmetry
  4. Charged spinors -- $SL(2N,C)$ hyperunification
  5. Gauging $SL(2N,C)$
  6. Tetrads linked to gravity
  7. Lagrangians
  8. Vector and axial-vector fields
  9. Total gauge multiplet
  10. Tetrad Condensation -- Lifting non-compact directions
  11. Dynamical tetrads
  12. Hyperflavor-blind vacuum
  13. Emerging gravity
  14. Preliminaries
  15. Radiative corrections
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Fermion-bubble diagrams generating the Einstein--Cartan term. Left: a mixed $V_{T}$--$V_{e}$ insertion produces $e\wedge e\wedge dT$. Right: two $V_{T}$--$V_{e}$ insertions produce $e\wedge e\wedge (gT\wedge T)$. Their sum reconstructs $e\wedge e\wedge R[T]$.