Projected Yang-Mills Theory and the MacDowell-Mansouri Analogy

Abstract: We study $SO(2n)$ gauge theories with a degenerate curvature kinetic operator defined by an orthogonal complex structure. The kinetic image is $\su(n)$, while the adjoint vacuum has stabilizer $U(n)$. We distinguish fixed projectors, unrestricted adjoint Higgs fields, and Higgs fields constrained exactly to the orbit $SO(2n)/U(n)$. For the constrained theory with a positive Higgs kinetic coefficient, the coset connection is auxiliary and the central connection is null. A nonlinear Dirac analysis establishes a regular weak-curvature branch with $2(n^2-1)$ physical degrees of freedom in four dimensions, locally equivalent to $SU(n)$ Yang--Mills theory. Normal Higgs displacements in the unrestricted theory can change the kinetic rank, so this count does not extend to that field space. We also characterize possible algebraic branches. We express the Stelle--West Higgs--gravity action through a common Clifford involution, and introduce projected Pontryagin theories and a full-rank equivariant completion.