Existence and stability of nonzero auxiliary-coset branches

Determine whether nonzero branches of the auxiliary coset connection \(X\), including branches disconnected from the regular \(X=0\) branch, exist and are stable while satisfying both the algebraic coset-connection equation and the remaining connection equation of the exactly orbit-constrained projected Yang–Mills theory.

Background

In the exactly constrained theory with positive Higgs kinetic coefficient, the coset connection $X$ is auxiliary. On the regular weak-curvature branch, its algebraic equation eliminates it through the solution $X=0$, reducing the theory locally to $SU(n)$ Yang–Mills theory.

The algebraic equation is nonlinear because the projected curvature contains the term ${}_{su}(X\wedge X)$. The paper develops eigenvalue, Cartan-root, fixed-point, and bifurcation criteria for possible nonzero solutions, but emphasizes that solving the pointwise algebraic equation is insufficient: a candidate branch must also satisfy the coupled equation for the remaining connection $B$. The existence and stability of such branches, especially those not connected to $X=0$, are therefore left unresolved.