Bezoutian Decoupling for Conformal Yang--Mills Multiplets in $(A)dS$

Abstract

Metsaev exhibited generic and decoupled formulations of conformal Yang--Mills theory in $(A)dS_6$, $(A)dS_8$, and $(A)dS_{10}$, and conjectured the corresponding relations in higher even dimensions. Motivated by the low-dimensional matrices in Appendix C, we identify and diagonalize a natural all-$N$ Bezoutian continuation of the three generic Gram forms. The monic Hankel cutoff, the evaluation pattern in Appendix C, and the conjectured mass nodes determine this continuation uniquely. Its vector and radical Gram-form diagonalization identities hold for every $D=d+1=2N+4$, $N\geq1$. The continuation is the Bezout matrix of $z\prod_{i=1}^N(z-ρi(2N+1-i))$ and $1$; evaluation at its simple roots produces the Vandermonde congruence appearing in the field redefinition. The same calculation gives closed formulas for the inverse, determinant, and inertia, and identifies the normalization weights with Johnson-graph multiplicities. Any nonlinear coefficient algebra already specified in the generic formulation is transported by this change of basis. The result does not construct such an algebra in arbitrary dimension. At $N=4$, we impose associativity, Frobenius invariance, and a fixed flat specialization; these algebraic constraints still admit a one-parameter family of pairwise distinct products in a fixed generic basis.