Hironaka decompositions in supersymmetric theories: an exploration

We ask when tree-level supersymmetric cohomology admits a Hironaka decomposition: a unique expansion in finitely many secondary operators with polynomial coefficients in algebraically independent bosonic primaries. Such a decomposition organizes the cohomology into towers with a common set of unrestricted dressings. For diagonal models with $N$ labels, $d\geq1$ bosonic species and $f$ fermionic species, we construct this decomposition and obtain $(N!)^{d-1}2^{fN}$ secondaries. We also prove that the scalar-branch ring of ABJM theory admits a Hironaka decomposition at every rank and positive integer Chern–Simons level, and construct explicit bases for the $SU(2)\times SU(2)$ Klebanov–Witten scalar ring. We also do the same for the Higgs branches of two class-S theories. However, not all supersymmetric cohomologies admit a Hironaka decomposition. In the Wess–Zumino model with $W=x^2y^2$, every positive-degree scalar annihilates the nonzero class $xy$, while $x^n$ is in cohomology for every $n\geq 1$. These two properties rule out a Hironaka decomposition over scalar primaries.