Hironaka decompositions in supersymmetric theories: an exploration
Abstract
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≥1 bosonic species and f fermionic species, we construct this decomposition and obtain (N!)d−12fN 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)× 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=x2y2, every positive-degree scalar annihilates the nonzero class xy, while xn is in cohomology for every n≥ 1. These two properties rule out a Hironaka decomposition over scalar primaries.
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BibTeX
@misc{gadde2026hironaka,
title = {{Hironaka decompositions in supersymmetric theories: an exploration}},
author = {Abhijit Gadde and Rajat Raj and Shivansh Tomar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36094},
eprint = {2610.36094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36094},
}
RIS
TY - UNPB
TI - Hironaka decompositions in supersymmetric theories: an exploration
AU - Abhijit Gadde
AU - Rajat Raj
AU - Shivansh Tomar
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36094
UR - https://arxiv.org/abs/2610.36094
AB - 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≥1 bosonic species and f fermionic species, we construct this decomposition and obtain (N!)^d−12^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)× 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≥ 1. These two properties rule out a Hironaka decomposition over scalar primaries.
ER -