Casimir multiplets suffice: strong generation in 𝒩=4 SU(N) SYM
Abstract
The Schur sector of four-dimensional 𝒩=4 super Yang−−Mills theory contains infinitely many protected operators, while the invariant polynomials of a single adjoint scalar are organized by finitely many Casimir degrees. I prove that the small 𝒩=4 short multiplets built from these invariants strongly generate the associated chiral algebra, thereby establishing the SU(N) case of the strong-generation conjecture of Beem et al. For every N≥2, all field components associated with degrees 2,…,N suffice, in every conformal weight and every degree of the standard relative quantum BRST cohomology. The proof brings together cyclic trace cohomology, quantum lifting, and an arbitrary-degree quartic−−odd transport identity. Exact field relations and a finite-rank Newton obstruction exclude every additional lowest space, including non-scalar flavor representations; the required normal-ordering corrections and BRST primitives are derived explicitly. The resulting completeness theorem explains how a finite family of supersymmetric multiplets organizes an unbounded quantum spectrum. Every Schur class is a normal polynomial in their components and derivatives, even when its gauge-theory representative involves fermions and arbitrarily many derivatives. This supplies a uniform set of local inputs for meromorphic operator products, bounds protected state multiplicities before cancellations in an index, and gives an algebraic account of the finite-rank reorganization of protected trace operators.
Related
BibTeX
@misc{he2026casimir,
title = {{Casimir multiplets suffice: strong generation in $\mathcalN=4$ $SU(N)$ SYM}},
author = {Yifan He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24594},
eprint = {2610.24594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24594},
}
RIS
TY - UNPB
TI - Casimir multiplets suffice: strong generation in 𝒩=4 SU(N) SYM
AU - Yifan He
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.24594
UR - https://arxiv.org/abs/2610.24594
AB - The Schur sector of four-dimensional 𝒩=4 super Yang−−Mills theory contains infinitely many protected operators, while the invariant polynomials of a single adjoint scalar are organized by finitely many Casimir degrees. I prove that the small 𝒩=4 short multiplets built from these invariants strongly generate the associated chiral algebra, thereby establishing the SU(N) case of the strong-generation conjecture of Beem et al. For every N≥2, all field components associated with degrees 2,…,N suffice, in every conformal weight and every degree of the standard relative quantum BRST cohomology. The proof brings together cyclic trace cohomology, quantum lifting, and an arbitrary-degree quartic−−odd transport identity. Exact field relations and a finite-rank Newton obstruction exclude every additional lowest space, including non-scalar flavor representations; the required normal-ordering corrections and BRST primitives are derived explicitly. The resulting completeness theorem explains how a finite family of supersymmetric multiplets organizes an unbounded quantum spectrum. Every Schur class is a normal polynomial in their components and derivatives, even when its gauge-theory representative involves fermions and arbitrarily many derivatives. This supplies a uniform set of local inputs for meromorphic operator products, bounds protected state multiplicities before cancellations in an index, and gives an algebraic account of the finite-rank reorganization of protected trace operators.
ER -