Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite N

Abstract

We study gauge-invariant two-matrix polynomials of degree L, corresponding to the states in the su(2) sector of 𝒩=4 super Yang-Mills theory with gauge group U(N) or SU(N) at finite N. For the U(N) theory, the state space has the structure of an algebra 𝒜_N(L), which is a direct sum of permutation centraliser algebras, i.e. sub-algebras of the group algebra ℂ[S_L] of S_L. For N ≥ L, 𝒜_N (L) = 𝒜 (L) is independent of N, and the one-loop dilatation operator acts on 𝒜(L) with N as a parameter. For N < L, 𝒜_N (L) is a subspace of 𝒜(L) defined using a finite-N cutoff on Young diagrams with L boxes. We establish two results on the eigenspaces of the one-loop dilatation operator. Our first main result is that the characteristic polynomial of the one-loop dilatation operator on 𝒜_N(L) divides the characteristic polynomial from 𝒜(L), continued to N < L. Our second result is that each eigenstate of the one-loop dilatation operator in the SU(N) theory generates an infinite tower of eigenstates in the U(N) theory, through the multiplication of single-letter traces. These towers define an eigenspace factorisation of the U(N) theory labelled by the number of such traces. Further, we conjecture that the space of SU(N)-invariant polynomials in two traceless matrices at finite N is isomorphic to the intersection of 𝒜_N(L) with the subspace of ℂ[S_L] spanned by derangements. This intersection conjecture is supported by dimension counting. Finally, we propose a one-loop dilatation operator on the intersection, thus providing a direct permutation framework for the SU(N) spectral problem at finite N. The computed spectra based on the proposal agree with the prediction from the eigenspace factorisation in the U(N) theory.

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BibTeX RIS

BibTeX

@misc{padellaro2026polynomial,
  title = {{Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite $N$}},
  author = {Adrian Padellaro and Sanjaye Ramgoolam and Ryo Suzuki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10745},
  eprint = {2610.10745},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10745},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite N
AU  - Adrian Padellaro
AU  - Sanjaye Ramgoolam
AU  - Ryo Suzuki
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10745
UR  - https://arxiv.org/abs/2610.10745
AB  - We study gauge-invariant two-matrix polynomials of degree L, corresponding to the states in the su(2) sector of 𝒩=4 super Yang-Mills theory with gauge group U(N) or SU(N) at finite N. For the U(N) theory, the state space has the structure of an algebra 𝒜_N(L), which is a direct sum of permutation centraliser algebras, i.e. sub-algebras of the group algebra ℂ[S_L] of S_L. For N ≥ L, 𝒜_N (L) = 𝒜 (L) is independent of N, and the one-loop dilatation operator acts on 𝒜(L) with N as a parameter. For N < L, 𝒜_N (L) is a subspace of 𝒜(L) defined using a finite-N cutoff on Young diagrams with L boxes. We establish two results on the eigenspaces of the one-loop dilatation operator. Our first main result is that the characteristic polynomial of the one-loop dilatation operator on 𝒜_N(L) divides the characteristic polynomial from 𝒜(L), continued to N < L. Our second result is that each eigenstate of the one-loop dilatation operator in the SU(N) theory generates an infinite tower of eigenstates in the U(N) theory, through the multiplication of single-letter traces. These towers define an eigenspace factorisation of the U(N) theory labelled by the number of such traces. Further, we conjecture that the space of SU(N)-invariant polynomials in two traceless matrices at finite N is isomorphic to the intersection of 𝒜_N(L) with the subspace of ℂ[S_L] spanned by derangements. This intersection conjecture is supported by dimension counting. Finally, we propose a one-loop dilatation operator on the intersection, thus providing a direct permutation framework for the SU(N) spectral problem at finite N. The computed spectra based on the proposal agree with the prediction from the eigenspace factorisation in the U(N) theory.
ER  -

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