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arXiv · 2607.04143

Finite-$N$ Operator Algebras and the Hilbert Space of Bilocal Holography

Byoungjoon Ahn·Robert de Mello Koch·Vatsal Garg·Antal Jevicki·Junggi Yoon
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Abstract

We give an operator-algebraic and representation-theoretic description of the Hilbert spaces of finite-$N$ bilocal holography. This work is a sequel to the finite-$N$ Hilbert space construction of arXiv:2602.20788 [hep-th]. The central result is the establishment of an invariant dual-pair operator algebra: before imposing the singlet constraint the Fock space carries commuting actions of the color group and of a bilocal Lie algebra, while the projection to the singlet sector selects a single irreducible representation of the invariant Lie algebra, which we call a master algebra. The finite-$N$ trace relations, beginning with the quadratic identities studied here, are shown to become representation-theoretic identities of the selected irreducible representation. We summarize the orthogonal, symplectic and unitary cases, identify the corresponding finite-$N$ constraints, compute the singlet Casimirs, and explain how finite traces and partition functions are obtained through characters of the resulting irreducible representations. This provides a novel, previously unknown mathematical description of the singlet space.

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Byoungjoon Ahn, Robert de Mello Koch, Vatsal Garg, Antal Jevicki, Junggi Yoon. 2026-07-05. Finite-$N$ Operator Algebras and the Hilbert Space of Bilocal Holography. https://arxiv.org/abs/2607.04143

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