Antal Jevicki

A bstract: 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 . 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 reduction has complementary operator-algebraic and invariant-theory manifestations: quadratic identities characterize the selected representation, while the familiar trace relations appear as null-state identities within that 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.

A bstract: For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite N trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is established, while for bosonic theories a space of freely acting primaries multiplied by a finite set of secondaries emerges. The Hilbert space of states obey finite N cut off bounds, implying finiteness of traces and entropy.

A bstract: Continuing the formulation of finite N Hilbert spaces in emergent theories we study in this work S _N symmetric collective models. For the case of N bosons in d dimensions, which map to matrix models with commuting matrices, we describe a complete algorithm and give a detailed case study reproducing the expected primaries and determining secondary invariants at each bidegree (a Hironaka decomposition). The method is based on null spaces (of the full collective theory) which are seen to yield all the independent trace relations, reducing the construction to linear algebra. As a stringent check, of our algorithm, we have verified that the system of invariants generates a subset of gauge invariant operators with no redundancies. This results in a reduction of the Hilbert space, in particular the gauge invariant secondary invariants realize an emergent Fock space with finite- N occupation-numbers.

A bstract: We study the Hilbert space structure of gauge-invariant operators emergent in large- N multi-matrix quantum mechanics. Building on the framework of [1], we identify a class of light single-trace operators that behave like free creation operators at low energy but saturate beyond a critical excitation level, ceasing to generate new states. This q -reducibility is a direct consequence of finite N trace identities and leads to a dramatic truncation of the high-energy spectrum of the emergent theory. The resulting number of independent degrees of freedom is far smaller than naïve semiclassical expectations, providing a concrete mechanism for how nonperturbative constraints shape the ultraviolet behaviour of emergent theories.

We develop analytical and numerical methods for the matrix thermofield in the large N limit. Through the double collective representation on the Schwinger-Keldysh contour, it provides thermodynamical properties and finite temperature correlation functions, for large N matrix quantum systems.

The space of invariants for a single matrix is generated by traces containing at most N matrices per trace. We extend this analysis to multi-matrix models at finite N. Using the Molien-Weyl formula, we compute partition functions for various multi-matrix models at different N and interpret them through trace relations. This allows us to identify a complete set of invariants, naturally divided into two distinct classes: primary and secondary. The full invariant ring of the multi-matrix model is reconstructed via the Hironaka decomposition, where primary invariants act freely, while secondary invariants satisfy quadratic relations. Significantly, while traces with at most N matrices are always present, we also find invariants involving more than N matrices per trace. The primary invariants correspond to perturbative degrees of freedom, whereas the secondary invariants emerge as non-trivial background structures. The growth of secondary invariants aligns with expectations from black hole entropy, suggesting deep structural connections to gravitational systems.

A bstract: For matrix model and QFT, we discuss how dual gravitational geometry emerges from matrix degrees of freedom (specifically, adjoint scalars in super Yang-Mills theory) and how operator algebra that describes an arbitrary region of the bulk geometry can be constructed. We pay attention to the subtle difference between the notions of wave packets that describe low-energy excitations: QFT wave packet associated with the spatial dimensions of QFT, matrix wave packet associated with the emergent dimensions from matrix degrees of freedom, and bulk wave packet which is a combination of QFT and matrix wave packets. In QFT, there is an intriguing interplay between QFT wave packet and matrix wave packet that connects quantum entanglement and emergent geometry. We propose that the bulk wave packet is the physical object in QFT that describes the emergent geometry from entanglement. This proposal sets a unified view on two seemingly different mechanisms of holographic emergent geometry: one based on matrix eigenvalues and the other based on quantum entanglement. Further intuition comes from the similarity to a traversable wormhole discussed as the dual description of the coupled SYK model by Maldacena and Qi: the bulk can be seen as an eternal traversable wormhole connecting boundary regions.

We construct a model of Pauli spin operators with all-to-all 4-local interactions by replacing Majorana fermions in the SYK model with spin operators. Equivalently, we replace fermions with hard-core bosons. We study this model numerically and compare the properties with those of the SYK model. We observe a striking quantitative coincidence between the spin model and the SYK model, which suggests that this spin model is strongly chaotic and, perhaps, can play some role in holography. We also discuss the path-integral approach with multi-local fields and the possibility of quantum simulations. This model may be an interesting target for quantum simulations because Pauli spins are easier to implement than fermions on qubit-based quantum devices.