On the large N convergence of matrix models

Abstract: In this paper, the large N behavior of a supersymmetric matrix model is compared with its exact continuum description. We concentrate on the large N limit of a supersymmetric matrix model describing a supermembrane with central charge on a toroidally compactified target space. We analyze, on the one hand, the supermembrane model formulated on a differentiable compact torus without boundary, with structure group given by the area-preserving diffeomorphisms, and, on the other hand, the associated regularized $SU(N)$ model. We emphasize in our analysis the structure of the constraints of the regularized model, which generate the $SU(N)$ algebra and reproduce the area-preserving diffeomorphism algebra in the large N limit, together with the topological information associated with the central charge of the model. We explain the role of the central charge in the compactified supermembrane and how it allows a top-down $SU(N)$ regularization. It is known that the regularized model has discrete spectrum. We prove, in the semiclassical approximation of the models, that in the large N limit the eigenvalues of the Hamiltonian of the supersymmetric matrix model are in one to one correspondence with, and converge exactly to, the eigenvalues of the Hamiltonian of the supermembrane (M2-brane) with central charge. Finally, we discuss some physical consequences of this result.

• The paper establishes that the matrix-regularized supermembrane with central charge converges spectrally to its continuum limit in the large N regime.
• It employs a semiclassical expansion to analyze both the Hamiltonian and global harmonic APD sectors, ensuring a discrete spectrum through topological constraints.
• The findings validate the matrix regularization method as a reliable framework for probing nonperturbative aspects of M-theory and quantum gravity.

Large N Convergence in Supersymmetric Matrix Models: The Supermembrane with Central Charge

Introduction and Motivation

The analysis of large $N$ limits in matrix models has remained fundamental to the pursuit of non-perturbative formulations of M-theory and related quantum gravity frameworks. In particular, supersymmetric matrix quantum mechanics—pioneered in the context of the BFSS model—provides a potential microscopic description of M-theory, where the large $N$ matrices are conjectured to encode the full dynamics of M2-branes. The accurate recovery of the continuum target theory from the discrete regularized (matrix) model in the $N\to\infty$ limit is essential for the validity of these approaches.

A profound challenge arises in the class of matrix models built to regularize the supermembrane (M2-brane), especially when compactified on a torus and subject to nontrivial topological (central charge) constraints. Topological subtleties and the nontrivial interplay between local (Hamiltonian) and global (harmonic) sectors of the area-preserving diffeomorphism (APD) group complicate establishing spectral convergence between the regularized and continuum formulations.

This work provides an explicit and rigorous analysis of the large $N$ limit for a matrix-regularized supermembrane model with central charge—where the compactification on a torus and the irreducible wrapping (i.e., a fixed nonzero central charge) are crucial ingredients.

Supermembrane on $M_9 \times T^2$ with Central Charge

The supermembrane theory generalizes the fundamental string paradigm to higher-dimensional objects. When formulated on an eleven-dimensional spacetime with compactification on a torus $T^2$, the embedding coordinates $(X^m, X^r)$ are subject to nontrivial winding constraints. Imposing the central charge condition restricts the theory to topological sectors characterized by irreducible wrapping over toroidal cycles. The first Chern class is fixed to a nonzero integer $n$, and the moduli describing the compact target are encoded via the winding matrix and the complex structure $\tau$.

A salient consequence of the central charge condition is the splitting of APDs into a local (Hamiltonian) sector—subject to matrix regularization—and a global harmonic sector which preserves topological data and is non-dynamical after gauge fixing. The compactification endows the theory with a discrete spectrum, contrasting with the continuous spectrum of the uncompactified or non-central charge sectors [mpgm11].

The Hamiltonian features bosonic, fermionic, and coupling terms, with explicit dependence on the covariant derivatives determined by the (fixed) harmonic background. Supersymmetry is partially broken, resulting in an $\mathcal{N}=1$ sector due to the fixed topological configuration.

Matrix Regularization and the $N$0 Model

The regularization strategy follows the Hoppe method, replacing functions on the membrane with $N$1 Hermitian matrices and the Poisson bracket with matrix commutators, leading to a finite-dimensional $N$2 structure. The APD algebra structure constants converge to their continuum counterparts in the large $N$3 limit for fixed energy levels (APD mode labels).

Key technical aspects:

• Only the Hamiltonian (local) sector of APDs is matrix-regularized; the harmonic background, selected by the central charge condition, remains fixed and unregularized.
• The matrix basis is constructed from Heisenberg-Weyl operators $N$4 and $N$5, with appropriate normalization to ensure correct algebraic convergence.
• The treatment of global APD constraints is incorporated at the regularized level, where they manifest as conditions determining the compact momenta zero modes in terms of the remaining dynamical variables.

A notable achievement is the demonstration of the discreteness of the regularized Hamiltonian's spectrum in the central charge sector, a property not present in the flat-space model and essential for any quantum gravity candidate [mpgm11].

Semiclassical Analysis and Spectral Convergence

To provide analytic control, the authors exploit the semiclassical regime by expanding the fields around static classical solutions determined by the irreducibly wrapped configuration. All quadratic fluctuations (bosonic, fermionic) are treated exactly. Canonical pairs are constructed for both non-compact and compact fluctuations after imposing and solving the constraints.

Both the continuum and matrix-regularized (finite $N$6) models reduce, in the quadratic approximation, to a set of supersymmetric harmonic oscillators with mode-dependent frequencies. For each APD mode $N$7:

• The continuum (non-regularized) oscillator frequencies are

$N$8

• The regularized frequencies at finite $N$9 are given by

$N\to\infty$0

The large $N\to\infty$1 limit ($N\to\infty$2 at fixed mode index $N\to\infty$3) yields pointwise convergence of the regularized frequencies to their continuum counterparts, thus guaranteeing one-to-one correspondence of the eigenvalues in the semiclassical spectrum.

Fermionic and bosonic contributions cancel in the zero-point energy mode by mode due to supersymmetry, removing divergences present in the purely bosonic analysis—an essential improvement over earlier treatments.

Treatment of Topological Constraints and Global APD Sectors

A central technical point is the careful handling of the global APD constraints and the harmonic sector:

• The zero modes of the compact momentum variables are not free parameters in the matrix model with central charge; rather, they are completely fixed by the remaining fluctuations and the imposed topological conditions.
• The presence and proper treatment of these global constraints are necessary for a consistent large $N\to\infty$4 correspondence. This aspect had been either neglected or incompletely treated in previous works.

Implications, Outlook, and Future Directions

The paper establishes that, at least at the semiclassical level, the matrix-regularized supermembrane with central charge exhibits strong spectral convergence to its continuum supermembrane analog as $N\to\infty$5. This validates the $N\to\infty$6-regularized model as a robust approximation for probing the quantum theory in compactified, topologically nontrivial sectors.

Theoretical implications encompass:

• Justification of the matrix regularization procedure when topological (central charge) constraints are taken into account, reinforcing its role in non-perturbative quantum gravity and M-theory programs.
• Revelation that the interplay between the APD harmonic and Hamiltonian sectors, and the precise imposition of global constraints, is crucial for convergence and physical consistency.
• The discrete spectrum in the large $N\to\infty$7 limit provides a stable sector for nonperturbative analysis, in contrast to the pathological continuous spectrum for the uncompactified supermembrane.

Open problems and prospects:

• Extension beyond the quadratic (semiclassical) regime. The full interacting theory involves residual nonlinearities and more intricate constraint structure, and the spectral convergence at strong coupling is nontrivial.
• Analysis of correlation functions and finer observables in the large $N\to\infty$8 limit.
• Potential application to other sectors of matrix models, such as those relevant for emergent spacetime and gauge/gravity duality scenarios.

Conclusion

This work demonstrates, on rigorous semiclassical grounds, that the supersymmetric $N\to\infty$9 matrix model for a supermembrane with central charge and toroidal compactification converges in spectrum to the corresponding continuum (non-regularized) theory as $N$0. The convergence is precise, mode-by-mode, provided the topological constraints are accurately incorporated and the global APD sectors treated properly. The results clarify longstanding issues in the nonperturbative formulation of M2-brane dynamics and solidify the matrix model regularization as a powerful framework in the study of nontrivial sectors of M-theory (2606.07504).