Lorentz-Covariant Supersymmetric Matrix Model

• The paper presents a Lorentz-covariant supersymmetric matrix model that preserves full spacetime symmetry and extends nonperturbative formulations of M-theory.
• It employs a novel RVPD symmetry and matrix regularization of Nambu brackets to consistently incorporate both bosonic and fermionic degrees of freedom.
• The analysis classifies various BPS states and connects the framework to D2-brane matrix theory, paving the way for extensions to higher brane models.

A supersymmetric Lorentz-covariant matrix model is a matrix-based nonperturbative formulation of M-theory or related high-dimensional supersymmetric theories that maintains manifest Lorentz invariance and supersymmetry at the level of the regularized action, field content, and gauge symmetry. These models aim to go beyond light-cone or gauge-fixed constructions—such as the original BFSS matrix model—by realizing full spacetime symmetry (SO(1,10) for M2-branes in 11D, for example) and extended local/global supersymmetry, incorporating both bosonic and fermionic degrees of freedom in a unified matrix framework. Recent developments center on the consistent regularization of membrane worldvolume Nambu brackets, the design of gauge symmetries compatible with Lorentz and supersymmetric algebras, and the systematic classification of BPS configurations, all within a matrix model context that can be extended (in principle) to higher branes.

1. Background and Lorentz Covariant Matrix Model Paradigm

Traditional matrix models such as BFSS are formulated in the light-cone gauge, breaking manifest Lorentz invariance but providing a nonperturbative definition for M-theory. Achieving a Lorentz-covariant formulation requires a matrix regularization scheme in which spacetime indices are treated democratically, and all physical states transform consistently under the full Lorentz group. An outstanding challenge in the context of membranes (e.g., M2-branes) has been the consistent discretization of the worldvolume Nambu bracket, which governs the dynamics of the bosonic degrees of freedom and is tightly interwoven with the gauge invariance associated with volume-preserving diffeomorphisms (VPD).

A new approach, established in (Katagiri, 8 Apr 2025), introduces a Lorentz-covariant gauge restriction using a fixed Lorentz vector $C_I$, leading to a restricted class of VPDs ("restricted VPD" or RVPD) that ensure closure of the algebraic structure necessary for consistent matrix regularization. This method is extended to the supersymmetric case in (Katagiri, 10 Aug 2025), which provides a fully Lorentz-covariant, supersymmetric matrix model for M2-branes.

2. RVPD Symmetry, Nambu Bracket Decomposition, and Matrix Regularization

The central technical advance enabling the supersymmetric Lorentz-covariant matrix model is the realization that the Nambu bracket,

$$\{X^I, X^J, X^K\} = \varepsilon^{ijk} (\partial_{\sigma^i} X^I)(\partial_{\sigma^j} X^J)(\partial_{\sigma^k} X^K),$$

can be decomposed as

$$\{A, B, C\} = \{\tau(A,B),C\} + (\partial_{\sigma^3} C)\{A,B\} + \Sigma(A,B;C),$$

where $$\tau(A, B) = (\partial_{\sigma^3}A) B - (\partial_{\sigma^3}B) A$$ and $\{A,B\}$ is a two-dimensional Poisson bracket.

Imposing $\partial_{\sigma^3} \tau(Q_1, Q_2) = 0$ and $\{Q_1, Q_2\} = 0$ on the variation parameters restricts allowed deformations to RVPDs, whose generators

$\tau(Q_1^R, Q_2^R)$

form a closed algebra under commutation. The restricted Nambu bracket can then be matrix-regularized by mapping

$\{A, B\} \to -i [A, B],$

allowing

$\delta_R X^I = [\tau(Q_1^R, Q_2^R), X^I].$

This regularization bypasses previous obstructions with the Leibniz rule and the Fundamental Identity, enabling a mathematically consistent Lorentz-covariant matrix description of membrane dynamics (Katagiri, 8 Apr 2025, Katagiri, 10 Aug 2025).

3. Supersymmetric Extension: Field Content and Symmetry Structure

The supersymmetric Lorentz-covariant M2-brane matrix model is constructed by recasting the full supermembrane action in terms of Nambu brackets, followed by matrix regularization of both bosonic and fermionic fields. The essential elements include:

• Matrix variables $X^I$ for bosonic membranes, with $I$ labeling $SO(1,10)$ indices.
• Fermionic matrix variables $\theta$ as $SO(1,10)$ spinors, with supersymmetry acting as $\delta_S X^I = i \bar\epsilon \Gamma^I \theta$ (schematically).
• The bosonic action term is

$S = \int d^3\sigma \frac{1}{2} \{X^I, X^J, X^K\}^2$

with appropriate fermionic and supersymmetry-coupling terms—expressed using triple brackets and their matrix-regularized analogues.

• Local $\kappa$-symmetry, originally ensuring the matching of bosonic and fermionic degrees of freedom, is modified to a restricted form (denoted $\tilde{\kappa}$) that commutes with RVPD, and the algebra closes into a novel symmetry structure (Katagiri, 10 Aug 2025).

The algebra of RVPD and restricted $\kappa$-symmetry supports consistent classical and quantum dynamics, enabling BPS state analysis and facilitating the preservation of Lorentz covariance.

4. BPS Spectrum and Classification of Solutions

Within this framework, BPS states—configurations preserving a fraction of the supersymmetry—are systematically classified using the RVPD/$\tilde{\kappa}$ symmetry algebra:

• Particle-like (purely time-dependent) solutions are 1/2-BPS, preserving 16 supercharges.
• Noncommutative membranes with a single nonzero commutator $[X^1, X^2] = i$ are 1/4-BPS, preserving 8 supercharges.
• Multiple noncommuting sectors ($[X^1, X^2]$, $[X^3, X^4]$, ..., nonzero) yield 1/8-, 1/16-, and 1/32-BPS states for 4, 6, and 8 noncommuting dimensions, respectively.
• The 10-dimensional noncommutative configuration is non-BPS.

These results are obtained via analysis of the projection conditions on the supersymmetry parameter $\epsilon$, induced by the structure of the triple commutators, e.g., $\epsilon = \Gamma_{012} \epsilon$ for the 1/4-BPS state.

5. Relation to D2-Brane Matrix Theory, Fundamental Identity, and Higher Branes

The supersymmetric Lorentz-covariant M2-brane matrix model naturally generalizes D2-brane (super-Yang-Mills) matrix theory by promoting the usual 2-bracket structure to a 3-bracket, regulated according to Lorentz-covariant and algebraic requirements. The construction preserves the Fundamental Identity (a generalization of the Jacobi identity to triple brackets), crucial for physical consistency and closure of the gauge symmetry.

The embedding in M-theory is thus made explicit: for D2-branes, matrix regularization leads to $U(N)$ gauge symmetry; for M2-branes, the regularized Nambu bracket yields a global symmetry algebra that encodes membrane dynamics in 11D. The approach provides a robust pathway toward Lorentz-covariant matrix models for higher branes such as M5-branes, where further generalization to 4-brackets and beyond would be required (Katagiri, 10 Aug 2025).

6. Algebraic and Geometric Structure

The RVPD algebra possesses a well-defined composition law: $$[\tau(Q_1, Q_2), \tau(H_1, H_2)] = \tau([\tau(Q_1, Q_2), H_1], H_2) + \tau(H_1, [\tau(Q_1, Q_2), H_2]),$$ reminiscent of infinite-dimensional Lie (or $w_\infty$-type) algebras. Matrix regularization then produces a noncommutative geometry, where the triple commutators reflect the geometric and supersymmetric content of fuzzy membranes or branes.

The presence of Lorentz-covariant Nambu brackets ensures spacetime symmetry is manifest, and the closure with $\tilde{\kappa}$-symmetry forms the basis for the BPS analysis and for the control of gauge invariances and redundancies.

7. Significance and Outlook

Supersymmetric Lorentz-covariant matrix models resolve longstanding problems in nonperturbative M-theory by marrying manifest spacetime symmetry, local and global supersymmetry, and consistent matrix regularization of the underlying worldvolume brackets and gauge symmetries. The framework supports classification of BPS solutions, provides a direct uplift of D2-brane (matrix) theory to M2-branes, and establishes a clear algebraic structure essential for the construction of higher brane models.

This approach overcomes the limitations of earlier light-cone (BFSS-type) models by systematically controlling the gauge symmetry and ensuring full Lorentz invariance. Extensions to models with additional symmetry breaking, noncommutative deformations, or further generalizations to M5-branes are natural directions made accessible by this methodology (Katagiri, 8 Apr 2025, Katagiri, 10 Aug 2025).