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Quantum Stability at One Loop for BPS Membranes in a Lorentz-Covariant RVPD Matrix Model

So Katagiri

TL;DR

This work proves the first rigorous one-loop quantum stability for BPS membranes in a Lorentz-covariant M2-brane matrix model built with Restricted Volume-Preserving Deformations (RVPD). By combining RVPD with a restricted $\kappa$-symmetry, the BRST algebra closes without an infinite ghost tower, enabling a clean BRST gauge-fixing and a finite one-loop determinant structure. For BPS backgrounds realizing noncommutative planes up to eight dimensions, bosonic and fermionic fluctuations cancel mode-by-mode, while the residual ghost determinant remains positive and finite; in contrast, the ten-dimensional membrane is non-BPS and tachyonic at one loop. These results establish RVPD as a viable covariant regularization route for membranes and suggest a roadmap toward M5-brane matrix models and connections to BFSS and BLG/ABJM frameworks.

Abstract

We establish the first rigorous one-loop proof of quantum stability for BPS membranes in the Lorentz-covariant M2-brane matrix model with Restricted Volume-Preserving Deformations (RVPD). Exploiting the closure of restricted $κ$-symmetry with RVPD ensures that the BRST complex terminates without an infinite ghost tower, keeping the gauge-fixed measure analytically controllable. Our main result establishes that the 2D, 4D, 6D, and 8D noncommutative membranes remain stable, while the 10D configuration inevitably develops a tachyonic mode. Our analysis unifies the treatment of zero-modes, connects the effective action to central charges, and clarifies relations to BFSS, BLG/ABJM, and prospective M5-brane matrix models, providing a roadmap for RVPD-based extensions.

Quantum Stability at One Loop for BPS Membranes in a Lorentz-Covariant RVPD Matrix Model

TL;DR

This work proves the first rigorous one-loop quantum stability for BPS membranes in a Lorentz-covariant M2-brane matrix model built with Restricted Volume-Preserving Deformations (RVPD). By combining RVPD with a restricted -symmetry, the BRST algebra closes without an infinite ghost tower, enabling a clean BRST gauge-fixing and a finite one-loop determinant structure. For BPS backgrounds realizing noncommutative planes up to eight dimensions, bosonic and fermionic fluctuations cancel mode-by-mode, while the residual ghost determinant remains positive and finite; in contrast, the ten-dimensional membrane is non-BPS and tachyonic at one loop. These results establish RVPD as a viable covariant regularization route for membranes and suggest a roadmap toward M5-brane matrix models and connections to BFSS and BLG/ABJM frameworks.

Abstract

We establish the first rigorous one-loop proof of quantum stability for BPS membranes in the Lorentz-covariant M2-brane matrix model with Restricted Volume-Preserving Deformations (RVPD). Exploiting the closure of restricted -symmetry with RVPD ensures that the BRST complex terminates without an infinite ghost tower, keeping the gauge-fixed measure analytically controllable. Our main result establishes that the 2D, 4D, 6D, and 8D noncommutative membranes remain stable, while the 10D configuration inevitably develops a tachyonic mode. Our analysis unifies the treatment of zero-modes, connects the effective action to central charges, and clarifies relations to BFSS, BLG/ABJM, and prospective M5-brane matrix models, providing a roadmap for RVPD-based extensions.

Paper Structure

This paper contains 48 sections, 1 theorem, 227 equations.

Table of Contents

  1. Introduction
  2. Lorentz-Covariant M2-Brane Matrix Model with RVPD Gauge Symmetry
  3. Notation and Conventions
  4. Supermembrane Action
  5. Reformulation Using the Nambu Bracket
  6. Restricted VPD (RVPD)
  7. Restricted $\kappa$-Symmetry
  8. Matrix-Regularized Supermembrane Action
  9. Classical BPS Solutions
  10. Counting of Degrees of Freedom
  11. BRST Ghosts
  12. Introduction of BRST Ghosts
  13. Nilpotency of the BRST Transformations
  14. Perturbative Expansion
  15. Expansion of the Matrix Model
  16. ...and 33 more sections

Key Result

Lemma 1

(Faddeev--Popov measure for RVPD). Upon separating the RVPD zero modes, the Jacobian associated with the Gaussian change of variables factorizes as

Theorems & Definitions (1)

  • Lemma 1