A Massive Study of M2-brane Proposals

Abstract: We test the proposals for the worldvolume theory of M2-branes by studying its maximally supersymmetric mass-deformation. We check the simplest prediction for the mass-deformed theory on N M2-branes: that there should be a set of discrete vacua in one-to-one correspondence with partitions on N. For the mass-deformed Lorentzian three-algebra theory, we find only a single classical vacuum, casting doubt on its M2-brane interpretation. For the mass-deformed ABJM theory, we do find a discrete set of solutions, but these are more numerous than predicted. We discuss possible resolutions of this puzzling discrepancy. We argue that the classical vacuum solutions of the mass-deformed ABJM theory display properties of fuzzy three-spheres, as expected from their gravitational dual interpretation.

• The paper demonstrates that the ABJM theory yields discrete vacuum solutions consistent with fuzzy geometry predictions, despite an unexpected surplus in vacua.
• The mass-deformed BF membrane model exhibits only one classical vacuum, questioning its efficacy in capturing M2-brane physics.
• The study highlights the need for further exploration, including quantum effects and alternative deformations, to bridge the classical-quantum gap in M2-brane models.

Analytical Overview of the Study on M2-brane Proposals

This paper provides a critical examination of two primary proposals for the worldvolume theory of M2-branes in M-theory: the ABJM theory and the BF membrane model based on a Lorentzian three-algebra. It employs a strategy of studying their respective mass-deformed versions to probe their validity and characteristic features.

In investigating the ABJM proposal, which utilizes an $N=6$ $U(N) \times U(N)$ Chern-Simons gauge theory, the study found a set of discrete vacuum solutions that aligned with theoretical expectations concerning fuzzy geometries—a point reinforced by their gravitational dual interpretations. However, a numerical discrepancy was observed wherein the vacua exceeded the expected count based on partitions of $N$. This surplus suggests either an inherent prediction error in the classical vacuum structure or a requirement for refinements in the theoretical model.

For the mass-deformed BF membrane model, the results were less promising. The study revealed only a singular classical vacuum state, failing to match M2-brane expectations. This outcome casts doubt on the Lorentzian three-algebra model's veracity as an accurate depiction of M2-branes. The paper speculates that quantum effects might resolve this mismatch, although such speculation remains unsubstantiated.

The paper implies profound implications for our understanding of M2-branes and their theoretical frameworks. The findings suggest that while the ABJM model shows certain promising characteristics, particularly in regards to reproducing fuzzy-three sphere properties, it requires further exploration to rectify the vacuum count discrepancy. On the other hand, the BF membrane model may need substantial revisions or evasive conceptual shifts to align with quantum M2-brane physics.

Speculatively, future inquiries could explore reconciling the classical-quantum gap in these theories, potentially leveraging lattice simulations for more detailed insights. Moreover, the consideration of other possible deformations within the ABJM or alternative formulations of the membrane theory, which preserve crucial symmetries like $SO(8)$, may progress the ongoing quest for a comprehensive model of M2-branes.

In conclusion, this study stands as a meticulous effort in the validation and refinement of theoretical M2-brane models− a necessary step in progressing M-theory and our understanding of brane dynamics within the framework of string theory. By addressing the limitations of existing proposals, it sets the stage for further theoretical advancements and computational developments in this domain.