nLab membrane matrix model

Idea

Applying a certain regularization prescription to the action functional of the superstring sigma-model it turns into a theory whose fields are Lie algebra elements – matrices for the case of matrix Lie algebras – known as the IKKT matrix model.

A similar regularization to the action functional of the membrane (the M2-brane) would have to involve objects replacing matrices which instead of a binary Lie bracket $[-,-]$ have a trinary Nambu-like bracket $[-,-,-]$. (Park-Sochichiu 08, Sato09, DeBellis-Saemann-Szabo 11).

Given the relation of the IKKT matrix model to type IIB string theory it is naturally and often expected that making sense of this might give a way to understand M-theory (whence the title for instance of (Sato 09)).

For this to work a clear understanding of the nature of these trinary brackets is required. After the success of the BLG model for the M2-brane several authors proposed that these are to be thought of as Flippov “3-Lie algebras” (Park-Sochichiu 08, Sato09, DeBellis-Saemann-Szabo 11). However, by the discussion there, following (MFFMER 08) it seems natural to think of “3-Lie algebras” as really being Lie 2-algebras equipped with a binary invariant polynomial.

Accordingly, a membrane matrix model should be a generalization of the IKKT matrix model from Lie algebras to Lie 2-algebras. This proposal is explored in (Ritter-Saemann 13).

Related concepts

• M2-brane, M-theory

matrix models for brane dynamics:

D-brane	matrix model
D0-brane	BFSS matrix model, BMN matrix model
D(-1)-brane	IKKT matrix model
D4-brane	nuclear matrix model

M-brane	matrix model
D2-brane	membrane matrix model

See also:

• K-matrix model

References

• Matsuo Sato, Covariant formulation of M-theory I, Int. J. Mod. Phys. A 24 (2009) 5019 (arXiv:0902.1333)

• J.-H. Park, C. Sochichiu, Taking the square root of Nambu-Goto action and obtaining Filippov-Lie algebra gauge theory action, Eur. Phys. J. C 64 (2009) 161 (arXiv:0806.0335)

• J. DeBellis, Christian Saemann, Richard Szabo, Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model, JHEP 1104 (2011) 075

• Patricia Ritter, Christian Saemann, Lie 2-algebra models (arXiv:1308.4892)