∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
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).
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:
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)
Last revised on March 8, 2021 at 06:13:08. See the history of this page for a list of all contributions to it.