nLab
membrane matrix model

Context

String theory

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

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).

References

Revised on September 27, 2016 15:13:26 by Urs Schreiber (89.204.138.165)