### Lie 3-Algebras on the Membrane (?)

#### Posted by Urs Schreiber

A conversation over conference # dinner I just had revolved around the observation that even though since the 1990s, due to work by Kontsevich, Zwiebach and others (see for instance Kajiura and Stasheff), it has been clear that string theory is fundamentally governed by $A_\infty$ and $L_\infty$-algebraic structures, this insight is not reflected in some of the work done in the community where one would expect it to be relevant.

From that point of view a recent phenomenon may be noteworthy, which those involved modestly address as the *membrane mini revolution* (a recent impression by Thomas Klose is here). It started with an article by Bagger and Lambert (see Jacques Distler’s useful review) in which the authors managed to construct an $N=2$ supersymmetic version of the worldvolume theory of the M-theory membrane. In their description they use a trinary skew (or partly skew) linear bracket. The authors addressed this bracket as a *3-algebra* [sic].

A *Lie* $n$-algebra is an $L_\infty$-algebra concentrated in the lowest $n$ degrees. An *$n$-Lie* algebra is a vector space with an $n$-ary skew bracket on it satisfying a Jacobi-like condition. Up to a potential issue of grading (see below), $n$-Lie algebras are special cases of $L_\infty$-algebras, as proven in

A. S. Dzhumadil’daev
*Wronskians as $n$-Lie multiplications*

(arXiv).

I am grateful to Calin Lazaroiu for this and the following reference.

The published evidence for the relevance of the homtopy-theoretic interpretation of the trinary Lambert-Bagger bracket remains somwehwat inconclusive (see Jacques Distler’s useful second review). In

de Medeiros, Figueroa-O’Farill, Méndez-Escobar, Ritter
*On the Lie-algebraic origin of metric 3-algebras*

(arXiv)

it says on p. 3 about this:

All this prompts one to question whether the 3-algebras appearing in the constructions [1-3,10,11] play a fundamental role in M-theory or, at least insofar as the effective field theory is concerned, are largely superfluous.

The authors then go on to discuss all these trinary brackets entirely in terms of pairs consisting of an ordinary Lie algebra and a representation.

Apart from usefulness issues of the 3-algebraic perspective, it is noteworthy that the Bagger-Lambert trinary bracket is in general not consistent with $L_\infty$-algebra grading conventions (as for instance in Lada-Stasheff p. 7), for no grading one puts on the underlying vector space $V$ – at least not unless one assumes that there are secretly two differently graded copies of $V$ in the game. One can consider the definition of $L_\infty$-algebras without the grading, in particular if there is just a single arity of brackets involved, as in the above articles. In this ungraded sense then the Bagger-Lambert trinary bracket, at least for the case that it is totally skew, really is an example of a Lie 3-algebra.
On the other hand, it makes me wonder that this trinary bracket is in general taken to be skew only in the first two arguments, as described down on p. 2 of de Medeiros et al. One could potentially accomodate for this by an $L_\infty$-algebra proper (with grading, that is) by having two copies of the underlying vector space, one of them shifted in degree down by one (which would mean we’d end up with an $L_\infty$-algeb*roid*).

## Re: Lie 3-Algebras on the Membrane (?)

Urs wrote:

I don’t understand this stuff at all and don’t pretend to. So, I wonder if maybe they have a Lie 3-algebra that’s not ‘semistrict’ — i.e., where skew-symmetry holds only up to chain homotopy.