Lie 3-Algebras on the Membrane (?)

November 6, 2008

Posted by Urs Schreiber

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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 $L_\infty$ -algebroid).

Posted at November 6, 2008 10:50 PM UTC

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

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Urs wrote:

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.

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.

Posted by: John Baez on November 7, 2008 4:44 PM | Permalink | Reply to this

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

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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.

Yes, good point.

More generally, I see a couple of potential ways to take the structures they seem to be seeing and promote them to proper higher Lie algebraic structures.

I was about to post a comment listing these ideas, now I can do it in reply to your comment:

first of all I want to emphasize that even the action of the bosonic membrane already has a Lie 3-algebra governing it, namely the Chern-Simons Lie 3-algebra. The holonomy of differential form data with values in the Chern-Simons Lie 3-algebra is the Chern-Simons action functional. As you know, this is described in a bit of detail in our article, suggestively titled

$L_\infty$ -algebra connections and applications to String- and Chern-Simons parallel transport.

The bigger picture behind this is described here.

So the ordinary Chern-Simons term already is inherently Lie 3-algebraic. Just as the ordinary Wess-Zumino-Witten term is inherently Lie 2-algebraic – and for the same general reasons: $n$ -branes couple to $n$ -gerbes and more generally $(n+1)$ -bundles with connection whose local connection forms are Lie $(n+1)$ -algebra valued. As we know.

Here it is important that we are really talking about $L_\infty$ -algebras proper, i.e. with the right $\mathbb{N}$ -grading on everything, so that we can think of Lie algebras internal (either weakly or non-weakly) to $\infty$ -vector spaces and obtain $\infty$ -groups integrating all these Lie $n$ -algebraic structures eventually.

So for that reason it seems that any supersymmetric extension of the membrane action should involve an extension of the ordinary Chern-Simons Lie 3-algebraic structure.

As the quote I gave above indicates, it is with the present state of the art of the mini-revolutionists somewhat hard to tell which properties of their “3-algebras” are really essential to the program, and which are “superfluous”. But still, one can try to start guessing how the structure they see could be fixed to yield Lie $n$ -algebraic structure proper (with the right grading).

To start with, as we pointed out, one should think about what the lack of skew symmetry means. As you say, it can possibly have two origins: possibly some of the generators are in different degree, or possibly we should be talking about weak Lie $n$ -algebras with coherently weakened skew-symmetry, as in Dmitry Roytenberg’s work #.

And it could be both. It could be that the trinary bracket on the membrane is really to be thought of as the weakly-skew-symmetric Jacobiator.

To check, we need to get our hands dirty and play around with some examples. Here is maybe a good one:

open de Medeiros et al. at page 7 and see their example 6 there.

One notices: this example is just a slight variation on the theme of the string Lie 2-algebra.

For suppose their generator $u$ is secretly shifted up in degree by one. That would fix the grading of the trinary bracket they give on the left, and make this example essentially the String Lie 2-algebra with one extra generator $u$ floating around, but somewhat decoupled to the rest.

Of course the Chern-Simons Lie 3-algebra which gives the Chern-Simons term is induced by the String Lie 2-algebra, as we explain in $L_\infty$ -connections…. So that looks suggestive.

More generally, how might one go about making $L_\infty$ -algebraic sense (with correct gradings) of definition 1 on page 4?

One option might be: assume that the vector space $V$ on which this is based is really the result of forgetting that we started with two copies, $V \oplus V[1]$ .

If you then assume that the formulas given for the trinary bracket are secretly to be read as the trinary bracket evaluated on three elements of $V$ with the output being secretly an element in $V[1]$ , then suddenly the degrees work out right. And we are beginning to see structures familiar for Lie 2-algebras with non-trivial Jacobiator taking values in the “vector space of morphisms”.

One could play with this in various directions, but that needs somebody who knows both the super physics which is supposed to be modeled here and at the same time knows how to handle categorified Lie algebras.

I know a potential candidate. Today I had had very nice discussion with Calin Lazaroiu on this stuff. He says in a few weeks we may know more…

Posted by: Urs Schreiber on November 7, 2008 5:33 PM | Permalink | Reply to this

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

Dzhumadil’daev shows that every n-Lie algebra, i.e. an n-ary bracket that satisfies the generalized Nambu identity
also satisfies the L_\ifnty identity
but NOT vice versa. Thus the Bagger-lambert
or mini-revolutionists find their identity
too restrictive, perhaps we should adopt the Linfty-point of view

Posted by: jim stasheff on November 11, 2008 12:50 AM | Permalink | Reply to this

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

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Lie 3-algebras versus 3-Lie algebras versus…

There is an extensive literature on 3-algebras and 3-Lie algebras in the non-categorical sense, and also for $n \gt 3.$ . That is, a vector space $V$ with a trilinear operation satisfying certain identities. Cf. Nambu algebras. Unfortunately, I have not found a search engine that can handle the - in those phrases. In the Lie case, the identity is ‘a Jacobi-like’ condition. However, there are two prominent variants of that condition. One is that the adjoint $L$ of the $n$ -ary operation $B: V^{\otimes n} \to V$ given as $L: V^{\otimes n-1} \to End(V)$ takes values in $Der(V)$ , meaning $L(x_1,\dots,x_{n-1})B(y_1,\dots,y_n) = \sum B(\dots, L(x_1,\dots,x_{n-1}) y_i,\dots).$

The other prominent variant is the $L_\infty$ -condition with all operations except $B$ being 0.

An excellent survey is the little known

Alexandre Vinogradov and Michael Vinogradov, On multiple generalizations of Lie algebras and Poisson manifolds, Secondary calculus and cohomological physics, Moscow (1997), 273–287,

I haven’t checked Dzhumadil’daev’s proof that the first implies the second.

In realtion to the first, notice that in

Paul de Medeiros, José Figueroa-O’Farrill, Elena Méndez-Escobar, Patricia Ritter On the Lie-algebraic origin of metric 3-algebras

3-algebras are described in terms of a Lie algebra $g$ and a representation on $V$ . In fact, the Lie algebra is the image $L (V^{\otimes n-1} )$ above.

A similar structure in the $L_\infty$ -context appears with $V = V_0 \oplus V_{\pm 1}$ where one part (which depends on conventions) being the space of fields and the other, the space of ‘gauge parameters’.

R. Fulp, T. Lada, and J. Stasheff, Sh-Lie Algebras Induced by Gauge Transformations

Posted by: jim stasheff on November 7, 2008 6:39 PM | Permalink | Reply to this

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

With respect to Nambu brackets another thing that deserves to be mentioned is this:

in the old days when the supermembrane maztrix model was popular, it was given in Hamiltonian form essentially by taking a Poisson bracket on the 2-dimensional surface of the membrane and then “quantizing” that Poisson bracket to a matrix commutator.

Back then some people had tried to find a “covariant” matrix model, where the Poisson bracket on the spatial slice of the membrane is replaced by some trinary Nambu Bracket living on the entire 3-dimensional worldvolume of the membrane. See for instance

Towards covariant matrix theory.

The plan then was to find a “quantization” of this Nambu-Bracket in terms of some trinary bracket and build a “covariant supermembrane matrix mechanics” from that. Somehow.

I am not aware that this idea ever led to any concrete result, but the following may be noteworthy in the present context:

As shown in

Baez, Hoffnung, Rogers, Categorified Symplectic Geometry and the Classical String

there is indeed a notion of geometric quantization for the Nambu bracket, and it does yield a Lie 2-algebra, given by a nontrivial trinary bracket.

Potentially something to keep in mind when thinking about “3-algebras” on the membrane…

Posted by: Urs Schreiber on November 7, 2008 7:07 PM | Permalink | Reply to this

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

Urs wrote: given by a nonlinear trinary bracket.

Non-linear over what algebra?

Can anyone direct me to a source (including page if possible)for the relation(s) satisfied by the trilinear in Baggers et al
or their co-mini-revolutionists?

Posted by: jim stasheff on November 8, 2008 2:17 PM | Permalink | Reply to this

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

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Urs wrote: given by a nonlinear trinary bracket.

Ah, wait, that’s a typo of course. Everything in sight here is linear. I must have meant to say something like “nontrivial” when my fingers typed “nonlinear”. I have fixed the above comment.

Posted by: Urs Schreiber on November 8, 2008 9:01 PM | Permalink | Reply to this

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

Can anyone direct me to a source (including page if possible)for the relation(s) satisfied by the trilinear in Baggers et al or their co-mini-revolutionists?

The fundamental identity is listed e.g. on p 3 of 0712.3738. In the appendix they prove equivalence with Andreas Gustafsson’s bilinear brackets.

Posted by: Thomas Larsson on November 9, 2008 8:58 AM | Permalink | Reply to this

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

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Jim kindly points out to me the recent preprint

M. Rausch de Traubenberg, Some results on cubic and higher order extensions of the Poincaré algebra

which is concerned with another (apparently) slight variation on this theme:

the author is motivated by the structure of (ordinary) super Lie algebras, where an ordinary Lie algebra $g$ (the even part) acts on a vector space $S$ (the odd part) which is equipped with a symmettric bilinear form $S \otimes S \to g$ which is “ $g$ -equivariant” (the Jacobi identity on $g\otimes S\otimes S$ ).

In definition 3.1 on p. 14 the author generalizes this by allowing the bilinear $S \otimes S \to g$ to be replaced with higher order maps $S \otimes S \otimes \cdots \otimes S \to g$ .

Posted by: Urs Schreiber on November 12, 2008 5:34 PM | Permalink | Reply to this

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

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Check out this paper:

Tamar Friedman, Orbifold singularities, the LATKe, and Yang–Mills with Matter.

In particular, note the definition of ‘Lie algebra of the third kind’ or ‘LATKe’ starting on the bottom of page 7.

Posted by: John Baez on December 2, 2008 4:05 AM | Permalink | Reply to this

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

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Check out this paper:

Tamar Friedman, Orbifold singularities, the LATKe, and Yang-Mills with Matter.

Thanks!

And so we have yet another re-discovery of $n$ -Lie algebras.

Posted by: Urs Schreiber on December 2, 2008 8:26 AM | Permalink | Reply to this

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

seems to be a physics papwer and even then
not aware of the (so-called) `revolution’
we’ve discussed

who should clue her in?
though at this time of year, I do appreciate the LATKe reference

Posted by: jim stasheff on December 2, 2008 9:45 PM | Permalink | Reply to this

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November 6, 2008