The number “two” may appear to be very special in theoretical physics, but maybe it has had it’s day…

By this we mean that much of physics is described in terms of “binary objects”: Lie brackets, commutators, metrics, rank two curvature tensors, quadratic Lagrangians, two dimensional world sheets of strings, Poisson structures, symplectic two forms, Laplacians and I am sure the list goes on. However, it has become increasingly clear in recent years that “higher objects” play an important role in theoretical physics as well as modern geometry.

For example, it has become increasingly clear that n-aray generalisations of Lie algebras play a role in physics. The Sh Lie algebras of Stasheff and the (not completely unrelated) n-Lie algebras of Filipov are great examples here. In one form or another, they can be found behind the BV-antifield formalism, Zwiebach’s closed string field theory, Kontsevich’s deformation quantisation of Poisson manifolds, Nambu’s generalised mechanics and the Bagger–Lambert–Gustavsson (BLG) description of multiple

stacked M2 branes.

The last one has been of interest to me lately.

So, M-theory was introduced by Witten in 1995 as a non-perturbative unification of the various superstring theories. Here, the fundamental objects are not strings but extended membranes of dimension 2 and 5, the so called M2 and M5 branes. Since then progress has been slow. No-one really knows what M-theory is and there is no proper understanding of the dynamics of interacting branes.

Then in Bagger & Lambert [2] in 2006 and independently Gustavsson [3] in 2007 construct and effective action for the low energy description of a stack of two M2 branes. The novel feature here is that 3-Lie algebras play a role here. A 3-Lie algebra should be thought of as a “Lie algebra” but with a tribracket not a bibracket. Details should not worry us.

The theory has the fields take their values in a 3-Lie algebra and their is a novel gauge symmetry. However, the original BLG-model can be recast as a conventional gauge theory, the ABJM theory [1]. So it starts to look that maybe 3-Lie algebras are some weird artificial artefact of M2 branes.

(There are many, many papers on the arXiv dealing with modifying the original BLG model. This usually can be understood in terms of 3-Lie algebra. I won’t say any more right now.)

But then very recently, Lambert & Papageorgakis [4] provided evidence that the effective description on M5 branes would also require 3-Lie algebras. However, they have not yet produced an action, which would be essential if the more or less standard methods of quantisation were to be applied.

This is fascinating. M-theory seems to be deeply tied to the theory of n-aray algebras, and in particular 3-aray algebras. There are may open questions here, both from a physics and mathematics point of view. In all it looks like n-aray algebra are here to stay.

——————-

**REFERENCES**

[1] Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, and Juan Maldacena. N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. *JHEP*, 10:091, 2008.

[2] Jonathan Bagger and Neil Lambert. Modeling multiple M2’s. *Phys. Rev*., D75:045020, 2007.

[3] Andreas Gustavsson. Algebraic structures on parallel M2-branes. *Nucl. Phys*., B811:66–76, 2009.

[4] Neil Lambert and Constantinos Papageorgakis. Nonabelian (2,0) Tensor Multiplets and 3-algebras. 2010,

arXiv:1007.2982 [hep-th].

I have followed this M2 “minirevolution” a bit to see, of course, if it connects to higher gauge theory, but I am still not quite sure what to make of it.

The trouble is that this so-called “3-algebra” of Bagger-Lambert does not admit any grading that would make it an L-infinity algebra, and so its homotopy-theoretic meaning, if any, is not clear. In the Lazaroiu-Saemann-Zejak-article

http://arxiv.org/abs/0901.3905

that you also cite in your oo-Lie-algebroid-article, the would-be L-oo algebras are considered without grading. While then the Bagger-Lambert structure is trivially an example, I have yet to see an argument why that disrespect for a grading leads to a useful notion. The fact that proper L-oo algebras are graded structures with brackets respecting the grading has a deep meaning that makes all the difference for the role of the structures in homotopy theory.

Given this, I find it a bit curious that it is the Bagger-Lambert structure that triggered such a large amount of attention on the possibility of higher algebraic structures in membrane theory. We know for well that for instance the gauge-coupling term of the membrane is the volume holonomy of a Chern-Simons 2-gerbe that is controled by genuine Lie 3-algebra data (wrote a series of article with Jim Stasheff and Hisham Sati on this Lie 3-algebraic aspect). There is a whole field of math now that describes the corresponding world-volume Chern-Simons theory using 3-categorical tools, etc. So in parts of the community the insight that “3 is the new 2” is an old hat. And more importantly: it is a well-setablished old hat, where the higher algebraic structure is really clear. Whereas for Bagger-Lambert 3-algebras it is not.

So while I enjoy seeing that string/M-theorists take renewed interest in the old observation that strings are controled by 2-categorical structures and membranes by 3-categorical structures, it puzzles me a bit why that excitement bases itself among all the possible well-established examples on the Bagger-Lambert-observation, whose proper higher algbraic interpretation, if it exists, seems to be as yet unknown.

Just a thought. Maybe you can give me another perspective on these matters.

Thanks Urs,

I too am not sure what to make of ungraded Loo-algebras.

Right now I have not looked enough into the BLG-like models and their related 3-algebras to say much more. A proper homotopy-theory understanding would be great.

Your higher gauge theory is something else I had intentions of reading up on. Too many interesting things out there are not enough time!

🙂