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D2-branes from M2-branes, three-algebras, and associators

Today, I certainly recommend you the hep-th paper by

Sunil Mukhi, Constantinos Papageorgakis
who investigate a new type of a Higgs mechanism in a new type of theory relevant for M-theory membranes. A condensate of a 3-algebra-valued eighth scalar is claimed to convert a topological field theory from M2-branes to the conventional Yang-Mills theory defined on D2-branes.

The authors wisely posted the paper as the first paper on the hep-th archive. They did the right thing because the paper is arguably the most important one.


Because I haven't yet written about these cool things, let me say a few words. The main resource I recommend you about this new theory is a paper by
Jonathan Bagger and Neil Lambert
that defines a very promising candidate for a theory describing N M2-branes because it has the required supersymmetries, conformal symmetry, SO(8) R-symmetry, and - according to the Mukhi & Papageorgakis paper - also the correct Yang-Mills correct limit after the Higgsing. That looks like a really non-trivial body of evidence for such an unusual theory.

For the sake of order, I won't link to the papers by Basu and Harvey (2004), Bagger and Lambert (2006), and Gustavsson (2007) who, especially in the latter case, also deserve credit. Instead, I will only discuss the story as presented by Bagger and Lambert (2007) mentioned above. David Berman has been playing with similar things. So was I. And Shiraz Minwalla was very helpful for the Mukhi & Papageorgakis new paper.

Mysterious triple structures of M-theory

Conventional physics uses quadratic Lagrangians, two-dimensional worldsheets, second-rank tensors under Yang-Mills groups, commutators between two objects, and similar structures based on the number "2" all the time. We know them quite well.

Still, it looks likely that there exists a whole realm of wisdom that remains mostly hidden in a cloud of mystery. Even though a great deal of the physics is known, we don't know of any simple covariant descriptions of M-theory in 11 dimensions, multiple M2-branes, and multiple M5-branes. We know how to study many physical phenomena in their context but our degree of understanding simply doesn't seem to match the Yang-Mills, worldsheet, free theories discussed in the previous paragraph.

There exist hints that these largely unknown structures might be based on the number "3" in a similar way as the known theories are based on the number "2". This comment looks extremely vague but there are many reasons to see this prophesy. Exceptional groups frequently appearing in M-theory have cubic invariants. Membrane worldvolumes have three, not two dimensions. The number of degrees of freedom of an M5-brane seems to scale with the third, not second power of N. And all these insights could be relevant for the third superstring revolution just like D-branes and Yang-Mills theories were for the second. ;-)

We understand the low-energy limit of a single M2-brane and a single M5-brane. In the former case, the theory has 8 transverse dimensions (2+1 + 8 = 11 as in M-theory). One of the dimensions can be electromagnetically dualized to a gauge field in 2+1 dimensions, obtainining a gauge theory in 2+1 dimensions with 7 additional transverse scalars, a description of D2-branes in type IIA string theory. That's how the Yang-Mills terms for the D2-brane gauge field is generating from the eleventh dimension of M-theory that gets compactified.

The non-Abelian case generalizing the simple construction above to the case of multiple M2-branes or D2-branes is not understood. Or at least, it wasn't understood until recently, until the end of 2007. ;-)

You might think of many ways how the number "2" in the well-known theories should be replaced by "3". Certain people keep on constructing 2-groups, gerbes, and similar superconstructions that never work at all. The Bagger-Lambert-Gustavsson construction is different because it actually seems to have all the required symmetries and the correct Yang-Mills limit after the Higgsing!

So how does the Bagger-Lambert-Gustavsson construction work? It will sound as a sort of kindergarten game but believe me, I am serious. ;-)

Well, the commutator [A,B] uses two letters. It vanishes if A,B commute with one another. The generalization to three objects must clearly be [A,B,C]. It vanishes if the associativity holds, so choose
< A,B,C > = (A.B).C - A.(B.C)
That's the associator, extending the commutator. ;-) For certain reasons, it is more useful to use the completely antisymmetrized associator
[A,B,C] = < A,B,C > +- 5 other terms
A trace form (behaving "democratically" with respect to three objects in a product) and a Hermitean conjugation must exist. The trace of A and the antisymmetrized associator [B,C,D] gives you an antisymmetrized object with four indices. Now you can write down a lot of mutated versions of well-known equations with an additional index, including the mutated Jacobi identity.

Mutated Lie rules and actions

The parameter Lambda of gauge transformation in this three-algebra realm have two "adjoint" indices instead of one. The mutated gauge fields and mutated gauge transformations look pretty much identical as in the Yang-Mills case if you write them in terms of two indices attached to every gauge field. The three-algebras are "more powerful" than the normal Lie algebras but one can construct a Lie algebra "imprint" of every three-algebras - a potential Lie algebras that can occur in D-brane Yang-Mills limits of it.

An expert could think that all of these games are childish and probably won't lead to sensible theories. The first shock occurs on page 7 of Bagger & Lambert where a supersymmetric action based on these things is found. It literally looks like the three-generalizations of the usual formulae. For example, the supersymmetry variation of a fermion has the normal term proportional to "X" but also an additional non-linear term proportional to the associator [X_I, X_J, X_K]. Wow. The 4-index mutated structure constant tensors appear elsewhere.

The final Lagrangian looks as expected. There is no kinetic term for the gauge field, normal kinetic terms for the scalars and fermions, a mutated Yukawa term with Psi.X.X.Psi replacing the usual Yukawa Psi.X.Psi term, and a Chern-Simons action for the gauge field. The final Lagrangian has no free parameters at all, assuming that certain quantization rules constrain the mutated structure constants f^{bcd}_a.

Now, what are these bizarre three-algebras? Do they exist at all? In fact, there is at least one non-trivial one, with four "colors" and f_{abcd} being proportional to the completely antisymmetrized epsilon_{abcd}. However, Bandres, Lipstein, Schwartz argue that it is difficult - at least for them - to generalize the four-color SO(4) into a more general case. Not even otherwise natural structure constants of the octonions seem to satisfy the required mutated Jacobi identity.

Higgsing and applications to M5-branes and M-theory

After you absorb all these novel three-algebra insights, you should read the new paper by Mukhi and Papageorgakis, fully deriving the D2-brane supersymmetric action from a mutated Higgsing of the M2-brane three-algebra action. That's pretty fascinating unless there is a hidden catch somewhere. The four-color Bagger-Lambert SO(4) algebra is eventually broken to the diagonal SU(2) inside SU(2) x SU(2), ending up with the U(2) supersymmetric Yang-Mills theory including the center-of-mass degrees of freedom.

Mukhi and Papageorgakis had to correct a confusion in the literature - some people expected, by counting the moduli, that the SO(2) 3-algebra theory should have led to the SU(3) theory for three M2-branes. Shiraz Minwalla importantly told them that this couldn't be the case because such a reduction would imply that there exists no non-trivial theory for two M2-branes.

I haven't known the supersymmetric action but I have already worked on a similar setup a year or two years ago and I think that I slightly know how an analysis of M5-branes using these M2-brane degrees of freedom can be made, including the mutated 3-version of the 't Hooft limit. Tomorrow, Berman, Tadrowski, Thompson will study open membranes and fivebranes in this framework. One day later, Mark van Raamsdonk will frustratingly argue that the moduli space of the BL theory has an unwanted O(2) quotient (missing dimension) and superconformal primaries can't be easily constructed.

So if some readers don't know what topics are hot right now, let me declare three-algebras of M2-branes to be the hottest topic right now. ;-)

And that's the memo.

Update: Newer articles about the research direction contain the phrase "membrane minirevolution" (click).

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