## Wednesday, April 09, 2008

### Membrane minirevolution goes on

Bagger & Lambert and Gustavsson have initiated something that I will call the Membrane Minirevolution so far.

Yesterday, Lambert and Tong figured out where the missing 16th dimension of the moduli space goes and they study the global properties of the moduli space. They were led to believe that the theory describes membranes on a R8/Z2 orbifold and it is the low-energy limit of an SO(5) Yang-Mills theory, not O(4) theory.

On Wednesday, Distler, Mukhi, Papageorgakis, Van Raamsdonk solve a homework exercise that Jacques Distler planned to do on his blog. Consequently, they solve pretty much the same problem as Lambert and Tong. They conclude that the moduli space depends on the level k. It is the quotient of R8 x R8 by the dihedral group with 4k elements called D_{2k}. The high-k limit, before the cyclic subgroup of D_{2k} in the denominator approaches U(1), effectively compactifies M-theory, to generate D2-branes in type IIA string theory!

It is painful I didn't get this picture before them because I masterminded the relevant section of the paper [14] where the same limit applied to the conical compactification of string theory is used to deduce a quiver model for the (2,0) theory and little string theory. At any rate, their picture finally explains why D2-branes (and compactification) should emerge from a vev in the M2-brane theory (vev is a priori very different from a compactification!) - the main reason why I think that their paper is more correct than Lambert & Tong. Other things about lost U(1) in the moduli space etc. also look more sensible now. The large-k limit of Chern-Simons theory is the "classical theory" and if you look at the Jacques et al. paper, you see that in this limit, i.e. classically, one gets a U(1) in the denominator and the moduli space loses a dimension - that's what the people have naively seen.

Things start to make sense although the case of many M2-branes is still badly incomplete. Is there a hope in this direction?

Back on Tuesday, Morozov proposed to relax the condition of the complete antisymmetry of f_{abcd}, the mutated structure constants. It allows him to suggest a new form of the triple commutator, involving normal commutators and traces. Such a picture can be extended to many membranes but he hasn't checked the supersymmetry of his model yet so it may very well be gone.