Tuesday, February 10, 2009

Covariant M(atrix) Theory

Matsuo Sato (PDF) proposes a new approach to the covariant M(atrix) theory.

Tsukuba, the home to the 3/4 of the IKKT model.

As the author's name indicates, Sato loves the Japanese (IKKT) matrix model for type IIB string theory. My opinion whether this matrix model is correct and nontrivially incorporates all physical observables of type IIB string theory has changed many times throughout those 12 years. It's been one of the most fluctuating yet potentially "answerable" questions about M(atrix) theory.

I haven't heard any convincing, complete answers to this question and my current opinion about it is fuzzy.

While the timeless IKKT matrix model formally comes from the action of many D-instantons, and its bosonic part involves a quartic expression in "X", namely a squared commutator, Matsuo Sato proposes a similar timeless matrix model whose bosonic part is a sixth-order expression in "X", a sum of squared triple commutators.

This should be seen as an imported idea from the recent membrane minirevolution that has dealt with 3-algebras. While the squared triple commutator defines the bosonic part of the action, he guesses that a Psi.[X,X,Psi] term completes the fermionic terms. He realizes that it's important to check SUSY but he hasn't done so.

The U(N) 3-algebra (defined by the triple commutators) is obtained from the normal U(N) Lie Algebra by adding 1+1 extra generators, so that the signature is Lorentzian. A simple "vev" should lead to the generation of the BFSS matrix model, including other backgrounds such as matrix string theory, he argues.

I will have to check whether it makes sense because similar methods to make the BFSS matrix model "timeless" have been tried in the past and none of them looked manifestly Lorentz-covariant, if I remember well. If it works well, it could be really great, even though my hopes that a similar model could provide us with a "manifestly background-independent" formulation of all of M-theory - to use a popular adjective - is limited by a long history of similar attempts that have failed in the past.


After some thinking, I am surprised that Sato didn't analyze the "d=3" case, in his formalism, which should generate the timeful BFSS matrix model for type IIB string theory. In the decompactification limit, he should see the superconformal 2+1-dimensional field theory - which should be the same theory as he has, one based on triple commutators, but with 2+1 spatial dimensions.

Incidentally, this BLG superconformal theory in 2+1 dimensions can be simply dimensionally reduced to 0+0 dimensions to get a supersymmetric model, so there's no doubt that it can be made supersymmetric.

The "d=3" (BFSS for type IIB) and "d=0" (IKKT for type IIB) should be related by an equivalence that should become manifest in Sato's model if it works. He seems to be extremely sloppy about the periodicity of various variables and most other details that are necessary to check the physics of individual vacua.

The price of Lorentzian 3-algebras

In the fast comments, JB has a very relevant comment with two appropriate references, claiming that the Lorentzian 3-algebras don't really bring us any new nontrivial physics (unlike the algebras in ABJM models) - they're either equivalent to free, or old super Yang-Mills theories. Well, the papers seem OK but they have only been done in 2+1 dimensions.

In 0+0 dimensions, the work hasn't been done and the answer may be different. Also, there might be more subtle operations to be done with the Sato model than just Higgsing, and those could generate more interesting models. Also, I need to stress that in M(atrix) theory, we don't really want to generate "entirely new theories".

In the light-cone gauge, the vacua are described by ordinary Yang-Mills theories and it is questionable whether we want something entirely new, besides a more compact "unifying umbrella" above all these theories. If you could derive all the old theories from such a new umbrella, it's likely that you could also treat the umbrella differently a derive new theories (ways to describe additional vacua previously unaccessible to matrix modeling).


  1. There is a new article in which it is claimed that it was independently developed of Matsuo Sato. It seems to be much more general and includes the supersymmetric case.


    What do you think about it?

  2. Dear Daniel, I wanted to write about the Lee-Park paper today but it was too similar to the topic of the Covariant Matrix Theory which appeared here, very recently, so I decided to skip it on the blog.

    I am not quite understanding "what" was independently developed by Sato. The IKKT model? I am not aware of such a dual authorship. If you mean Sato's paper, indeed, it was developed by Sato.

  3. I'm sorry, "claiming" and "Independtly deeveloped" are exagerations. I was just refering to the note added on page 5, because the ideas used in the two papers are very similar.

    Actualy, I was thinking more about in updating this post, since this might be interesting to that other guy, and other people, who are very skeptical about CMT.

  4. Sorry, Daniel, but I have no idea what you're talking about and I don't have enough energy and desire to decode it now.

    Who is claiming that? What is he claiming? What was developed by Sato? Page 5 of which paper? Who is skeptical about CMT (besides me and most other people)? Why is it important or relevant? It's just too many questions.

  5. Eh. I said nothing at all with much relevant content. I really meant that Lee and Park, in the end of section 1.2 of their paper, added a note telling that Sato's CMS paper has a similar approach to theirs. I was also surprised that both got to similar conclusions at the same time.

    And when I wrote about the skepticals, I refered to your observations in the section "Update": "The "d=3" (BFSS for type IIB) and "d=0" (IKKT for type IIB) should be related by an equivalence that should become manifest in Sato's model if it works."

    I guess that's partialy addressed, isn't it? In the conclusion, he says that 2.1 is classicaly equivalent to BFSS, and he also get IKKT for IIB at 4.12.