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August 6, 2005

Noncritical M-Theory?

One of the interesting talks I didn’t talk about in my posts about Strings 2005 was by Petr Hořava. Now that the paper is out, I should try to make amends.

Hořava and Keeler propose a “noncritical M-theory” which is the double scaling limit of a 2+1 dimensional nonrelativistic (spinless) fermion in an inverted 2D harmonic oscillator potential. The idea is that certain vacua of this theory correspond to the type 0A and 0B noncritical string theories in 1+1 dimensions.

The 0A theory is obtained as follows. Work in polar coordinates, and expand the fermion in creation and annihilation operators {a q(ω),a q (ω)}=δ q,qδ(ωω) \{ a_q(\omega), a^\dagger_{q'}(\omega') \} = \delta_{q,q'} \delta(\omega-\omega') where qq\in\mathbb{Z} is the angular momentum in the plane. They propose that the OA vacuum with RR charge, qq, is obtained by taking the double-scaling limit N,ε F0,Nε F=μfixed N\to\infty,\qquad \varepsilon_F\to 0,\qquad N\varepsilon_F = \mu\, \text{fixed} and filling the Fermi sea for angular momentum qq

(1)
a q(ω)|0A=0, ω>μ a q (ω)|0A=0, ω<μ \array{ \arrayopts{\colalign{ right left}} a_q(\omega) |0A\rangle = 0,& \omega \gt -\mu\\ a_q^\dagger(\omega) |0A\rangle = 0,& \omega \lt -\mu\\ }

while leaving it empty for angular momentum qqq'\neq q. a q(ω)|0A=0,ω a_{q'}(\omega) |0A\rangle = 0, \forall\omega

The 0B vacuum is obtained by working in Cartesian coordinates, where the energies of the two (inverted) oscillators are separately conserved. Pick a particular value of ω 2\omega_2, and fill the Fermi sea for those oscillators with that value of ω 2\omega_2, while leaving the Fermi sea empty for other values of ω 2\omega_2.

Effectively, you reduce the 2D inverted harmonic oscillator to a 1D one, which is the 0B theory.

On the other hand, this 2+1 D Fermion theory has lots of other interesting vacua that one can contruct in a double scaling limit. Hořava and Keeler discuss some interesting examples.

In particular, the 2+1 dimensional “noncritical M-theory” vacuum arises from filling up the Fermi sea, as in (1), for all qq\in \mathbb{Z}.

What’s not clear to me, once one has decided to admit states of the 2+1 D theory with rather peculiar normalizability properties, such as these, where one should stop.

For instance, nothing prevents me from considering an n+1n+1 dimensional nonrelativistic spinless fermion in an nn dimensional inverted harmonic oscillator potential. In appropriate cylindrical or Cartesian coordinates, I would again recover the 1+1 dimensional 0A/0B string, by an obvious generalization of the above construction. But this theory would have yet-more “interesting” vacua.

Another paper, also featuring free fermions, just appeared. It was the subject of Vijay’s talk. I should try to say something about that one soon.

Update:

Luboš has his own, somewhat sceptical, comments on this paper. He’s rather bothered by the lack of direct connection with other string theory backgrounds. This doesn’t bother me, particularly. I’m more interested in whether there’s a mapping back to a consistent spacetime picture for these theories.
Posted by distler at August 6, 2005 1:59 AM

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Re: Noncritical M-Theory?

Jacques,

I find this paper very interesting. It reminds me of earlier ideas to use the 3D Ising model to describe strings. Does this comparison make sense?

Posted by: Wolfgang on August 9, 2005 10:01 AM | Permalink | Reply to this

Re: Noncritical M-Theory?

Offhand, I don’t see a connection as, in the 3D Ising case, you probably want a 2+1 D Lorentz (or 3D Euclidean) invariant background.

These backgrounds are not Lorentz-invariant. In the 1+1 D case, there’s a nontrivial dilaton gradient, and a nontrivial “tachyon” potential. In the other backgrounds, the spacetime interpretation isn’t always so clear, but I expect that, in all cases, it’s not 2+1 D Lorentz-invariant.

Still, an interesting thought …

Posted by: Jacques Distler on August 9, 2005 10:23 AM | Permalink | PGP Sig | Reply to this

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