Musings

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 $$\{ a_q(\omega), a^\dagger_{q'}(\omega') \} = \delta_{q,q'} \delta(\omega-\omega')$$ where $q\in\mathbb{Z}$ is the angular momentum in the plane. They propose that the OA vacuum with RR charge, $q$, is obtained by taking the double-scaling limit $$N\to\infty,\qquad \varepsilon_F\to 0,\qquad N\varepsilon_F = \mu\, \text{fixed}$$ and filling the Fermi sea for angular momentum $q$

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

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 $\omega_2$, and fill the Fermi sea for those oscillators with that value of $\omega_2$, while leaving the Fermi sea empty for other values of $\omega_2$.

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 $q\in \mathbb{Z}$.

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 $n+1$ dimensional nonrelativistic spinless fermion in an $n$ 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.