Musings

Another View

Dang! I forgot to mention that Itzhaki and McGreevy have a radically different spacetime interpretation of the large-N model of fermions in a harmonic oscillator potential, discussed in my previous post.

Just as the model of N fermions in the inverted harmonic oscillator potential is dual to the 1+1-dimensional noncritical string theory (the 0B noncritical string theory, to be precise), the theory in an uninverted harmonic oscillator potential can also be interpreted in terms of a weird sort of noncritical string theory.

In their version, the string theory has an unbroken spatial translation-invariance (and the Hamiltonian of the quantum mechanics problem is the generator of these spatial translations). The dilaton gradient is in the time-direction which, moreover, is compact. The string coupling is a pure phase, $g_s\sim e^{2i\phi}$, but the genus expansion is really controlled by $1/\mu_0$, where $\mu_0$ is the coefficient of the Liouville term, $\mu_0 e^{2i\phi}$ in the worldsheet Lagrangian.

What, you ask, is the physics of world where every point in space lies on a closed timelike curve? Well, their universe has two ends, at $x=\pm\infty$. If you fix the boundary conditions at $x=-\infty$, you can ask for the probabilities for different things to appear on the other boundary, at $x=+\infty$.

At least, that’s their interpretation of the physics when the filled Fermi sea has the topology of a disk. I have no idea how they would interpret the configurations which involve disconnected droplets, concentric rings, or whatever, each of which have quite elegant interpretations in AdS/CFT.

Anyway, if you think that this interpretation of the large-N gauged harmonic oscillator is too contrived and baroque, you haven’t been hanging out here long enough.