- the free matrix model for a single hermitean N x N matrix with a harmonic oscillator potential
- the free fermions, and excitations of a Fermi droplet which are chiral bosons
- a crazy dual string theory
Let's start with the matrix model. It is a quantum mechanical model with the Lagrangian Tr(D_t(X)^2 - X^2), roughly speaking. Because its U(N) symmetry is gauged, we require the physical states to be U(N) invariant.
Nevertheless, it is a free theory that we can easily solve. There are N^2 harmonic oscillators, they have some ground state, and they can be excited by the matrix-valued creation operators. In order to get gauge invariant excited states, we must combine the creation (and perhaps annihilation) operators into traces. It's a simple combinatorial task to see what happens. Most of the traces are linearly dependent, at least if they ask on the gauge-invariant Hilbert space.
An alternative solution is provided by free fermions. Because the wavefunction is U(N) invariant, it depends on the eigenvalues only, and using the Vandermonde determinant redefinition of the wavefunction, the system may be converted to free fermions in a certain phase space; the coordinates in the phase space are the eigenvalues of "x" and its canonical momentum "p". The fermions tend to fill the interior of a disk in the phase space and form a Fermi droplet.
One can study the perturbation of the Fermi surface (a wiggly disk), and these perturbations rotate in the clockwise direction - it's the usual rotation in the phase space of a harmonic oscillator. Therefore, they are some sort of chiral bosons, and these chiral bosons may be related to the creation operators above, and some of these relations seem to be relevant for a less trivial case of M(atrix) theory that I'm now trying to solve in a different way.
Of course, the most nontrivial side of this "triality" is a string theory, and I am still not following what the string theory exactly is. At any rate, it has an imaginary time-like dilaton gradient in the periodic time dimension (it's something that I believe is relevant for one of my long-term projects related to N=2 and N=(2,1) strings on del Pezzo surfaces as a theory of everything, and a similarity like that is encouraging). That string theory, conjectured to be dual to the simple matrix harmonic oscillator above, has some strange behavior of the amplitudes - for example, they truncate at some genus - and because I don't really understand it too well (and what it's good for and in what sense it's unique), we will have to refer to the original paper, if you're interested.