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The vacuum is gauge invariant (a singlet of U(N)). Any state in the dynamics can
be accessed by raising operators acting on the vacuum. However, we need to impose the
Gauss’ law constraint. A fermion ψ
a
b
has an upper and lower U(N) index. These need to
be fully contracted to form a singlet. These contractions are in the form of traces. For
example, a single trace state acting on the vacuum is given by
|ki = tr(ψ
k
) |0i (2.2)
Using the cyclic property of the trace and the fermionic character of ψ it can be easily
shown that
tr(ψ
k
) = (−1)
k−1
tr(ψ
k
) (2.3)
so that only traces with an odd number of fields ψ are allowed. Each of these traces has
fermi statistics and has energy k = 2s + 1, where s = 0, 1, . . . . At infinite N each trace is
supposed to correspond to a different ‘particle’.
If we normalize the energies in units of 1/2, we get a single particle state for each half
integer (s + 1/2). This is the same spectrum of states as a free chiral fermion in 1 + 1
dimensions (a c = 1/2 CFT) on a circle with NS boundary conditions [13] (antiperiodic
boundary conditions χ(θ + 2π) = −χ(θ)).
This suggests that this is a matrix model for a c = 1/2 CFT in a similar way that
a bosonic gauged matrix model gives rise to a c = 1 chiral boson in 1 + 1 dimensions.
The main insight of this map to the chiral boson is that in the bosonic matrix model one
can reduce the dynamics to the eigenvalues of the matrix X. When the Gauss’ law is
implemented the eigenvalues act as fermions. This dynamics for bosons is usually best
described in terms of a first order dynamics
S =
Z
dt tr
¯
ZiD
t
Z −m
¯
ZZ
(2.4)
where Z = X + iP and
¯
Z are complex and the Gauss’ law constraint requires them to
commute. The corresponding fermions are in the phase space of X and the ground state
can be described by a droplet in phase space. This is a familiar story of the quantum hall
effect. The eigenvalue dynamics is the effective field theory of free electrons in the lowest
Landau level in 2 + 1 dimensions, which has been slightly deformed by a binding potential
proportional to x
2
+ y
2
. The traces tr(Z
k
) are collective excitations of the droplet with
angular momentum k on the edge.
Unlike the bosonic matrix model, in the fermionic setup we can not choose a gauge
where we diagonalize the fermionic degrees of freedom. The interpretation of an edge
dynamics is harder to do and will not be pursued here.
Because the bosonic system can also be interpreted in terms of first quantized fermions
in a bulk 2+1 system, we can also write the wave functions in terms of Slater determinants
of single particle states. These wave functions are governed by Schur polynomials of the
matrices. These are obtained by traces in irreducible representations of U(N), which are
labeled by Young diagrams. The map from multi-traces to Schur functions in the bosonic
model is non-trivial [14]. It is also obtained from character expansions of the symmetric
– 3 –