  We present a 2+1 dimensional quantum gauge theory with correlated fermions
that is exactly solvable by bosonization. This model describes a system of
Luttinger liquids propagating on two sets of equidistant lines forming a grid
embedded in two dimensional continuum space; this system has two dimensional
character due to density-density interactions and due to a coupling to
dynamical photons propagating in the continuous embedding space. We argue that
this model gives an effective description of partially gapped fermions on a
square lattice that have density-density interactions and are coupled to
photons. Our results include the following: after non-trivial renormalizations
of the coupling parameters, the model remains well-defined in the quantum field
theory limit as the grid of lines becomes a continuum; the photons in this
model are massive due to gauge-invariant normal-ordering, similarly as in the
Schwinger model; the exact excitation spectrum of the model has two gapped and
one gapless mode.
