  We study the vacuum structure of three-dimensional $SU(N)$ gauge theory
coupled to a Chern-Simons term at level $k$ and to $F$ fundamental Dirac
fermions. We use a large $N$ holographic description based on a D3/D7 system in
type IIB string theory compactified on a supersymmetry breaking circle. The
multiple vacua of the theory and the transitions between them are nicely
captured by the dual holographic background. The resulting phase diagram, which
we derive both at leading and first subleading orders in the $1/N$ expansion,
shows a rich structure where topological field theories, non-linear sigma
models and first-order phase transitions appear.
