  We consider the dynamics of a probe fermion charged under a U(1) Maxwell
field and a two form potential $B_{(2)}$ in a five dimensional gravity
background. The gravity background is constructed from a new solution we find
of type IIB supergravity. This new solution is expected to be dual to
non-commutative Yang-Mills theory in the 't Hooft limit with global U(1)
currents. We study the zero frequency, near horizon behavior of the fermion,
where the equations of motion reduce to that of two interacting fermions in
AdS$_2$ with an electric field. We show that the operator dimensions in the
AdS$_2$ space are complex, leading to the two components of the retarded
Green's function in the dual theory to be complex conjugates of each other. In
order to preserve unitarity, this result implies there are no zero frequency
quasinormal modes in our system. This has important implications for
generalizations of recent holographic Fermi liquid setups with AdS$_2$ regions,
as it suggests that infinite lifetime excitations can have energies above/below
the chemical potential. Therefore, the Fermi energy may not be uniquely set by
the chemical potential. Furthermore, since the gravity background breaks
rotational symmetry along the spatial directions of the dual Yang-Mills theory,
we do not expect the Fermi surface to be spherical in shape in momentum space.
