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We consider the dynamics of a probe fermion charged under a U(1) Maxwellfield and a two form potential $B_{(2)}$ in a five dimensional gravitybackground. The gravity background is constructed from a new solution we findof type IIB supergravity. This new solution is expected to be dual tonon-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 inAdS$_2$ with an electric field. We show that the operator dimensions in theAdS$_2$ space are complex, leading to the two components of the retardedGreen's function in the dual theory to be complex conjugates of each other. Inorder to preserve unitarity, this result implies there are no zero frequencyquasinormal modes in our system. This has important implications forgeneralizations of recent holographic Fermi liquid setups with AdS$_2$ regions,as it suggests that infinite lifetime excitations can have energies above/belowthe chemical potential. Therefore, the Fermi energy may not be uniquely set bythe chemical potential. Furthermore, since the gravity background breaksrotational 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.

Non-Unitary Fermionic Quasinormal Modes at Zero Frequency