  Robustness of the Luttinger theorem for fermionic liquids is examined in
holography. The statement of the Luttinger theorem, the equality between the
fermion charge density and the volume enclosed by the Fermi surface, can be
mapped to a Gauss's law in the gravity dual, a la Sachdev. We show that various
deformations in the gravity dual, such as inclusion of magnetic fields, a
parity-violating theta-term, dilatonic deformations, and higher-derivative
corrections, do not violate the holographic derivation of the Luttinger
theorem, as long as the theory is in a confining phase. Therefore a robustness
of the theorem is found for strongly correlated fermions coupled with strongly
coupled sectors which admit gravity duals. On the other hand, in the deconfined
phase, we also show that the deficit appearing in the Luttinger theorem is
again universal. It measures a total deficit which measures the charge of the
deconfined ("fractionalized") fermions, independent of the deformation
parameters.
