  Luttinger's theorem is a major result in many-body physics that states the
volume of the Fermi surface is directly proportional to the particle density.
In its "hard" form, Luttinger's theorem implies that the Fermi volume is
invariant with respect to interactions (as opposed to a "soft" Luttinger's
theorem, where this invariance is lost). Despite it's simplicity, the
conditions on the fermionic self energy under which Luttinger's theorem is
valid remains a matter of debate, with possible requirements for its validity
ranging from particle-hole symmetry to analyticity about the Fermi surface. In
this paper, we propose the minimal requirements for the application of a hard
Luttinger's Theorem to a generic fermionic system of arbitrary interaction
strength by invoking the Atiyah-Singer index theorem to quantify the
topologically-robust behavior of a generalized Fermi surface. We show that the
applicability of a hard Luttinger's theorem in a D-dimensional system is
directly dependent on the existence of a (D-1)-dimensional manifold of gapless
chiral excitations at the Fermi level, regardless of whether the system
exhibits Luttinger or Fermi surfaces (i.e., manifolds of zeroes of the Green's
function and inverse Green's function, respectively). The exact form of the
self-energy which guarantees validity of a hard Luttinger's theorem is derived,
and agreement with current experiments, numerics, and theories are discussed.
