  In this paper, we construct for the first time the projective elliptic genera
for a compact oriented manifold equipped with a projective complex vector
bundle. Such projective elliptic genera are rational q-series that have
topological definition and also have analytic interpretation via the fractional
index theorem in Mathai-Melrose-Singer (2006) without requiring spin condition.
We prove the modularity properties of these projective elliptic genera. As an
application, we construct elliptic pseudodifferential genera for any elliptic
pseudodifferential operator. This suggests the existence of putative
rotation-equivariant elliptic pseudodifferential operators on loop space whose
equivariant indices are elliptic pseudodifferential genera.
