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In this paper, we construct for the first time the projective elliptic generafor a compact oriented manifold equipped with a projective complex vectorbundle. Such projective elliptic genera are rational q-series that havetopological definition and also have analytic interpretation via the fractionalindex theorem in Mathai-Melrose-Singer (2006) without requiring spin condition.We prove the modularity properties of these projective elliptic genera. As anapplication, we construct elliptic pseudodifferential genera for any ellipticpseudodifferential operator. This suggests the existence of putativerotation-equivariant elliptic pseudodifferential operators on loop space whoseequivariant indices are elliptic pseudodifferential genera.

Projective elliptic genera and elliptic pseudodifferential genera