# Projective elliptic genera and elliptic pseudodifferential genera

## TL;DR

This paper introduces the first construction of projective elliptic genera for compact manifolds with projective vector bundles, establishing their modularity and linking them to elliptic pseudodifferential operators.

## Contribution

It develops the concept of projective elliptic genera with topological and analytic interpretations, extending the theory to include elliptic pseudodifferential genera without spin restrictions.

## Key findings

- Constructed projective elliptic genera for manifolds with projective bundles.
- Proved modularity properties of these genera.
- Linked elliptic pseudodifferential operators to elliptic pseudodifferential genera.

## Abstract

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.