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Projective vector bundles (or gerbe modules) are generalizations of vector bundles in the presence of a gerbe on manifolds. Given a projective vector bundle, we will first show how to use it to twist the Witten genus to get modular invariants, which we call projective elliptic genera. Then we will give two applications: (1) given any pseudodifferential operator, we will construct modular invariants generalizing the Witten genus, which corresponds to the Dirac operator; (2) we will enhance the Hori map in T-duality to the graded Hori map and show that it sends Jacobi forms to Jacobi forms. This represents our joint works with Mathai.