A mirror conjecture for projective bundles

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We propose, motivate and give evidence for a relation between the $\mathcal D$-modules of the quantum cohomology of a smooth complex projective manifold $X$ and a projective bundle $\PP(\oplus L_i)$ over $X$.
This is the published version. Section 3.1 improvements and the discussion on the small quantum cohomology of a projective bundles will appear in a different paper

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