On vector bundles for a Morse decomposition of $L({\mathbb{C}\mathrm{P}}^n)$

We give a description of the negative bundles for the energy integral on the free loop space $L{\mathbb{C}\mathrm{P}}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.