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

TL;DR

This paper describes the negative bundles in the free loop space of complex projective space using circle vector bundles over Stiefel manifolds and computes their mod p Chern classes.

Contribution

It provides a new description of negative bundles in loop space using geometric bundles and calculates their mod p Chern classes.

Findings

Negative bundles characterized via circle vector bundles over Stiefel manifolds

Computed mod p Chern classes of homotopy orbit bundles

Enhanced understanding of Morse decomposition in loop space

Abstract

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.