String cohomology groups of complex projective spaces

Abstract

Let $X$ be a space and write $LX$ for its free loop space equipped with the action of the circle group $\mathbb{T}$ given by dilation. The equivariant cohomology $H^{\ast }\left(L{X}_{h\mathbb{T}};\mathbb{Z}∕p\right)$ is a module over $H^{\ast }\left(B\mathbb{T};\mathbb{Z}∕p\right)$. We give a computation of this module when $X=ℂP^r$ for any positive integer $r$ and any prime number $p$. The computation does not use the fact that $ℂP^r$ is formal, nor does it use the Jones isomorphism and negative cyclic homology.