We extend some of the recently uncovered algebraic structures on the free
loop space (by Chas and Sullivan) to: i) subspaces of holomorphic maps when
the target is a projective space, and ii) maps between two closed manifolds.
As an application we determine the homology of second fold free loop spaces
of complex projective spaces. We also use these methods and others to get
a description of the homology of projective "non-linear sigma models" (i.e.
maps of Riemann surfaces into $CP^n$).
This work is joint with Paolo Salvatore.