  We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the
complex projective plane. We prove that they are homotopy equivalent to
explicitly described algebraic subspaces of the configuration spaces of $n$
points. We compute the rational homotopy type of these embedding spaces and
their cohomology with rational coefficients. Our approach relies on the
comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration
space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the
symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$
embedded symplectic balls.
