Topology of spaces of equivariant symplectic embeddings

We compute the homotopy type of the space of $\mathbb{T}^n$-equivariant symplectic embeddings from the standard $2n$-dimensional ball of some fixed radius into a $2n$-dimensional symplectic-toric manifold $(M, \, \sigma)$, and use this computation to define a $\mathbb{Z}_{\ge 0}$-valued step function on $\mathbb{R}_{\ge 0}$ which is an invariant of the symplectic-toric type of $(M, \, \sigma)$. We conclude with a discussion of the partially equivariant case of this result.