Delooping of high-dimensional spaces of string links

We study a connection between a multivariable version of the Goodwillie-Weiss’ calculus of functors and derived mapping spaces of k-fold bimodules over a family of operads. As our main application, under the assumption $d_i+3\le n$ for all $i\in \{1,\dots ,k\}$, we produce explicit deloopings of high-dimensional spaces of string links modulo immersions from ${\bigsqcup }_i{\mathbb{R}}^{d_i}$ to ${\mathbb{R}}^n$ and their polynomial approximations.