One can define differential forms $\Omega^*(LM)$ on the (smooth) free loop space $LM$ by giving $LM$ the structure of a diffeological space or a Chen differential space. A diffeological space $X$ is defined via plots, which are maps $U\to X$ from open $U\subset \mathbb R^n$ for all $n$, subject to some properties. In the case of $LM$, a plot is a map $U\to LM$ such that the adjoint $U\times S^1 \to LM$ is a map of smooth manifolds. Then a de Rham form is defined via its "pullbacks" through all plots, subject to compatibility with smooth maps of open euclidean sets.
Let $S^1\vee S^1$ be the figure eight space and $\mathrm{Map}(8) = \mathrm{Map}(S^1\vee S^1, M)$ the space of figure eights in $M$. In Prop. 4.3 arXiv:1911.06202: "String topology and configuration spaces of two points" - Florian Naef, Thomas Willwacher, the authors use the symbol $\Omega^*(\mathrm{Map}(8))$ for the differential forms on the space of figure eights in $M$, but do not give a definition. How do you define these differential forms? Since $S^1\vee S^1$ is not a manifold, one cannot proceed exactly as above. The definition of these differential forms should be such that the concatenation map $\mathrm{Map}(8) \to LM$ and the inclusion $\mathrm{Map}(8)\to LM\times LM$ yield pullback maps on de Rham forms.