$\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\String}{\mathrm{String}}\newcommand{\U}{\mathrm{U}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\O}{\mathrm{O}}\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\Mp}{\mathrm{Mp}}$We can describe the classifying spaces of the groups $\O\_n$, $\SO\_n$, $\U\_n$, $\Sp\_n$, $\Sigma\_n$, and $A\_n$ as follows:


* $\mathrm{B}\O\_n$, $\mathrm{B}\U\_n$, $\mathrm{B}\Sp\_n$ are the Grassmanians of $n$-planes $\Gr\_n(\mathbb{R}^\infty)$, $\Gr\_n(\mathbb{C}^\infty)$, and $\Gr\_n(\mathbb{H}^\infty)$;
* $\mathrm{B}\SO\_n$ is the Grassmanians of *oriented* $n$-planes $\oGr\_n(\mathbb{R}^\infty)$;
* $\mathrm{B}\Sigma\_n$ is the unordered configuration space $\UConf\_n(\mathbb{R}^\infty)$ of $n$ points on $\mathbb{R}^\infty$;
* $\mathrm{B}A\_n$ is the space $\oUConf\_n(\mathbb{R}^\infty)$ whose points are [$n$ points on $\mathbb{R}^\infty$ together with an orientation of their spanned $n$-plane](https://mathoverflow.net/questions/30113);


Are there similar "geometric" descriptions for $\mathrm{B}\Spin\_n$ and $\mathrm{B}\String\_n$?


What about $\mathrm{B}\SU\_n$ (is there a [more explicit description than "the $3$-connected cover of $\mathrm{B}\U\_n$"](https://math.stackexchange.com/a/2850463)?), [$\mathrm{B}\Mp\_n$](https://en.wikipedia.org/wiki/Metaplectic_group), [$\mathrm{B}\widetilde{A}\_n$](https://groupprops.subwiki.org/wiki/Double_cover_of_alternating_group), and [$\mathrm{B}\mathcal{A}\_n$](https://arxiv.org/abs/1605.09192)?



Geometric models for the classifying spaces of the spin and string covers of the orthogonal, symplectic, and symmetric groups
For $\def\B{{\rm B}} \def\bB{{\bf B}} \def\Spin{{\rm Spin}} \def\String{{\rm String}} \B\Spin(n)$, simply equip the $n$-planes with a [spin structure](https://mathoverflow.net/questions/122748/what-is-a-spinor-structure/122798#122798), as originally proposed by Stolz and Teichner.


For $\B\String(n)$, equip the $n$-planes with a string structure,
as described in a [paper by Douglas and Henriques](http://andreghenriques.com/PDF/TringWP.pdf).


Also, I am not sure what the intended difference between $\B$ and $\bB$ is,
but if $\bB$ does refer to the stack version of these classifying spaces,
then the above constructions continue to work perfectly well
for stacks: to a smooth manifold $S$ assign the groupoid respectively 2-groupoid of vector bundles with base $S$ and whose fibers are equipped with a spin respectively string structure.


This yields the stacks $\bB\Spin(n)$ and $\bB\String(n)$, and applying the shape functor to these stacks yields the classifying spaces $\B\Spin(n)$ and $\B\String(n)$.


