Geometric models for the classifying spaces of the spin and string covers of the orthogonal, symplectic, and symmetric groups

Question

$\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\String}{\mathrm{String}}\newcommand{\U}{\mathrm{U}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\O}{\mathrm{O}}\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\Mp}{\mathrm{Mp}}$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;

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 $\mathrm{B}\U_n$"?), $\mathrm{B}\Mp_n$, $\mathrm{B}\widetilde{A}_n$, and $\mathrm{B}\mathcal{A}_n$?

Answer 1

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, 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.

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 $\B\String(n)$.