nLab
smooth super infinity-groupoid

this entry is under construction

Idea

A smooth super $\mathrm{\infty }$-groupoid is a super ∞-groupoid equipped with smooth cohesion.

This includes supermanifolds and deloopings of super Lie groups.

Definition

We take a smooth super $\mathrm{\infty }$-groupoid to be a smooth ∞-groupoid but not over the base topos ∞Grpd of bare ∞-groupoids, but over the base topos Super∞Grpd of super ∞-groupoids.

Recall from the discussion at super ∞-groupoid that there is a canonical line object

$$ℛ\in \mathrm{SuperSet}\hookrightarrow \mathrm{Super}\mathrm{\infty }\mathrm{Grpd}$$\mathcal{R} \in SuperSet \hookrightarrow Super\infty Grpd

which as a presheaf on the site of superpoints is given by

$$ℛ:{ℝ}^{0\mid q}\mapsto ({\Lambda }_q{)}_{\mathrm{even}},$$\mathcal{R} : \mathbb{R}^{0|q} \mapsto (\Lambda_q)_{even} \,,

where ${\Lambda }_q$ is (the underlying set of) the Grassmann algebra on $q$ generators, and $({\Lambda }_q{)}_{\mathrm{even}}$ is (the underlying set of) its even part.

Definition

Write $\mathrm{sCartSp}$ for the internal site in $\mathrm{SuperSet}\hookrightarrow$ Super∞Grpd whose

• objects are the products ${ℝ}^{}$s for $n\in \mathbb{N}$;

• morphisms are those morphisms of supersets that are smooth with respect to the canonical supermanifold structure on ${ℝ}^n$ (see the section As manifolds over the base topos on superpoints) ].

• coverage is given by the internal good open covers.

(…)

Definition

Let

$$\mathrm{SmoothSuper}\mathrm{\infty }\mathrm{Grpd}:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{sCartSp},\mathrm{Super}\mathrm{\infty }\mathrm{Grpd})$$SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)

be the (∞,1)-topos of internal (∞,1)-sheaves in Super∞Grpd over the internal site $\mathrm{sCartSp}$.

Note

In particular $\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ is an $(\mathrm{\infty },1)$-topos over the base topos Super∞Grpd

$$\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{Super}}}{\leftarrow }}{\underset{{\Gamma }_{\mathrm{Super}}}{\to }}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}.$$Smooth Super \infty Grpd \stackrel{\overset{Disc_{Super}}{\leftarrow}}{\underset{\Gamma_{Super}}{\to}} Super \infty Grpd \,.

Properties

Claim

We have that

$$\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}\simeq {\mathrm{Sh}}_{(\mathrm{\infty },1)}({\mathrm{CartSp}}_{\mathrm{smooth}}\times \mathrm{SuperPoint},\mathrm{\infty }\mathrm{Grpd}).$$Smooth Super\infty Grpd \simeq Sh_{(\infty,1)}(CartSp_{smooth} \times SuperPoint, \infty Grpd) \,.

By the discussion of externalization at internal site.

Proposition

$\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}$ is a cohesive (∞,1)-topos over Super∞Grpd.

$$\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{{\Pi }_{\mathrm{Super}}}{\to }}{\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{Super}}}{\leftarrow }}{\stackrel{\stackrel{{\Gamma }_{\mathrm{Super}}}{\to }}{\underset{{\mathrm{coDisc}}_{\mathrm{super}}}{\leftarrow }}}}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\leftarrow }}}}\mathrm{\infty }\mathrm{Grpd}.$$Smooth Super \infty Grpd \stackrel{\overset{\Pi_{Super}}{\to}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\to}}{\underset{coDisc_{super}}{\leftarrow}}}} Super \infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd \,.

In fact we have a commutative diagram of cohesive (∞,1)-toposes

$$\begin{array}{ccc}\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}& \stackrel{\stackrel{{\Pi }_{\mathrm{Super}}}{\to }}{\stackrel{\stackrel{{\mathrm{Disc}}_{\mathrm{Super}}}{\leftarrow }}{\stackrel{\stackrel{{\Gamma }_{\mathrm{Super}}}{\to }}{\underset{{\mathrm{coDisc}}_{\mathrm{super}}}{\leftarrow }}}}& \mathrm{Super}\mathrm{\infty }\mathrm{Grpd}\\ \downarrow \uparrow & & \downarrow \uparrow \\ \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}& \stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{\leftarrow }}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{\leftarrow }}}}& \mathrm{\infty }\mathrm{Grpd}\end{array}$$\array{ Smooth Super \infty Grpd &\stackrel{\overset{\Pi_{Super}}{\to}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\to}}{\underset{coDisc_{super}}{\leftarrow}}}} & Super \infty Grpd \\ \downarrow \uparrow && \downarrow \uparrow \\ Smooth \infty Grpd & \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }

Structures

We discuss realizations of the general abstract structures in a cohesive (∞,1)-topos realized in $\mathrm{Smooth}\mathrm{Super}\mathrm{\infty }\mathrm{Grpd}$.

Exponentiated super $L_{\mathrm{\infty }}$-algebras

A super L-∞ algebra $𝔤$ is an L-∞ algebra internal to $\mathrm{Sh}(\mathrm{SuperPoint})$.

The Lie integration of $𝔤$ is …

Applications

• D'Auria-Fre formulation of supergravity

References

For general references see the references at super ∞-groupoid .

A discussion of smooth super $\mathrm{\infty }$-groupoids is in section 3.5 of

• Urs Schreiber, differential cohomology in a cohesive topos.