nLab smooth super infinity-groupoid

this entry is under construction

Idea

The notion of smooth super $\infty$-groupoid or smooth super geometric homotopy type is the combination of of super ∞-groupoid and smooth ∞-groupoid. The cohesive (∞,1)-topos of smooth super-$\infty$-groupoids is a context that realizes higher supergeometry.

Smooth super $\infty$-groupoids include supermanifolds, super Lie groups and their deloopings etc. Under Lie differentiation these map to super L-∞ algebras.

Definition

We consider one of at least two possible definitions, that differ (only) slightly in some fine technical detail. The other is at super smooth infinity-groupoid.

We take a smooth super $\infty$-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.

Definition

Write $sCartSp$ for the full subcategory of that of supermanifolds on the super Cartesian spaces $\{\mathbb{R}^{p|q}\}_{p,q \in \mathbb{N}}$. Regard this as a site by taking the coverage the product coverage of the good open cover coverage of CartSp and the trivial coverage on superpoints.

Remark

So a covering family of $\mathbb{R}^{p|q}$ is of the form

$$\{ U_i \times \mathbb{R}^{0|q} \longrightarrow \mathbb{R}^{p|q} \}_{i}$$

for

$$\{ U_i \longrightarrow \mathbb{R}^{p} \}_{i}$$

a differentiably good open cover of $\mathbb{R}^{p}$.

Definition

Let

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

be the (∞,1)-topos of (∞,1)-sheaves over the site $sCartSp$, def. .

Proposition

The (∞,1)-topos $Smooth \infty Grpd$ of def. is a cohesive (∞,1)-topos over ∞Grpd.

This and the stronger statement that it is in fact it is actually cohesive over Super∞Grpd is discussed below, see cor. .

Properties

Cohesion over smooth $\infty$-groupoids and over super $\infty$-groupoids

Proposition

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

$$Smooth Super \infty Grpd \stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} Super \infty Grpd \,.$$

Proof

By definition of the coverage on $sCartSp$ in def. , the proof of the cohesion of Smooth∞Grpd = $Sh_\infty(CartSp)$ goes through verbatim for each fixed superpoint and that gives precisely the claim.

Proposition

Super∞Grpd is infinitesimally cohesive over ∞Grpd.

By the discussion at Super∞Grpd.

Corollary

$SmoothSuper\infty Grpd$ is cohesive and in

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

$$\array{ Smooth Super \infty Grpd &\stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}} & Super \infty Grpd \\ \downarrow \uparrow && \downarrow \uparrow \\ Smooth \infty Grpd & \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} & \infty Grpd }$$

where the right vertical adjoints exhibit infinitesimal cohesion.

Structures

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

Exponentiated super $L_\infty$-algebras

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

The Lie integration of $\mathfrak{g}$ is …

Applications

• D'Auria-Fre formulation of supergravity

• The brane bouquet of Green-Schwarz action functionals for super $p$-brane sigma-models.

Related concepts

• cohesive (∞,1)-topos

  • discrete ∞-groupoid

  • Euclidean-topological ∞-groupoid

  • smooth ∞-groupoid

  • synthetic differential ∞-groupoid

  • super ∞-groupoid

  • smooth super ∞-groupoid

    • Super Gerbes

  • super formal smooth infinity-groupoid

  • synthetic differential super ∞-groupoid

References

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

Discussion of smooth super $\infty$-groupoids:

• Urs Schreiber, section 4.5 of: differential cohomology in a cohesive topos

• Hisham Sati, Urs Schreiber, Section 3.1.3 of: Proper Orbifold Cohomology (arXiv:2008.01101)

• Urs Schreiber, Introduction to Higher Supergeometry, lecture at Higher Structures in M-Theory 2018, Durham Symposium

  published in parts as: Higher Structures in M-Theory (with Branislav Jurčo, Christian Saemann, Martin Wolf), Fortschritte der Physik (2019) (arXiv:1903.02807, doi:10.1002/prop.201910001)