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superalgebra and (synthetic ) supergeometry
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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.
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.
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.
So a covering family of $\mathbb{R}^{p|q}$ is of the form
for
a differentiably good open cover of $\mathbb{R}^{p}$.
Let
be the (∞,1)-topos of (∞,1)-sheaves over the site $sCartSp$, def. .
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. .
$Smooth Super \infty Grpd$ is a cohesive (∞,1)-topos over Super∞Grpd.
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.
Super∞Grpd is infinitesimally cohesive over ∞Grpd.
By the discussion at Super∞Grpd.
$SmoothSuper\infty Grpd$ is cohesive and in
In fact we have a commutative diagram of cohesive (∞,1)-topos
where the right vertical adjoints exhibit infinitesimal cohesion.
We discuss realizations of the general abstract structures in a cohesive (∞,1)-topos realized in $Smooth Super \infty Grpd$.
A super L-∞ algebra $\mathfrak{g}$ is an L-∞ algebra internal to $Sh(SuperPoint)$.
The Lie integration of $\mathfrak{g}$ is …
The brane bouquet of Green-Schwarz action functionals for super $p$-brane sigma-models.
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)
Last revised on April 14, 2024 at 15:22:22. See the history of this page for a list of all contributions to it.