Contents

supersymmetry

# Contents

## Idea

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

Super smooth $\infty$-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) in some fine technical detail. The other is at smooth super infinity-groupoid.

###### Definition

Write

There are then “semidirect product” sites $CartSp \rtimes InfinPoint$ and $CartSp \rtimes SuperPoint$ (whose objects are Cartesian products of the given form inside synthetic differential supergeometry and whose morphisms are all morphisms in that context (not just the product morphisms)).

Set then

$FormalSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes InfPoint)$

for the collection of formal smooth ∞-groupoids (see there) and finally

$SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)$

for that of super smooth $\infty$-groupoid

## Properties

### Cohesion

###### Remark

The sites in question are alternatingly (co-)reflective subcategories of each other (we always display left adjoints above their right adjoints)

$\ast \stackrel{\longleftarrow}{\hookrightarrow} CartSp \stackrel{\hookrightarrow}{\longleftarrow} CartSp\rtimes InfPoint \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} CartSp \rtimes SuperPoint \,.$

Here

• the first inclusion picks the terminal object $\mathbb{R}^0$;

• the second inclusion is that of reduced objects; the coreflection is reduction, sending an algebra to its reduced algebra;

• the third inclusion is that of even-graded algebras, the reflection sends a $\mathbb{Z}_2$-graded algebra to its even-graded part, the co-reflection sends a $\mathbb{Z}_2$-graded algebra to its quotient by the ideal generated by its odd part, see at superalgebra – Adjoints to the inclusion of plain algebras.

###### Remark

Passing to (∞,1)-categories of (∞,1)-sheaves, this yields, via (∞,1)-Kan extension, a sequence of adjoint quadruples as follows:

$\array{ & && && &\longleftarrow& \\ & && &\hookrightarrow& &\hookrightarrow& \\ & &\longleftarrow& &\longleftarrow& &\longleftarrow& \\ \Delta \colon & \infty Grpd &\hookrightarrow& Smooth \infty Grpd &\hookrightarrow& FormalSmooth \infty Grpd &\hookrightarrow& SuperFormalSmooth \infty Grpd \\ & &\longleftarrow& &\longleftarrow& \\ & &\hookrightarrow& }$
###### Proposition

Passing to the adjoint triples of idempotent monads and idempotent comonads which this induces, then yields

Hence we get a process of adjoint modalities of the form

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

where “$\vee$” denotes inclusion of modal types. The first level is cohesion, the second is differential cohesion (elasticity), the third is a further refinement given by supergeometry, which takes further “square roots” of all infinitesimal generators.

###### Proof

All the sites are ∞-cohesive sites, which gives that we have an cohesive (infinity,1)-topos. The composite inclusion on the right is an ∞-cohesive neighbourhood site, whence the inclusion $Smooth\infty Gpd\hookrightarrow SuperFormalSmooth\infty Grpd$ exhibits differential cohesion.

With this the rightmost adjoint quadruple gives the Aufhebung of $\Re \dashv \Im$ by $\rightsquigarrow \dashv Rh$ and the further opposition $\rightrightarrows \dashv \rightsquigarrow$.

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$$\phantom{A}$site$\phantom{A}$$\phantom{A}$sheaf topos$\phantom{A}$$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$$\phantom{A}$Point$\phantom{A}$$\phantom{A}$Set$\phantom{A}$$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$CartSp$\phantom{A}$$\phantom{A}$SmoothSet$\phantom{A}$$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$$\phantom{A}$FormalCartSp$\phantom{A}$$\phantom{A}$FormalSmoothSet$\phantom{A}$$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$SuperFormalCartSp$\phantom{A}$$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

## References

Last revised on October 2, 2020 at 07:22:13. See the history of this page for a list of all contributions to it.