nLab rheonomy modality

supersymmetry

Applications

In higher supergeometry the bosonic modality $\stackrel{\rightsquigarrow}{(-)}$ which sends supermanifolds to their underlying ordinary bosonic smooth manifolds has a further right adjoint $\Rh$, see at super smooth infinity-groupoid.

This means that if $\mathbf{Fields}$ is a moduli stack of fields, for instance for supergravity, then $Rh(\mathbf{Fields})$ is such that for $\hat X$ any supermanifold with underlying manifold $X \to \hat X$, then maps

$\hat X \longrightarrow Rh(\mathbf{Fields})$

are equivalently maps

$X \longrightarrow \mathbf{Fields}$

hence are fields configurations on the underlying ordinary manifold $X$.

In the supergeometry formulation of supergravity this is what goes into the rheonomy superspace constraint which demands that on-shell super-field configurations $\hat X \to \mathbf{Fields}$ have to be uniquely determined by their restriction along $X \to \hat X$. Therefore the space $Rh(\mathbf{Fields})$ contains the rheonomic field configurations among all the field configurations modulated by $\mathbf{Fields}$.

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$