nLab rheonomy modality

See also

Context

Super-Geometry

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

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

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

are equivalently maps

XFields X \longrightarrow \mathbf{Fields}

hence are fields configurations on the underlying ordinary manifold XX.

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

See also

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }

Last revised on July 23, 2022 at 18:16:09. See the history of this page for a list of all contributions to it.