rheonomy modality

**superalgebra** and (synthetic ) **supergeometry**

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 $\mathbf{Fields}$.