bosonic modality

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

On super smooth infinity-groupoids there is a modal operator $\stackrel{\rightsquigarrow}{(-)}$ which projects onto the bosonic components of the supergeometry. On formal dual superalgebras this is given by passing to the *body*. In terms of physical fields this is the projection onto boson fields, which are hence the modal types of $\rightsquigarrow$, and so it makes sense to speak of the *bosonic modality*.

This has a left adjoint $\rightrightarrows$ (which on superalgebras passes to the even-graded sub-algebra) and hence together these form an adjoint modality which may be thought of as characterizing the supergeometry. See at *super smooth infinity-groupoid – Cohesion*. With $\rightrightarrows$ being opposite to $\rightsquigarrow$ thereby, it makes sense to call it the *fermionic modality*.

Notice that the fermionic currents in physics (e.g. the electron density current) are indeed fermionic bilinears, i.e. are indeed in the even subalgebras of the underlying superalgebra.

$\array{
fermions & \rightrightarrows &\stackrel{}{\dashv}& \stackrel{\rightsquigarrow}{} & bosons
}$

The right adjoint of the bosonic modality is the *rheonomy modality*.

Revised on December 22, 2015 10:08:16
by Nikolaj K?
(185.25.95.132)