nLab
bosonic modality

Context

Modalities, Closure and Reflection

Super-Geometry

Idea

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.

fermions bosons \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)