David Corfield supergeometry

Idea

If we understand the revolution as requiring a change to geometry, then along with homotopification (embodying the gauge principle) there is also the creation of supergeometry (allowing the representation of fermionic fields, building in the Pauli exclusion principle). (But note Kapranov’s claim that supergeometry is one step on the sphere spectrum path.)

The formalization of fermions is in supergeometry: fermion fields are sections of spinor bundles with odd-degree fibers in supergeometry.

Supergeometry can be understood via internalisation.

one can almost say that mathematicians and physicists mean different things when they speak about supergeometry. This contributes to the enduring feeling of wonder and mystery surrounding this subject. (Kapranov)

The entry point for a mathematician here could be found in the idea of taking natural “square roots” of familiar mathematical and physical quantities

So one can say that, in some approximate sense, derived geometry can be regarded as super-geometry but in an equivariant context, with respect to the action of the supergroup $Aut(\mathbb{A}^{0|1})$. However, this is a rather simplified point of view for several reasons. Even in the context of dg-algebras, the distinction between negative and positive grading is very important, it corresponds to the distinction between “geometry in the small” (intersections, singularities) and “geometry in the large” (stacks, homotopy type). The precise definitions impose “geometry in the large” in a more global “stacky” way.

11d geometry

Supertorsion-free structure on $\mathbb{R}^{10,1|32}$-manifold provides supergravity equations.

Kapranov extension

See nLab on superalgebra.

References

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)