#spinors

It's easy to write down explicit equations for many Calabi-Yau 3-folds (those with SU(3) holonomy), but very difficult to explicitly cast a manifold with holonomy metric G₂ or Spin(7). Robert Bryant wrote out an explicit Spin(7) and an explicit G₂ manifold in 1987: it took 49 simultaneous nonlinear PDE’s to exhibit the symmetry.

I’m currently reading David Hestenes’s book Space-Time Algebra, in which he argues that Clifford algebras are undervalued as a suitable language for mathematical physics. It’s coordinate-free, unlike the traditional physicist language, and gets more directly at the physically relevant math than the conventional mathematical approach via tensors.

Consider now a system of n units ι₁,ι₂,...,ιₙ such that the multiplication of any two of them is polar; that is, ιᵣ ιₛ = −ιₛιᵣ.

Thus, the existence of a spinor structure appears, on physical grounds, to be a reasonable condition to impose on any cosmological model in general relativity.

Parallel transport around a closed loop can change a vector. This effect is used to quantify the curvature of space, via the Riemann curvature tensor.