  The celebrated Green-Schwarz mechanism in heterotic string theory has been
suggested to secretly underly a higher gauge theoretic phenomenon, embodying a
higher Bianchi identity for a higher-degree analog of a curvature form of a
higher gauge field. Here we prove that the non-perturbative Horava-Witten
Green-Schwarz mechanism for heterotic line bundles in heterotic M-theory with
M5-branes parallel to MO9-planes on $A_1$-singularities is accurately encoded
in the higher gauge theory for higher gauge group of the equivariant homotopy
type of the Z/2-equivariant A-infinity-loop group of twistor space.
  In this formulation, the flux forms of the heterotic gauge field, the B-field
on the M5-brane, and of the C-field in the M-theory bulk are all unified into
the character image of a single cocycle in equivariant twistorial Cohomotopy
theory; and that cocycle enforces the quantization condition on all fluxes: the
integrality of the gauge flux, the half-shifted integrality of the C-field flux
and the integrality of the dual C-field flux (i.e., of the Page charge in the
bulk and of the Hopf-WZ term on the M5-brane). This result is in line with the
Hypothesis H that M-brane charges are quantized in J-twisted Cohomotopy theory.
  The mathematical essence of our proof is, first, the construction of the
equivariant twisted non-abelian character map via an equivariant twisted
non-abelian de Rham theorem, which we prove; and, second, the computation of
the equivariant relative minimal model of the Z/2-equivariant
Sp(1)-parametrized twistor fibration. We lay out the relevant background in
equivariant rational homotopy theory and explain how this brings about the
subtle flux quantization relations in heterotic M-theory.
