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The celebrated Green-Schwarz mechanism in heterotic string theory has beensuggested to secretly underly a higher gauge theoretic phenomenon, embodying ahigher Bianchi identity for a higher-degree analog of a curvature form of ahigher gauge field. Here we prove that the non-perturbative Horava-WittenGreen-Schwarz mechanism for heterotic line bundles in heterotic M-theory withM5-branes parallel to MO9-planes on $A_1$-singularities is accurately encodedin the higher gauge theory for higher gauge group of the equivariant homotopytype 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-fieldon the M5-brane, and of the C-field in the M-theory bulk are all unified intothe character image of a single cocycle in equivariant twistorial Cohomotopytheory; and that cocycle enforces the quantization condition on all fluxes: theintegrality of the gauge flux, the half-shifted integrality of the C-field fluxand the integrality of the dual C-field flux (i.e., of the Page charge in thebulk and of the Hopf-WZ term on the M5-brane). This result is in line with theHypothesis H that M-brane charges are quantized in J-twisted Cohomotopy theory.  The mathematical essence of our proof is, first, the construction of theequivariant twisted non-abelian character map via an equivariant twistednon-abelian de Rham theorem, which we prove; and, second, the computation ofthe equivariant relative minimal model of the Z/2-equivariantSp(1)-parametrized twistor fibration. We lay out the relevant background inequivariant rational homotopy theory and explain how this brings about thesubtle flux quantization relations in heterotic M-theory.

The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes