Schreiber The Character Map in Equivariant Twistorial Cohomotopy

An article that we have written:

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• Hisham Sati, Urs Schreiber:

  
  

  The character map in equivariant twistorial Cohomotopy

  implies the Green-Schwarz mechanism with heterotic M5-branes

  
  

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  • pdf (v2)

  • arXiv:2011.06533

Abstract. The celebrated Green-Schwarz mechanism in heterotic string theory has been suggested to secretly be a higher gauge theoretic phenomenon.

Here we prove that its lift to (i) the non-perturbative Hořava-Witten Green-Schwarz mechanism for (ii) heterotic line bundles in heterotic M-theory with (iii) M5-branes parallel to MO9-planes on (iii) A1-singularities is accurately encoded in the higher gauge theory for higher gauge group of the equivariant homotopy type of the $\mathbb{Z}_2$-equivariant $A_\infty$-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 heart of our proof is the construction of the equivariant twisted non-abelian character map via an equivariant twisted non-abelian de Rham theorem, which we prove; as well as the computation of the equivariant relative minimal model of the $\mathbb{Z}_2$-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.

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Followup to

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation

  arXiv:2008.08544

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