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adding references on the suggestion that an actual E8-principal bundle on 11d spacetime plays a role here:
D. Diaconescu, Gregory Moore, Edward Witten, $E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (arXiv:hep-th/0005090)
E. Diaconescu, Dan Freed, Greg Moore, Section 3 in: The $M$-theory 3-form and $E_8$-gauge theory, chapter in Haynes Miller, Douglas Ravenel (eds.) Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press 2007 (arXiv:hep-th/0312069, doi:10.1017/CBO9780511721489)
Allan Adams1, Jarah Evslin, The loop group of $E_8$ and K-theory from 11d, JHEP 02 (2003) (arXiv:hep-th/0203218)
Hisham Sati, $E_8$ Gauge Theory and Gerbes in String Theory, Adv. Theor. Math. Phys. 14:1-39, 2010 (arXiv:hep-th/0608190)
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