This is an article of ours
Hisham Sati, Urs Schreiber, Jim Stasheff,
Twisted differential string and fivebrane structures
Communications in Mathematical Physics
October 2012, Volume 315, Issue 1, pp 169-213
on differential string structures and differential fivebrane structures and their interpretation in terms of the Green-Schwarz mechanism in heterotic string theory. The cohomological theory is part of the story at differential cohomology in a cohesive topos.
Abstract In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, twisted differential string- andtwisted differential Fivebrane-structures_ that generalize the notion of Spin structure and Spin lifting gerbes and their differential refinement to smooth $\mathrm{Spin}$-connections. We show that all these structures can be encoded in terms of nonabelian cohomology and twisted nonabelian cohomology and twisted differential nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the $H_3$-field, as well as its magnetic dual version for the $H_7$-field define cocycles in twisted differential nonabelian cohomology that may be called, respectively, differential twisted $\mathrm{Spin}(n)$-, $\mathrm{String}(n)$- and $\mathrm{Fivebrane}(n)$- structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of $U(n)$ or $O(n)$. We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L-∞ algebra (L-∞ algebroid) valued differential form data provided by the differential refinements of these twisted cocycles.
differential cohomology in a cohesive topos
Twisted differential structures
Twisted Differential String and Fivebrane Structures
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