# Twisted Bianchi identity for D6-branes from M-theory?

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The descent of the M2/M5-brane charges to the D$p$-brane charges and their twisted Bianchi identites $d F_{p+2} =H_3 \wedge F_{p}$ is straightforward for $p \in \{0,2,4\}$, as spelled out rigorously for instance in section 4.2 of

Varghese Mathai, Hisham Sati,  Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP0403:016,2004 (hep-th/0312033)

The origin of the D6 brane charge $F_8$is more subtle, as its M-theory lift involves the field of gravity: $F_8 = \star_{10} F_2$. The correct Bianchi identity $d F_8 = H_3 \wedge F_6$ is claimed on the left of (4.13) in the above article, but it does not actually follow from the 11d derivation given there, as far as I see.

What is a quick way to derive the Bianchi identity $d F_8 = H_3 \wedge F_6$ in type IIA from 11d?

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