Flux Quantization on 10D Type IIA Superspace via Cyclification from 11D

We produce the dimensional reduction to 10D IIA supergravity (SuGra), via cyclification, of the remarkable result that full 11D SuGra is put on shell just by imposing the duality-symmetric Bianchi identities on C-field super-flux densities over supertorsion-free superspace. Generally, we highlight that when duality-symmetric superspace Bianchi identities are characterized by Whitehead bracket $L_{\mathrm{\infty }}$-algebras $\mathfrak{l}\mathcal{A}$ of a classifying space $\mathcal{A}$, their dimensional reduction is characterized by the cyclic loop space $\mathrm{C}\mathrm{y}\mathrm{c}(\mathcal{A})$. We promote this to a general mechanism of dimensional reduction on super-spacetime, compatible with the global (infrared) completion of supergravity theories by flux quantization in non-abelian cohomology with coefficients in $\mathcal{A}$ and $\mathrm{C}\mathrm{y}\mathrm{c}(\mathcal{A})$, respectively. In the case of 11D SuGra, the characteristic $L_{\mathrm{\infty }}$-algebra is $\mathfrak{l}{S}^4$ and hence we obtain that full on-shell 10D IIA SuGra is equivalent to $\mathfrak{l}\mathrm{C}\mathrm{y}\mathrm{c}(S^4)$-Bianchi imposed identities on NS/RR super-flux densities over supertorsion-free 10D super-spacetime. This implies that any space which is $\mathbb{R}$-rationally equivalent to $\mathrm{C}\mathrm{y}\mathrm{c}(S^4)$ classifies an admissible flux quantization law, which provides a global completion of 10D IIA SuGra that admits oxidation to 11D.