T-Duality from super Lie n-algebra cocycles for super p-branes

We compute the L_∞-theoretic dimensional reduction of the F1/Dp-brane super L_∞-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient L_∞-algebras are naturally related by an L_∞-isomorphism which we find to act on the super p-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between K⁰ and K¹, rationally. In particular this is a derivation of the Buscher rules for RR-fields (Hori's formula) from first principles. Moreover, we show that these L_∞-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced L_∞-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations.