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

We compute the $L_{\infty}$-theoretic double dimensional reduction of the F1/Dp-brane super $L_{\infty}$-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 $L_{\infty}$- algebras are naturally related by an $L_{\infty}$-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 $K^0$-cocycles in type IIA and $K^1$-cocycles in type IIB, 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 $L_{\infty}$-algebras are the homotopy quotients of the RR-charge coefficients by the “T-duality Lie 2-algebra”. We find that the induced $L_{\infty}$-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 show that this satisfies expected properties of a local model space for F-theory elliptic fibrations.

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Published 2 May 2019