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

Abstract

We compute the $L_{\infty }$-theoretic dimensional reduction of the F1/D$p$-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 $K^0$ and $K^1$, 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 observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations.

44 pages; v2: added more discussion of double dimensional reduction via cyclic L-infinity cohomology; v3: added derivation of Buscher rules for RR-fields, expanded on role of curved L-infinity algebras in double dimensional reduction

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