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

An article that we have written:

Domenico Fiorenza,
Hisham Sati,
Urs Schreiber:

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

Adv. Theor. Math. Phys. 22 5 (2018)

doi:10.4310/ATMP.2018.v22.n5.a3

arXiv:1611.06536

Abstract. We compute the L-infinity-theoretic double 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. 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 all the expected properties of a local model space for F-theory elliptic fibrations.