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

TL;DR

This paper derives a mathematical framework for T-duality in superstring theory using super Lie n-algebra cocycles, revealing new insights into the algebraic structure underlying T-duality and its relation to F-theory.

Contribution

It introduces an $L_$-algebraic approach to T-duality, deriving Buscher rules from first principles and modeling T-folds via homotopy quotients of RR-charge coefficients.

Findings

Derived T-duality as an $L_$-isomorphism between super Lie algebra cocycles.

Connected T-duality with rationalized K-theory and F-theory models.

Provided a local algebraic model for T-folds and F-theory elliptic fibrations.

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.