Super-\(\mathrm{Lie}_\infty\) T-Duality and M-Theory

Super \(L_\infty\)-algebras unify extended super-symmetry with rational classifying spaces for higher flux densities: The super-invariant super-fluxes which control super \(p\)-branes and their supergravity target super-spaces are, together with their (non-linear) Bianchi identities, neatly encoded...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Giotopoulos, Grigorios, Sati, Hisham, Schreiber, Urs
Format: Article
Language:English
Subjects:
Algebra
Avatars
Branes
Flux quantization
Homology
M theory
Oxidation
Relativity
Spacetime
Supergravity
Supersymmetry
Toruses
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Summary:Super \(L_\infty\)-algebras unify extended super-symmetry with rational classifying spaces for higher flux densities: The super-invariant super-fluxes which control super \(p\)-branes and their supergravity target super-spaces are, together with their (non-linear) Bianchi identities, neatly encoded in (non-abelian) super-\(L_\infty\) cocycles. These are the rational shadows of flux-quantization laws (in ordinary cohomology, K-theory, Cohomotopy, iterated K-theory, etc). We first review, in streamlined form while filling some previous gaps, double-dimensional reduction/oxidation and 10D superspace T-duality along higher-dimensional super-tori. We do so tangent super-space wise, by viewing it as an instance of adjunctions (dualities) between super-\(L_\infty\)-extensions and -cyclifications, applied to the avatar super-flux densities of 10D supergravity. In particular, this yields a derivation, at the rational level, of the traditional laws of "topological T-duality" from the super-\(L_\infty\) structure of type II superspace. At this level, we also discuss a higher categorical analog of T-duality involving M-branes. Then, by considering super-space T-duality along all 1+9 spacetime dimensions while retaining the 11th dimension as in F-theory, we find the M-algebra appearing as the complete brane-charge extension of the fully T-doubled/correspondence super-spacetime. On this backdrop, we recognize the "decomposed" M-theory 3-form on the "hidden M-algebra" as an M-theoretic lift of the Poincaré super 2-form that controls superspace T-duality as the integral kernel of the super Fourier-Mukai transform. This provides the super-space structure of an M-theory lift of the doubled/correspondence space geometry, which controls T-duality.
ISSN:2331-8422