Schreiber Super-Lie-infinity T-Duality and M-Theory

An article that we are finalizing at CQTS:

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• Grigorios Giotopoulos,$\;$ Hisham Sati $\;$and$\;$ Urs Schreiber:

  
  

  Super-Lie$_\infty$ T-Duality and M-Theory

  
  

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  • arXiv:2411.10260

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on super-space T-duality and its potential lifts to M-theory.

Abstract. 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 being the rational shadows of flux-quantization laws (in ordinary cohomology, K-theory, Cohomotopy, iterated K-theory, …).

We first review, in streamlined form and filling some previous gaps, double-dimensional reduction/oxidation and 10D superspace T-duality along higher-dimensional super-tori, 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. This yields in particular a derivation, at the rational level, of the traditional laws of “topological T-duality” from the super-$L_\infty$ structure of type II super-space.

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 appear as the D/NS5-brane 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).

Recalling that the hidden M-algebra appears also in a higher form of rational-topological T-duality where strings are replaced by M5-branes, we end with comments on the M-algebra as a Kleinian local model space for U-duality-covariant superspace supergravity.

Following up on:

Fiorenza, Sati, 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

• Higher T-duality of super M-branes

  Adv. Theor. Math. Phys. 24 3 (2020)

  doi:10.4310/ATMP.2020.v24.n3.a3