nLab exceptional tangent bundle

Idea

A generalization of the notion of tangent bundle in exceptional generalized geometry.

For 11-dimensional supergravity the bosonic body of its typical fiber is

$$\mathbb{R}^{d-1,1} \oplus \wedge^2 (\mathbb{R}^{d-1,1})^\ast \oplus \wedge^5 (\mathbb{R}^{d-1,1})^\ast$$

where the second summand corresponds to M2-brane charges and the third to M5-brane charges.

This happens to coincide with the bosonic body of the M-theory super Lie algebra, see there for more.

Related concepts

• generalized tangent bundle

(…)

References

The notion is already recognizable in

• Bernard de Wit, Hermann Nicolai, (2) in: Hidden Symmetries, Central Charges and All That, Class. Quant. Grav. 18 (2001) 3095-3112 [arXiv:hep-th/0011239, doi:10.1088/0264-9381/18/16/302]

via KK-reduction of the “M-algebra”.

It became more widely appreciated under the heading “exceptional generalized geometry” with:

• Christopher M. Hull, Generalised Geometry for M-Theory, JHEP 0707:079 (2007) [arXiv:hep-th/0701203, doi:10.1088/1126-6708/2007/07/079]

The actual term “exceptional tangent bundle” appears, in the context of exceptional field theory, for instance in

• David Berman, Chris D. A. Blair, (4.4) of: The Geometry, Branes and Applications of Exceptional Field Theory, International Journal of Modern Physics A 35 30 (2020) 2030014 [arXiv:2006.09777, doi:10.1142/S0217751X20300148]

and the term “exceptional tangent space” (or rather “exceptional generalised tangent space”) appears for instance in

• Paulo P. Pacheco, Daniel Waldram: §2.2 in: M-theory, exceptional generalised geometry and superpotentials, JHEP 0809: 123 (2008) [arXiv:0804.1362, doi:10.1088/1126-6708/2008/09/123]

• André Coimbra, Charles Strickland-Constable, Daniel Waldram: $E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory, J. High Energ. Phys. 2014 54 (2014) [arXiv:1112.3989, doi:10.1007/JHEP02(2014)054]

Argument that the M-algebra constitutes the maximal ($n=11$) (super-)exceptional tangent space:

• Grigorios Giotopoulos, Hisham Sati, Urs Schreiber: The M-algebra completes the hierarchy of Super-Exceptional Tangent Spaces [arXiv:2411.03661]