Cohomology and Extensions
Formal Lie groupoids
A hyperbolic Kac-Moody Lie algebra in the E-series
… E6, E7, E8, E9, E10, , …
As U-duality group of 1d supergravity
is conjectured (West 01) to be the U-duality group (see there) of 11-dimensional supergravity compactified to 0 dimensions.
|supergravity gauge group (split real form)||T-duality group (via toroidal KK-compactification)||U-duality||maximal gauged supergravity|
|1|| S-duality||10d type IIB supergravity|
|SL O(1,1)||9d supergravity|
|SU(3) SU(2)||SL||8d supergravity|
|E9||2d supergravity||E8-equivariant elliptic cohomology|
(Hull-Townsend 94, table 1, table 2)
Fundamental representation and brane charges
The first fundamental representation of , usually denoted , is argued, since (West 04), to contain in its decomposition into representations of the representations in which the charges of the M-branes and other BPS states transform.
According to (Nicolai-Fischbacher 03, first three rows of table 2 on p. 72, West 04, Kleinschmidt-West 04) and concisely stated for instance in (Wess 11, (2.17)), the level decomposition of under starts out as so:
Here the -truncation happens to coincide with the bosonic body underlying the M-theory super Lie algebra and via the relation of that to BPS charges in 11-dimensional supergravity/M-theory, the direct summands here have been argued to naturally correspond to
Relation to supergravity
Literature discussing U-duality and in the context of exceptional generalized geometry of 11-dimensional supergravity.
Original articles include the following:
The observation that seems to neatly organite the structures in 11-dimensional supergravity/M-theory is due to
- Peter West, and M theory, Class. Quant. Grav., 18:4443–4460, 2001.
A precursor to West 01 is
as explained in (Henneaux-Julia-Levie 10).
Further developments include
Peter West, , ten forms and supergravity, JHEP0603:072,2006 (arXiv:hep-th/0511153)
Fabio Riccioni, Peter West, The origin of all maximal supergravities, JHEP 0707:063,2007 (arXiv:0705.0752)
Fabio Riccioni, Peter West, E(11)-extended spacetime and gauged supergravities, JHEP 0802:039,2008 (arXiv:0712.1795)
Fabio Riccioni, Duncan Steele, Peter West, The E(11) origin of all maximal supergravities - the hierarchy of field-strengths, JHEP 0909:095 (2009) (arXiv:0906.1177)
E. Bergshoeff, I. De Baetselier, T. Nutma, E(11) and the Embedding Tensor (arXiv:0705.1304, poster)
Discussion of the semidirect product of with its -representation, and arguments that the charges of the M-theory super Lie algebra and in fact further brane charges may be identified inside originate in
and was further explored in
Axel Kleinschmidt, Peter West, Representations of and the role of space-time, JHEP 0402 (2004) 033 (arXiv:hep-th/0312247)
Paul Cook, Peter West, Charge multiplets and masses for , JHEP 11 (2008) 091 (arXiv:0805.4451)
Peter West, origin of Brane charges and U-duality multiplets, JHEP 0408 (2004) 052 (arXiv:hep-th/0406150)
Peter West, Generalised geometry, eleven dimensions and (arXiv:1111.1642)
Relation to exceptional field theory is discussed in
Relation to Borcherds superalgebras is discussed in
Pierre Henry-Labordere, Bernard Julia, Louis Paulot, Borcherds symmetries in M-theory, JHEP 0204 (2002) 049 (arXiv:hep-th/0203070)
Marc Henneaux, Bernard Julia, Jérôme Levie, , Borcherds algebras and maximal supergravity (arxiv:1007.5241)
Jakob Palmkvist, Tensor hierarchies, Borcherds algebras and , JHEP 1202 (2012) 066 (arXiv:1110.4892)