nLab D=12 supergravity

Idea

There is a sensible theory of supergravity in a total of 12 spacetime dimensions. Even though this requires an exotic non-Lorentzian signature of $(10,2)$ (hence with a “2-dimensional time”) it has been argued that this is a better starting point for obtaining low-dimensional supergravity theory by KK-compactification, since it yields some lower-dimensional theories that are missed when starting with 11-dimensional supergravity, notably type IIB supergravity in 10 dimensions, hence relates to F-theory as 11-dimensional supergravity relates to M-theory (e.g. Nishino 97b, Hewson 97). (A theory in $(9,3)$ signature has also been proposed in (Kriz 05).)

It is an oft-repeated folklore that the highest number of spacetime dimensions for supergravity to make sense is 11, realized by 11-dimensional supergravity. However, there are some assumptions that go into this conclusion. First of all, the argument goes that after KK-compactification to 4-dimensions there must not appear supermultiplets with mass-less fields of spin $\gt 2$, since another folklore argument states that quantum field theory in $3+1$ dimensions with fields of spin larger than 2 is inconsistent.

(This in turn needs further qualification: Consistent quantum field theory with an infinite tower of higher spin fields is consistent, this is called higher spin gauge theory arising as the vanishing string tension-limit of string field theory. Ever since this discovery, the modified folklore is that field theories with a finite number of higher spin fields is inconsistent.)

Since acting with a supersymmetry generator on elements of a supermultiplet increases spin by 1/2, this argument requires that there are at most $(2 - (-2)) \times 2 = 8$ super charges in (3+1)d, hence corresponding to N=8 d=4 supergravity.

This, in turn, requires, by the rules of KK-compactification, that

1. there be only a single supercharge in dimension $10+1$, since the irreducible real spin representation of $Spin(10,1)$ has real dimension 32, which branches as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(10,1)$;

2. there cannot be any supercharge in dimension $11+1$, since the irreducible real spin representation of $Spin(11,1)$ has real dimension 64, which branches as $\mathbf{64} \mapsto 16 \cdot \mathbf{4}$ under $Spin(3,1) \hookrightarrow Spin(11,1)$.

However, the second conclusion here is evaded by a change of spacetime signature: The irreducible real spin representation of Spin(10,2) still happens to be of dimension 32 and still branches as $\mathbf{32} \mapsto 8 \cdot \mathbf{4}$.

Properties

The $2+1$-brane in $10+2$ dimensions

There is supposed to be a consistent fundamental super p-brane on $10+2$-dimensional supergravity backgrounds, whose double dimensional reduction yields the M2-brane in 11-dimensional supergravity and further the superstrings not just of type IIA supergravity but also (?) of type IIB supergravity. The worldvolume of this p-brane has 4 spacetime dimensions with signature $(2,2)$. Therefore some authors refer to this as a “2+2”-brane, even though this does not mesh well with the naming convention of $p$-branes in Lorentzian signature. Since Lorentzian $p$-branes have $(p+1)$-dimensional worldvolume, the systematic naming here would be “2+1”-brane.

See (Blencowe-Duff 88, section 7, Hewson-Perry 96, Nishino 97b)

Related concepts

• D=11 supergravity

• F-theory

• D=14 supersymmetry

• bosonic M-theory

References

General

• Leonardo Castellani, Pietro Fré, F. Giani, K. Pilch, Peter van Nieuwenhuizen, Beyond $d=11$ Supergravity and Cartan Integrable Systems, Phys.Rev. D26 (1982) 1481 (spire:11999)

• Itzhak Bars, Supersymmetry, p-brane duality and hidden space-time dimensions, Phys. Rev. D54, 5203 (1996) (arXiv: hep-th/9604139).

• Itzhak Bars, S-Theory, Phys.Rev. D55 (1997) 2373-2381 (arXiv:hep-th/9607112)

• Hitoshi Nishino, Supergravity in 10 + 2 Dimensions as Consistent Background for Superstring, (arXiv:hep-th/9703214)

• Hitoshi Nishino, N=2 Chiral Supergravity in (10 + 2)-Dimensions As Consistent Background for Super (2 + 2)-Brane, Phys. Lett. B437 (1998) 303-314 (arXiv:hep-th/9706148)

• Stephen Hewson, An approach to F-theory, Nucl. Phys. B534 (1998) 513-530 (arXiv:hep-th/9712017)

• Hitoshi Nishino, Supergravity Theories in $D \geq 12$ Coupled to Super p-Branes, Nucl.Phys. B542 (1999) 217-261 (arXiv:hep-th/9807199)

• Stephen Hewson, On supergravity in $(10,2)$ (arXiv:hep-th/9908209)

• Tatsuya Ueno, BPS States in 10+2 Dimensions, JHEP 0012:006, 2000 (arXiv:hep-th/9909007)

• Leonardo Castellani, A locally supersymmetric SO(10,2) invariant action for D=12 supergravity, (arXiv:1705.00638)

On the $2+1$-brane in $10+2$ dimensions

• Miles Blencowe, Mike Duff, Supermembranes and the Signature of Space-time, Nucl. Phys. B310 (1988) 387-404 (spire:262142, 10.1016/0550-3213(88)90155-1, pdf)

• S. F. Hewson, M. J. Perry, The twelve dimensional super $(2+2)$-brane, Nucl.Phys. B492 (1997) 249-277 (arXiv:hep-th/9612008)

• Nishino 97b

On supergravity in $9 + 3$ dimensions

• Igor Kriz, Some remarks on fundamental physical F-theory, (arXiv:hep-th/0508046)

In bosonic M-theory

On organizing all these variants inside bosonic M-theory and KK-reduction on Cayley planes to actual M-theory:

• Michael Rios, Alessio Marrani, David Chester, The Geometry of Exceptional Super Yang-Mills Theories, Phys. Rev. D 99 (2019) 046004 [arXiv:1811.06101, doi:10.1103/PhysRevD.99.046004]

• Michael Rios, Alessio Marrani, David Chester, Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory, Phys. Lett. B 808 (2020) 135674 [arXiv:1906.10709, doi:10.1016/j.physletb.2020.135674]

reviewed in:

• Alessio Marrani, Exceptional super Yang-Mills in $27 + 3$ and worldvolume M-theory, talk at M-Theory and mathematics 2023, NYU Abu Dhabi (Jan, 2023) [web]