D=12 supergravity





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics


Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)


in grand unified theory

minimally extended supersymmetric standard model




dark matter candidates


auxiliary fields



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)(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)(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 >2\gt 2, since another folklore argument states that quantum field theory in 3+13+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))×2=8(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+110+1, since the irreducible real spin representation of Spin(10,1)Spin(10,1) has real dimension 32, which branches as 3284\mathbf{32} \mapsto 8 \cdot \mathbf{4} under Spin(3,1)Spin(10,1)Spin(3,1) \hookrightarrow Spin(10,1);

  2. there cannot be any supercharge in dimension 11+111+1, since the irreducible real spin representation of Spin(11,1)Spin(11,1) has real dimension 64, which branches as 64164\mathbf{64} \mapsto 16 \cdot \mathbf{4} under Spin(3,1)Spin(11,1)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 3284\mathbf{32} \mapsto 8 \cdot \mathbf{4}.


The 2+12+1-brane in 10+210+2 dimensions

There is supposed to be a consistent fundamental super p-brane on 10+210+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)(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 pp-branes in Lorentzian signature. Since Lorentzian pp-branes have (p+1)(p+1)-dimensional worldvolume, the systematic naming here would be “2+1”-brane.

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



On the 2+12+1-brane in 10+210+2 dimensions

On supergravity in 9+39 + 3 dimensions

In bosonic M-theory

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

Last revised on April 6, 2021 at 03:28:52. See the history of this page for a list of all contributions to it.