# nLab D=12 supergravity

Contents

### Context

#### Gravity

gravity, supergravity

supersymmetry

## Applications

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## 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.

## References

### 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.