4.6.4
N
= 8
• N
= 8
Supergravity multiplet
(
λ
=
−
2) :
(
−
2
,
−
3
2
8
,
−
1
28
,
−
1
2
56
,
0
70
,
1
2
56
,
1
28
,
3
2
8
,
2
)
.
This is TCP self conjugate.
Note that each of the supergravity multiplets contains
N
gravitinos, which are associated
to the gauge fields for local supersymmetry.
The detailed story of how this works is the
subject of
supergravity
.
When
N
>
8 any supermultiplet necessarily contains states with
26

helicity greater than two. Hence
N
>
8 is excluded for an interacting theory. As we will see,
the maximum number of conserved supercharges for an interacting theory in any spacetime
dimension is 32.
N
= 8 supergravity in four dimensions is maximally supersymmetric in this
sense. The values
N
= 3
,
5
,
6 are also possible, though they are of less interest. The value
N
= 7 is equivalent to
N
= 8 when one forms the supergravity multiplet as the
λ
=
−
3
/
2
multiplet plus its TCP conjugate.
4.7
R symmetries
R symmetries are transformations that act on the generators of the superalgebra so as
to preserve the algebra.
In other words, they belong to the automorphism group of the
superalgebra. In general, the supercharges
Q
A
α
transform nontrivially under R symmetries.
Thus, R-symmetry transformations
do not commute with susy transformations
. They may
or may not be true symmetries of a given susy field theory – depending on details. If they
aren’t, it would make more sense to call them R transformations rather than R symmetries,
but the latter is the more usual usage. Another way to finesse the issue is to speak of
R
charges
without specifying whether or not they are conserved.
4.7.1
Massive R symmetries
The
N
-extended massive
D
= 4 susy algebra – without central charges – has 4
N
super-
charges
Q
A
α
,
¯
Q
˙
αA
, which can be recast as a
Clifford Algebra
{
Γ
r
,
Γ
s
}
= 2
δ
rs
Γ
r
= (Γ
r
)
†
,
(4.38)
with 4
N
generators (
r, s
= 1
, . . . ,
4
N
).
The way to do this to take the oscillators
b
αA
,
b
†
αA
α
= 1
,
2
,
A
= 1
, . . . ,
N
introduced earlier and define Γ’s as double the “real” and
“imaginary” parts:
b
+
b
†
and
−
ib
+
ib
†
– giving 4
N
Γ’s.
As we already saw, there is
a representation of dim 2
2
N
.
This means that the Γ’s can be represented by 2
2
N
×
2
2
N
matrices. The Clifford algebra has a natural associated
SO
(4
N
) algebra generated by the
matrices
Γ
rs
≡
1
2
[Γ
r
,
Γ
s
]
r, s
= 1
,
2
, . . . ,
4
N
.
While the 2
2
N
Fock space states give an irrep of the susy algebra, they give a reducible
representation of
Spin
(4
N
). Specifically, the 2
2
N−
1
bosonic states and the 2
2
N −
1
fermionic
states each give spinorial irreps of
Spin
(4
N
).
The
SO
(4
N
) group has a natural action as an automorphism group of the susy algebra.
Specifically,
Γ
r
→
A
r
s
Γ
s
,
A
∈
SO
(4
N
)
(4.39)
preserves the Clifford algebra, and hence there is a corresponding action on the
Q
A
α
’s that
preserves the susy algebra.
However, this
SO
(4
N
) cannot be the symmetry of any field
27

theory – it contains (bosonic) generators that change spin and would violate the CM theorem.

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