![](https://csdnimg.cn/release/download_crawler_static/12302411/bg4.jpg)
as expected. One can similarly show that this algebra is
satisfied for the action of S
A
on the other two fields.
Therefore, the D ¼ 11 superparticle spectrum describes the
physical degrees of freedom of D ¼ 11 supergravity.
III. D = 11 PURE SPINOR SUPERPARTICLE
As for the D ¼ 10 case [4], we will obtain the D ¼ 11
pure spinor superparticle from the gauge-fixed Brink-
Schwarz-like superparticle (7) by introducing a new set
of variables ðΘ
α
;P
α
Þ and a new symmetry coming from the
following first-class constraints,
ˆ
D
α
¼ D
α
þ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffi
2
p
P
þ
p
ðΓ
m
Γ
þ
SÞ
α
P
m
; ð17Þ
where fS
A
;S
B
g¼δ
AB
and D
α
¼ P
α
þ Γ
m
αβ
Θ
β
P
m
. Using
the relation fΘ
α
;P
β
g¼iδ
α
β
, one can show that
fD
α
;D
β
g¼2iðΓ
m
Þ
αβ
P
m
. Let us check that these ones
are indeed first-class constraints:
f
ˆ
D
α
;
ˆ
D
β
g¼fD
α
;D
β
gþ
1
ffiffiffi
2
p
P
þ
ðΓ
m
Γ
þ
Þ
α
A
ðΓ
n
Γ
þ
Þ
β
A
P
m
P
n
¼ 2iðΓ
m
Þ
αβ
P
m
−
ffiffiffi
2
p
i
ffiffiffi
2
p
P
þ
Γ
m
αλ
Γ
n
βδ
Γ
þλδ
P
m
P
n
¼ 2iðΓ
m
Þ
αβ
P
m
−
i
P
þ
ðΓ
m
Γ
þ
Γ
n
Þ
αβ
P
m
P
n
: ð18Þ
Since Γ
m
Γ
þ
¼ −Γ
þ
Γ
m
þ 2η
mþ
, we obtain
f
ˆ
D
α
;
ˆ
D
β
g¼2iðΓ
m
Þ
αβ
P
m
−
i
P
þ
ð2η
þm
Γ
n
αβ
ÞP
m
P
n
þ
i
P
þ
Γ
þ
αβ
P
2
¼
i
P
þ
Γ
þ
αβ
P
2
: ð19Þ
Thus, the modified Brink-Schwarz-like action will be
S ¼
Z
dτ
˙
X
m
P
m
−
i
2
˙
S
A
S
A
þ eP
m
P
m
þ
˙
Θ
α
P
α
þ f
α
ˆ
D
α
;
ð20Þ
where we have added the usual kinetic term for the
variables ðΘ
α
;P
α
Þ and the last term takes into account
the new constraint through the fermionic Lagrange multi-
plier f
α
. The standard BRST method gives us the following
gauge-fixed action,
S¼
Z
dτ
˙
X
m
P
m
−
i
2
˙
S
A
S
A
−
1
2
P
m
P
m
þ
˙
Θ
α
P
α
þ
˙
cbþ
˙
ˆ
Λ
α
ˆ
W
α
;
ð21Þ
and the BRST operator
ˆ
Q ¼
ˆ
Λ
α
ˆ
D
α
þ cP
m
P
m
−
i
2P
þ
ð
ˆ
ΛΓ
þ
ˆ
ΛÞb; ð22Þ
once we choose the gauge e ¼ −
1
2
and f
α
¼ 0. The ghosts
c and
ˆ
Λ
α
come from gauge fixing the reparametrization
symmetry and the new fermionic symmetry, respectively.
Now, we will show that the cohomology of the BRST
operator
ˆ
Q is equivalent to the cohomology of a BRST
operator Q ¼ Λ
α
D
α
, where Λ
α
is a pure spinor. We will
show this claim in two steps. First, we show that the
ˆ
Q-cohomology is equivalent to Q
0
-cohomology, where
Q
0
¼ Λ
0α
ˆ
D
α
and Λ
0
Γ
þ
Λ
0
¼ 0. Finally, we will prove that
the Q
0
-cohomology is equivalent to the Q-cohomology.
Let us start by defining the operator Q
0
¼ Λ
α
0
ˆ
D
α
. Notice
that when Λ
α
0
is equal to
ˆ
Λ
α
or Λ
0α
, Q
0
becomes the first
term of
ˆ
Q or Q
0
, respectively. Now, let V be a state such that
Q
0
V ¼ðΛ
0
Γ
þ
Λ
0
ÞW, for some W. Because of the property
that Λ
0α
satisfies, V is annihilated by Q
0
. Also, using (19),
we find that ðQ
0
Þ
2
¼
i
2P
þ
P
m
P
m
ðΛ
0
Γ
þ
Λ
0
Þ. So, we con-
clude that Q
0
W ¼
i
2P
þ
P
m
P
m
V. We can then show that the
state
ˆ
V ¼ V − 2iP
þ
cW is annihilated by
ˆ
Q,
ˆ
Q
ˆ
V ¼
ˆ
QðV − 2iP
þ
cWÞ
¼
ˆ
QV − 2iP
þ
ð
ˆ
QcÞW þ 2iP
þ
cð
ˆ
QWÞ
¼ð
ˆ
ΛΓ
þ
ˆ
ΛÞW þ cP
m
P
m
V − 2iP
þ
−
i
2P
þ
ð
ˆ
ΛΓ
þ
ˆ
ΛÞW
þ 2iP
þ
c
i
2P
þ
P
m
P
m
V
¼ð
ˆ
ΛΓ
þ
ˆ
ΛÞW þ cP
m
P
m
V − ð
ˆ
ΛΓ
þ
ˆ
ΛÞW − cP
m
P
m
V
¼ 0; ð23Þ
where we have assumed that b annihilates physical states.
Now, let us show that if a state V is BRST trivial (in the Q
0
-
cohomology) we can find a state
ˆ
V ¼ V − 2iP
þ
cW which
is also BRST trivial (in the
ˆ
Q-cohomology). Let V be a
state which satisfies V ¼ Q
0
Ω þðΛ
0
Γ
þ
Λ
0
ÞY, for some Y.
It is clear that if Λ
α
0
¼ Λ
0α
we have that V is Q
0
exact and if
Λ
α
0
¼
ˆ
Λ
α
we have that the first term of
ˆ
Q is equal to
V − ð
ˆ
ΛΓ
þ
ˆ
ΛÞY. So, we see that
ˆ
QðΩ þ 2iP
þ
cYÞ¼
ˆ
QΩ þ 2iP
þ
ð
ˆ
QcÞY − 2iP
þ
cð
ˆ
QYÞ
¼ V − ð
ˆ
ΛΓ
þ
ˆ
ΛÞY þ cP
m
P
m
þ 2iP
þ
−
i
2P
þ
ð
ˆ
ΛΓ
þ
ˆ
ΛÞY
− 2iP
þ
c
W −
i
2P
þ
P
m
P
m
Ω
; ð24Þ
MAX GUILLEN PHYS. REV. D 97, 066002 (2018)
066002-4