5.4. Supersymmetric
99
where “dim-0” are the generators which commute with dilatations (see sect. 2.2), so the last column gives Lorentz⊗internal symmetry (at least). ν is the dimension of
the (irreducible) Lorentz spinor (1/2 that of the conformal spinor), N is the number of minimal supersymmetries, and D(>2) is the dimension of (Minkowski) spacetime
(with conformal group SO(D,2), so 2 less than the D in the previous table). The 2
choices for twice-odd D depend on whether we choose to represent the superconformal group on both primed and unprimed spinors. If so, there can be a separate N and
N′. Again, we have used * to indicate groups which are Wick rotations such that
the defining representation is pseudoreal instead of real. (SL* is sometimes denoted SU*.)
Unfortunately, as discussed in the previous section, the bosonic subgroup acting
on the conformal spinor part of the defining representation, as defined by the spinor metrics (plus the trace condition, when relevant) gives a group bigger than the conformal group unless D ≤ 6. However, we can still use D ≤ 6, and perhaps some of the qualitative features of D > 6, for our analysis of massless field equations. We then generalize our analysis of sect. 2.2 from conformal to superconformal. It’s sufficient to apply just the S-supersymmetry generators to just the Klein-Gordon operator. We
then find [2.6,5.19]:
1
1 p2
2 {pb , J ab} + 1
2 {pa , ∆} = pbM ab + pa
d − D−2
2
, (5.4.11)
2
→ /pq = /pd → q[αqβ] + ··· = d[αdβ] + ···
where the last expression means certain Lorentz pieces of dd plus certain terms containing Lorentz and internal symmetry generators. In particular, 1 γ
αβ d
16 abc
αdβ +
1 p
2 [aM bc] is the supersymmetric analog of the Pauli-Lubansky vector [5.20]. The vector equation is (2.2.8) again, derived in essentially the same way.
For the constraints we therefore choose [5.21]
A = 1p2 ,
2
Bα = γaαβpadβ ,
C
αβ
abc = 1 γ
d
p
16 abc
αdβ + 1
2 [aM bc]
,
D
b
a = M a pb + kpa
;
(5.4.12a)
or, in matrix notation,
A = 1p2 ,
2
B = /pd ,
100
5. PARTICLE
Cabc = 1 dγ
p
16
abcd + 1
2 [aM bc]
,
D
b
a = M a pb + kpa
;
(5.4.12b)
where out of (5.4.11) we have chosen A and D as for nonsupersymmetric theories
(sect. 2.2), B for unitarity (as explained above), and just the Pauli-Lubansky part of the rest (which is all of it for D=10), the significance of which will be explained below.
These constraints satisfy the algebra
{B, B} = 4/pA ,
[Da, Db] = −2MabA − p[aDb] ,
[Cabc, B] = −8γabcdA ,
[Cabc, Dd] = ηd[apbDc] ,
[Cabc, C
[a
[a
b
def ] = − 1 [δ
p
d(4γabc
δ γc]
4
[d
eCbc]f] − (abc ↔ def )] − 1
128
def − 2δ[d
e
f ])B
,
rest = 0 ,
(5.4.13)
with some ambiguity in how the right-hand side is expressed due to the relations
p · D = 2kA ,
/
pB = 2dA ,
dB = 2(tr I)A ,
1 p
dγ
6 [aCbcd] = − 1
16
abcdB
,
pcCcab = 2MabA + p[aDb] + 1 dγ
16
abB
.
(5.4.14)
(γab = 1γ
2 [aγb], etc.)
In the case of supersymmetry with an internal symmetry group (extended su-
persymmetry, or even simple supersymmetry in D=5,6,7), there is an additional con-
straint analogous to Cabc for superisospin:
Ca,int = 1dγ
8
aσintd + paM int
.
(5.4.15)
σint are the matrix generators of the internal symmetry group, in the representation to which d belongs, and Mint are those which act on the external indices of the
superfield.
5.4. Supersymmetric
101
Unfortunately, there are few superspin-0 multiplets that are contained within
spin-0, isospin-0 superfields (i.e., that themselves contain spin-0, isospin-0 component fields). In fact, the only such multiplets of physical interest in D>4 are N=1
Yang-Mills in D=9 and N=2 nonchiral supergravity in D=10. (For a convenient listing of multiplets, see [5.22].) However, by the method described in the previous section, spinor representations for the Lorentz group can be introduced. By including “γ-
matrices” for internal symmetry, we can also introduce defining representations for the internal symmetry groups for which they are equivalent to the spinor representations of orthogonal groups (i.e., SU(2)=USp(2)=SO(3), USp(4)=SO(5), SU(4)=SO(6),
SO(4)-vector=SO(3)-spinor⊗SO(3)′-spinor, SO(8)-vector=SO(8)′-spinor). Further-
more, arbitrary U(1) representations can be described by adding extra terms without introducing additional coordinates. This allows the description of most superspin-0 multiplets, but with some notable exceptions (e.g., 11D supergravity). However,
these equations are not easily generalized to nonzero superspins, since, although the superspin operator is easy to identify in the light-cone formalism (see below), the corresponding operator would be nonlocal in a covariant description (or appear always
with an additional factor of momentum).
We next consider the construction of mechanics actions. These equations describe
only multiplets of superspin 0, i.e., the smallest representations of a given supersymmetry algebra, for reasons to be described below. (This is no restriction in D=3,
where superspin doesn’t exist, and in D=4 arbitrary superspin can be treated by a
minor modification, since there superspin is abelian.) As described in the previous section, only spin-0 and spin-1/2 superfields can be described by classical mechanics, and we begin with spin-0, dropping spin terms in (5.4.12), and the generator D. The action is then given by (3.1.10), where [5.23]
zM = (xm, θµ) ,
πA = (pa, idα) ,
.
zM e A
M
(z) = (.x − i.θγθ, .θ) ,
iGi(π) = (A, Bα, Cabc) .
(5.4.16)
Upon quantization, the covariant derivatives become
π
M
A = ieA
∂M = i(∂a, ∂α + iγaαβθβ∂a) ,
(5.4.17)
which are invariant under the supersymmetry transformations
δx = ξ − iǫγθ , δθ = ǫ .
(5.4.18)
102
5. PARTICLE
The transformation laws then follow directly from (3.1.11), with the aid of (5.4.13) for the λ transformations.
The classical mechanics action can be quantized covariantly by BRST methods.
In particular, the transformations generated by B [5.24] (with parameter κ) close on those generated by A (with parameter ξ):
δx = ξp + iκ(γd + /
pγθ) ,
δθ = /
pκ ,
δp = 0 ,
δd = 2p2κ ,
δg = .ξ + 4iκ/
pψ
,
δψ = .κ .
(5.4.19)
Because of the second line of (5.4.14), the ghosts have a gauge invariance similar to the original κ invariance, and then the ghosts of those ghosts again have such an
invariance, etc., ad infinitum. This is a consequence of the fact that only half of θ
can be gauged away, but there is generally no Lorentz representation with half the
components of a spinor, so the spinor gauge parameter must itself be half gauge, etc.
Although somewhat awkward, the infinite set of ghosts is straightforward to find.
Furthermore, if derived from the light cone, the OSp(1,1|2) generators automatically contain this infinite number of spinors: There, θ is first-quantized in the same way as the Dirac spinor was second-quantized in sect. 3.5, and θ obtains an infinite number of components (as an infinite number of ordinary spinors) as a result of being a
representation of a graded Clifford algebra (specifically, the Heisenberg algebras of γα and ˜
γα). This analysis will be made in the next section.
On the other hand, the analysis of the constraints is simplest in the light-cone
formalism. The A, B, and D equations can be solved directly, because they are all of the form p · f = p+f− + · · · :
1
A = 0
→
p
2
− = −
p
,
2p
i
+
1
B = 0
→
γ−d = −
γ
2p
ipiγ−γ+d
,
+
1
D = 0
→
M
j
−i =
(M p
p
i
j + kpi)
,
M−+ = k ,
(5.4.20a)
+
where we have chosen the corresponding gauges
x+ = 0 ,
γ+θ = 0 ,
M+i = 0 .
(5.4.20b)
5.4. Supersymmetric
103
These solutions restrict the x’s, θ’s, and Lorentz indices, respectively, to those of the light cone. (Effectively, D is reduced by 2, except that p+ remains.)
However, a superfield which is a function of a light-cone θ is not an irreducible
representation of supersymmetry (except sometimes in D≤4), although it is a unitary one. In fact, C is just the superspin operator which separates the representations: Due to the other constraints, all its components are linearly related to
C+ij = p+Mij + 1 dγ
16
+γij d
.
(5.4.21)
Up to a factor of p+, this is the light-cone superspin: On an irreducible representation of supersymmetry, it acts as an irreducible representation of SO(D-2). In D=4 this
can be seen easily by noting that the irreducible representations can be represented in terms of chiral superfields ( ¯
d = 0) with different numbers of d’s acting on them,
and the d ¯
d in C just counts the numbers of d’s. In general, if we note that the full
light-cone Lorentz generator can be written as
∂
Jij = −ix[ipj] + 1θγ
+ M
2
ij ∂θ
ij
1
1
= −ix[ipj] −
qγ
dγ
16p
+γij q +
+γij d + M ij
+
16p+
1
= ˆ
Jij +
C
p
+ij
,
(5.4.22)
+
then, by expressing any state in terms of q’s acting on the Clifford vacuum, we see that ˆ
J gives the correct transformation for those q’s and the x-dependence of the Clifford vacuum, so C/p+ gives the spin of the Clifford vacuum less the contribution of the
qq term on it, i.e., the superspin. Unfortunately, the mechanics action can’t handle spin operators for irreducible representations (either for Mij or the superspin), so we must restrict ourselves not only to spin 0 (referring to the external indices on the superfield), but also superspin 0 (at least at the classical mechanics level). Thus, the remaining constraint C = 0 is the only possible (first-class) constraint which can make the supersymmetry representation irreducible. The constraints (5.4.12) are therefore necessary and sufficient for deriving the mechanics action. However, if we allow the trivial kind of second-class constraints that can be solved in terms of matrices, we can generalize to spin-1/2. In principle, we could also do superspin-1/2, but this leads to covariant fields which are just those for superspin-0 with an extra spinor index tagged on, which differs by factors of momentum (with appropriate index contractions) from the desired expressions. (Thus, the superspin operator would be nonlocal on the
104
5. PARTICLE
latter.) Isospin-1/2 can be treated similarly (and superisospin-1/2, but again as a tagged-on index).
The simplest nontrivial example is D=4. (In D=3, C+ij vanishes, since the trans-
verse index i takes only one value.) There light-cone spinors have only 1 (complex) component, and so does C+ij. For this case, we can (and must, for an odd number N
of supersymmetries) modify Cabc:
Cabc = 1 dγ
p
16
abcd + 1
2 [aM bc] + iH ǫabcdpd
,
(5.4.23)
where H is the “superhelicity.” We then find
1
C+ij = p+ǫij M + iH − i
[d
4p
a, ¯
da]
,
(5.4.24)
+
where M
b
ij = ǫij M and {da, ¯
db} = p+δa , and a is an SU(N) index. The “helicity”
h is given by M = −ih, and we then find by expanding the field over chiral fields φ
[5.25-27] ( ¯
dφ = 0)
ψ ∼ (d)nφ
→
H = h + 1(2n
4
− N) .
(5.4.25)
Specifying both the spin and superhelicity of the original superfield fixes both h and H, and thus determines n. Note that this requires H to be quarter-(odd-)integral for odd N. In general, the SU(N) representation of φ also needs to be specified, and the relevant part of (5.4.15) is
1
C b
b
b
+a
= p+ Ma − i
[d
δ [d
4p
a, ¯
db] − 14 a
c, ¯
dc]
,
(5.4.26)
+
and the vanishing of this quantity forces φ to be an SU(N)-singlet. (More general
cases can be obtained simply by tacking extra indices onto the original superfield, and thus onto φ.)
We next consider 10D super Yang-Mills. The appropriate superfield is a Weyl
or Majorana spinor, so we include terms as in the previous section in the mechanics action. To solve the remaining constraint, we first decompose SO(9,1) covariant
spinors and γ-matrices to SO(8) light-cone ones as
d+
dα = 21/4
,
d−
√
√
2p+
/
pT
Σ
0
Σ
0
2p−
/
pT
/
pαβ =
√
,
/
pαβ =
√
,
/
p †
†
T
− 2p−
0
Σ
0
Σ
/
pT
− 2p+
5.4. Supersymmetric
105
/
a
†
†
†
†
T /
bT + /
bT /
aT = /
aT /
bT + /
bT /
aT = 2aT · bT ,
/
pT * = Σ/
pT Σ ,
Σ = Σ† = Σ* ,
Σ2 = I
.
(5.4.27)
(We could choose the Majorana representation Σ = I, but other representations can
be more convenient.) The independent supersymmetry-covariant derivatives are then
∂
d+ =
+ p
∂θ+
+Σθ+
,
pT
,
p+ .
(5.4.28)
In order to introduce chiral light-cone superfields, we further reduce SO(8) to
SO(6)⊗SO(2)=U(4) notation:
√
d
b
a
pab
δa pL
0 I
d+ =
2
¯
,
/
pT =
,
Σ =
da
δ a
b ¯
pL
¯
pab
I
0
(¯
pab = 1 ǫabcdp
2
cd)
.
(5.4.29)
In terms of this “euphoric” notation, the constraints C+ij are written on the SO(6)-
spinor superfield as
1
ψ
b
b
c
δc ψa − 1δa ψc
1
ψc
C b
= i1
4
− i
[d
δ b[d
,
p
+a
¯
2
c ¯
b ¯
a, ¯
db] − 14 a
d, ¯
dd]
¯
+
ψc
−δ
4p
a ψb + 1 δ
ψc
ψc
4 a
+
1
ψ
−ψ
1
ψ
C
c
c
c
= i1
− i
[d
,
p
+
¯
2
¯
a, ¯
da]
¯
+
ψc
ψc
4p+
ψc
1
ψ
¯
c
ǫabcdψd
1
ψc
C
= i1
− i
d
,
(5.4.30)
p
+ab
¯
2
adb
¯
+
ψc
0
2p+
ψc
and the complex conjugate equation for C ab
+
. (Note that it is crucial that the original
SO(10) superfield ψα was a spinor of chirality opposite to that of dα in order to obtain soluble equations.) The solution to the first 2 equations gives ψ in terms of a chiral superfield φ,
ψa = daφ ,
(5.4.31a)
and that to the third equation imposes the self-duality condition [5.25-27]
1 ǫabcdd
2 ¯
φ .
(5.4.31b)
24
adbdcddφ = p+
This can also be written as
1 p
−1/2d φ = ¯
φ
2 +
→
dθ eθaπap+/2φ(θa) = [φ(¯
πa)] * .
(5.4.32)
This corresponds to the fact that in the mechanics action τ reversal on θα includes multiplication by the charge conjugation matrix, which switches θa with ¯
θa, which
equals −(2/p+)∂/∂θa by the chirality condition ¯
da = 0.
106
5. PARTICLE
These results are equivalent to those obtained from first-quantization of a me-
chanics action with dα = 0 as a second-class constraint [5.28]. (This is the analog of the constraint ˆ
γ = 0 of the previous section.) This is effectively the same as drop-
ping the d terms from the action, which can then be written in second-order form by eliminating p by its equation of motion. This can be solved either by using a chiral superfield [5.27] as a solution to this constraint in a Gupta-Bleuler formalism,
¯
daφ = 0
→
d ¯
aφ = 0
→
¯
φd+φ = 0