2
α
2
α
= ηac b
,
(4.1.9)
and thus the lagrangian
L = 1 φ δ(M 2)[
b∂
Aa(
2
αβ
✷ − (Mα b)2] φ = 12
✷ Aa − ∂a∂bAb) .
(4.1.10)
Similarly, for the gauge transformation
Λα = i Λ α
1
b
α
i
→
δAa = a M ∂
∂
.
(4.1.11)
2
α
b Λα
= −12 aΛα
As a result of the δ(M
2
αβ ) acting on QαΛα, the only part of Λ which survives is
the part which is an overall singlet in the matrix indices and explicit α index: in this case, Λ α
α
i
= δi λ → δAa = −∂aλ. Note that the φQ2φ term can be written as
a (Qφ)2 term: This corresponds to subtracting out a “gauge-fixing” term from the
“gauge-fixed” lagrangian φ(✷ − M2)φ. (See the discussion of gauge fixing in sect.
4.4.)
For a massless antisymmetric tensor we start with A[ij) = (A[ab], Aaα, A(αβ)) ap-
pearing as
1
φ = −1 ij
A
=
[i
2
ji
,
ij
√
⊗ j)
(4.1.12)
A
A
2
(and similarly for |Λα ), and truncate to just A[ab]. Then, from (4.1.8),
1 MαcM d a b
= 1Mαc ηda
b
+ ηdb a
2
α
2
α
α
= ηda c b + ηdb a c
+ ηd(aηb)c 1 α
2
α
,
(4.1.13)
and we have
L = 1Aab(
∂
α
.
(4.1.14)
4
✷ Aab + ∂c∂[aAb]c) , δAab = 12 [aΛb]α
For a massless traceless symmetric tensor we start with h(ij] = (h(ab), hαb, h[αβ]) satisfying hii = haa + hαα = 0, appearing as
1
φ = 1 ij
h
=
(i
j]
,
(4.1.15)
2
ji
,
ij
√
S
S
2
and truncate to (h(ab), h[αβ]), where h[αβ] = 1C
2
αβ ηabh(ab), leaving just an uncon-
strained symmetric tensor. Then, using (4.1.13), as well as
1 MγaM b α β = 1Cαβ (a b)
2
γ
2
− ηab α β
,
(4.1.16)
4.1. OSp(1,1|2)
55
and using the condition hαα = −haa, we find
L = 1hab
hab∂
c + 1hc
ha
4
✷ hab − 12
b∂cha
2
c∂a∂bhab − 14
a✷ hbb
,
δh
α
ab = − 1 ∂
.
(4.1.17)
2
(aΛb)α
This is the linearized Einstein-Hilbert action for gravity.
The massive cases can be obtained by the dimensional reduction technique, as
in (2.2.9), since that’s how it was done for this entire procedure, from the light-cone Poincaré algebra down to (4.1.6). (For the string, the OSp generators are represented in terms of harmonic oscillators, and MαmM is cubic in those oscillators instead of quadratic, so the oscillator expressions for the generators don’t follow from dimensional reduction, and (4.1.6) must be used directly with the MαmM terms.) Techni-
cally, pm = m makes sense only for complex fields. However, at least for free theories, the resulting i’s that appear in the papm crossterms can be removed by appropriate
redefinitions for the complex fields, after which they can be chosen real. (See the discussion below (2.2.10).) For example, for the massive vector we replace Am → iAm
(and then take Am real) to obtain
φ = a Aa + i m Am + α Aα ,
φ = Aa a − iAm m + Aα α
.
(4.1.18)
The lagrangian and invariance then become
L = 1Aa[(
A
2
✷ − m2)Aa − ∂a∂bAb] + 12 m✷ Am + mAm∂aAa
= 1F 2
(mA
4
ab − 12
a + ∂aAm)2
,
δAa = −∂aλ , δAm = mλ .
(4.1.19)
This gives a Stueckelberg formalism for a massive vector.
Other examples reproduce all the special cases of higher-spin fields proposed ear-
lier [4.3] (as well as cases that hadn’t been obtained previously). For example, for totally symmetric tensors, the usual “double-tracelessness” condition is automatic: Starting from the light cone with a totally symmetric and traceless tensor (in transverse indices), extending i → (a, α) and restricting to Sp(2) singlets, directly gives a totally symmetric and traceless tensor (in D-dimensional indices) of the same rank, and one of rank 2 lower (but no lower than that, due to the total antisymmetry in
the Sp(2) indices).
The most important feature of the BRST method of deriving gauge-invariant
actions from light-cone (unitary) representations of the Poincaré group is that it
56
4. GENERAL GAUGE THEORIES
automatically includes exactly the right number of auxiliary fields to make the action local. In the case of Yang-Mills, the auxiliary field (A−) was obvious, since it results directly from adding just 2 commuting dimensions (and not 2 anticommuting) to the
light cone, i.e., from making D − 2-dimensional indices D-dimensional. Furthermore, the necessity of this field for locality doesn’t occur until interactions are included (see sect. 2.1). A less trivial example is the graviton: Naively, a traceless symmetric D-dimensional tensor would be enough, since this would automatically include the
analog of A−. However, the BRST method automatically includes the trace of this
tensor. In general, the extra auxiliary fields with anticommuting “ghost-valued”
Lorentz indices are necessary for gauge-covariant, local formulations of field theories
[4.4,5]. In order to study this phenomenon in more detail, and because the discussion will be useful later in the 2D case for strings, we now give a brief discussion of general relativity.
General relativity is the gauge theory of the Poincaré group. Since local trans-
lations (i.e., general coordinate transformations) include the orbital part of Lorentz transformations (as translation by an amount linear in x), we choose as the group
generators ∂m and the Lorentz spin Mab. Treating Mab as second-quantized opera-
tors, we indicate how they act by writing explicit “spin” vector indices a, b, . . . (or spinor indices) on the fields, while using m, n, . . . for “orbital” vector indices on which Mab doesn’t act, as on ∂m. (The action of the second-quantized Mab follows from that of the first-quantized: E.g., from (4.1.8), (2.2.5), and the fact that (Mij)† = −Mij, we have MabAc = −ηc[aAb].) The spin indices (but not the orbital ones) can be contracted with the usual constant tensors of the Lorentz group (the Lorentz metric and γ matrices). The (antihermitian) generators of gauge transformations are thus
λ = λm(x)∂m + 1λab(x)M
2
ba
,
(4.1.20)
and the covariant derivatives are
∇
m
bc
a = ea
∂m + 1ω
M
2
a
cb
,
(4.1.21)
where we have absorbed the usual derivative term, since derivatives are themselves
generators, and to make the covariant derivative transform covariantly under the
gauge transformations
∇ ′
a = eλ∇ae−λ
.
(4.1.22)
Covariant field strengths are defined, as usual, by commutators of covariant derivatives,
[∇
c
cd
a, ∇b] = T ab ∇c + 1 R
M
2
ab
dc
,
(4.1.23)
4.1. OSp(1,1|2)
57
since that automatically makes them transform covariantly (i.e., by a similarity transformation, as in (4.1.22)), as a consequence of the transformation law (4.1.22) of the covariant derivatives themselves. Without loss of generality, we can choose
T c
ab
= 0 ,
(4.1.24)
since this just determines ω c
m
ab in terms of ea
, and any other ω can always be written
as this ω plus a tensor that is a function of just T . (The theory could then always be rewritten in terms of the T = 0 ∇ and T itself, making T an arbitrary extra tensor
with no special geometric significance.) To solve this constraint we first define
e
m
a = ea
∂m ,
[e
c
a, eb] = cab ec
.
(4.1.25)
c c
m
a
ab
can then be expressed in terms of ea , the matrix inverse em ,
e m
b
b
a
n
n
a
em = δa
,
em ea = δm
,
(4.1.26)
and their derivatives. The solution to (4.1.24) is then
ωabc = 1(c
2
bca − ca[bc])
.
(4.1.27)
The usual global Lorentz transformations, which include orbital and spin pieces
together in a specific way, are a symmetry of the vacuum, defined by
∇
m
m
a
= κ∂a
↔
ea
= κδa
.
(4.1.28)
κ is an arbitrary constant, which we can choose to be a unit of length, so that
∇ is dimensionless. (In D = 4 it’s just the usual gravitational coupling constant,
proportional to the square root of Newton’s gravitational constant.) As a result of general coordinate invariance, any covariant object (i.e., a covariant derivative or tensor with only spin indices uncontracted) will then also be dimensionless. The
subgroup of the original gauge group which leaves the vacuum (4.1.28) invariant
is just the usual (global) Poincaré group, which treats orbital and spin indices in the same way. We can also treat these indices in a similar way with respect to
the full gauge group by using the “vielbein” e m
a
and its inverse to convert between
spin and orbital indices. In particular, the orbital indices on all fields except the vielbein itself can be converted into spin ones. Also, since integration measures are antisymmetric, converting dxm into Ωa = dxme a
m
converts dDx into ΩD = dDx e−1,
58
4. GENERAL GAUGE THEORIES
where e = det(e m
a
). On such covariant fields, ∇ always acts covariantly. On the
other hand, in the absence of spinors, all indices can be converted into orbital ones.
In particular, instead of the vielbein we could work with the metric tensor and its inverse:
g
a
b
m
n
mn = ηabem en
,
gmn = ηabea eb
.
(4.1.29)
Then, instead of ∇, we would need a covariant derivative which knows how to treat
uncontracted orbital indices covariantly.
The action for gravity can be written as
S = −1
dDx e−1 R ,
R = 1 R ba
.
(4.1.30)
2
2
ab
This can be rewritten in terms of cabc as
e−1R = −∂
b
m(e−1eamca b) + e−1 − 1 (cab
cabcc
cabcc
2
b)2 + 1
8
abc − 14
bca
(4.1.31)
using
e−1eaf a = ∂m(e−1eamf a) + e−1cabbf a .
(4.1.32)
Expanding about the vacuum,
e m
m
m
a
= κδa + κD/2ha
,
(4.1.33)
where we can choose eam (and thus ham) to be symmetric by the λab transformation,
the linearized action is just (4.1.17). As an alternative form for the action, we can consider making the field redefinition
e m
m
a
→ φ−2/(D−2)ea
,
(4.1.34)
which introduces the new gauge invariance of (Weyl) local scale transformations
e m′
m
a
= eζea
,
φ′ = e(D−2)ζ/2φ .
(4.1.35)
(The gauge choice φ = constant returns the original fields.) Under the field redefinition (4.1.34), the action (4.1.30) becomes
S →
dDx e−1 2 D−1(
Rφ2
.
(4.1.36)
D−2 ∇aφ)2 − 1
2
We have actually started from (4.1.30) without the total-derivative term of (4.1.31), which is then a function of just e m
a
and its first derivatives, and thus correct even at
boundaries. (We also dropped a total-derivative term −∂
b
m( 1 φ2e−1eamc
2
a b) in (4.1.36),
4.1. OSp(1,1|2)
59
which will be irrelevant for the following discussion.) If we eliminate φ by its field equation, but keep surface terms, this becomes
S →
dDx e−1 2 D−1
D−2 ∇ · (φ∇φ)
=
dDx e−1 D−1
D−2 ✷ φ2
=
dDx e−1 D−1
D−2 ✷
φ 2 + 2 φ (φ − φ ) + (φ − φ )2
(4.1.37)
We can solve the φ field equation as
1
φ = φ
1 −
R
.
(4.1.38)
4 D−1✷ + R
D−2
(We can choose φ = 1, or take the κ out of (4.1.28) and introduce it instead through φ = κ−(D−2)/2 by a global ζ transformation.) Assuming φ falls off to φ fast enough
at ∞, the last term in (4.1.37) can be dropped, and, using (4.1.38), the action becomes
[4.6]
1
S → −1
dDx e−1 R
R
.
(4.1.39)
2
− R4D−1✷ + R
D−2
Since this action has the invariance (4.1.35), we can gauge away the trace of h or, equivalently, gauge the determinant of e m
a
to 1. In fact, the same action results from
(4.1.30) if we eliminate this determinant by its equation of motion.
Thus, we see that, although gauge-covariant, Lorentz-covariant formulations are
possible without the extra auxiliary fields, they are nonlocal. Furthermore, the nonlocalities become more complicated when coupling to nonconformal matter (such as
massive fields), in a way reminiscent of Coulomb terms or the nonlocalities in lightcone gauges. Thus, the construction of actions in such a formalism is not straightforward, and requires the use of Weyl invariance in a way analogous to the use of Lorentz invariance in light-cone gauges. Another alternative is to eliminate the trace of the metric from the Einstein action by a coordinate choice, but the remaining constrained (volume-preserving) coordinate invariance causes difficulties in quantization [4.7].
We have also seen that some properties of gravity (the ones relating to conformal
transformations) become more transparent when the scale compensator φ is intro-
duced. (This is particularly true for supergravity.) Introducing such fields into the OSp formalism requires introducing new degrees of freedom, to make the representation larger (at least in terms of gauge degrees of freedom). Although such invariances are hard to recognize at the free level, the extensions of sect. 3.6 show signs of performing such generalizations. However, while the U(1)-type extension can be applied
60
4. GENERAL GAUGE THEORIES
to arbitrary Poincaré representations, the GL(1)-type has difficulty with fermions.
We’ll first discuss this difficulty, then show how the 2 types differ for bosons even for the vector, and finally look again at gravity.
The U(1) case of spin 1/2 reproduces the algebra of sect. 3.5, since MA′B′ of
(3.6.12b) is exactly the extra term of (3.5.1):
Mij = 1γ
2 ij
→
¯
φδ(M
2
αβ )(✷ + Q2)φ = ¯
φδ( )eiγ+′ /
p/2i1 /pγ
2
−′ eiγ+′ /
p/2φ
= i1 (γ
γ
γ
4
−′ φ + i 1
2 −′ γ+′ /
pφ)δ( )γ+′ /
p(γ−′φ + i12 −′γ+′ /pφ)
= i1 ˆ¯
φ/
pγ ˆ
φ ,
(4.1.40)
2
−′
where γij = 1[γ
2
i, γj }, and we have used
0 = 1(γαβ + γα′β′)(γ
8
αβ + γα′β′ ) = (γαγα′ )2 + 4
→
γαγα′ = 2i .
(4.1.41)
(We could equally well have chosen the other sign. This choice, with our conven-
tions, corresponds to harmonic-oscillator boundary conditions: See sect. 4.5.) After eliminating γ+′φ by gauge choice or, equivalently, by absorbing it into γ−′φ by field redefinition, this becomes just ¯
ϕ/
pϕ. However, in the GL(1) case, the analog to (4.1.41)
is
0 = 1(
8 {γα, γβ′ } + {γα′ , γβ })({γα, γβ′ } + {γα′ , γβ }) = (γαγα′ )(γβ′ γβ )
,
γαγα′ + γα′γα = −4 ,
(4.1.42)
and to (4.1.40) is
✷ + Q2 = −p2 + 1(γαγ
(γα′γ
8
α′ )γ−′ (/
p − γ+p2) + 18
α)(/
p − γ+p2)γ−′ .
(4.1.43)
Unfortunately, φ and ¯
φ must have opposite boundary conditions γαγα′ = 0 or γα′γα =
0 in order to contribute in the presence of δ(M
2
αβ ), as is evidenced by the asymmetric
form of (4.1.43) for either choice. Consequently, the parts of φ and ¯
φ that survive
are not hermitian conjugates of each other, and the action is not unitary. (Properly speaking, if we choose consistent boundary conditions for both φ and ¯
φ, the action
vanishes.) Thus, the GL(1)-type OSp(1,1|2) is unsuitable for spinors unless further modified. In any case, such a modification would not treat bosons and fermions
symmetrically, which is necessary for treating supersymmetry. (Fermions in the usual OSp formalism will be discussed in more detail in sect. 4.5.)
