=
dxdc 1 Φ†KΦ ,
(4.4.1)
2
for some operator O, where the first Q, appearing in the covariant bracket, is understood to be the second-quantized one. In order to get ✷ −M2 as the kinetic operator for part of Φ, we choose
∂
∂
O = − c,
→
K = c(✷ − M2) − 2
M+ .
(4.4.2)
∂c
∂c
When expanding the field in c, ✷ −M2 is the kinetic operator for the piece containing all physical and ghost fields. Explicitly, (3.4.3b), when substituted into the lagrangian L = 1Φ†KΦ and integrated over c, gives
2
∂ L = 1φ†(✷ − M2)φ + ψ†M+ψ ,
(4.4.3)
∂c
2
and in the BRST transformations δΦ = iǫQΦ gives
δφ = iǫ(Q+φ − M+ψ) , δψ = iǫ Q+ψ − 1(
2 ✷ − M 2)φ)
.
(4.4.4)
φ contains propagating fields and ψ contains BRST auxiliary fields. Although the
propagating fields are completely gauge-fixed, the BRST auxiliary fields have the
gauge invariance
δψ = λ ,
M+λ = 0 .
(4.4.5)
The simplest case is the scalar Φ = ϕ(x). In this case, all of ψ can be gauged away by (4.4.5), since M+ = 0. The lagrangian is just 1ϕ(
2
✷ − m2)ϕ. For the massless
vector (cf. (4.2.4b)),
Φ = i Ai + ic − B ,
(4.4.6)
70
4. GENERAL GAUGE THEORIES
where we have again used (4.4.5). By comparing (4.4.3) with (4.2.3), we see that the φ2 term is extended to all J3, the ψ2 term has the opposite sign, and the crossterm is dropped. We thus find
L = 1(Ai)†
Aa
2
✷ Ai + 2B2 = 12
✷ Aa − iC✷ C + 2B2 ,
(4.4.7)
where we have written Aα = i(C, C), due to (3.4.4,13). This agrees with the result
(3.2.11) in the gauge ζ = 1, where this B = 1B. The BRST transformations (4.4.4),
2
using (4.2.5), are
δAa = iǫ∂aC ,
δC = 0 ,
δC = ǫ(2B − ∂ · A) ,
δB = i1ǫ
2 ✷ C
,
(4.4.8)
which agrees with the linearized case of (3.2.8).
We next prove the equivalence of this form of gauge fixing with the usual ap-
proach, described in sect. 3.2 [4.1] (as we have just proven for the case of the massless vector). The steps are: (1) Add terms to the original BRST auxiliary fields, which
vanish on shell, to make them BRST invariant, as they are in the usual BRST for-
mulation of field theory. (In Yang-Mills, this is the redefinition B → B in (3.2.11).) (2) Use the BRST transformations to identify the physical fields (which may include auxiliary components). We can then reobtain the gauge-invariant action by dropping
all other fields from the lagrangian, with the gauge transformations given by replacing the ghosts in the BRST transformations by gauge parameters.
In the lagrangian (4.4.3) only the part of the BRST auxiliary field ψ which appears in M+ψ occurs in the action; the rest of ψ is pure gauge and drops out of the action.
Thus, we only require that the shifted M+ψ be BRST-invariant:
ψ = ψ + Aφ ,
δM+ψ = 0 .
(4.4.9)
Using the BRST transformations (4.4.4) and the identities (from (3.4.3b))
Q2 = 0
→
[✷ − M2, M+] = [✷ − M2, Q+] = [M+, Q+] = 0 ,
Q+2 = 1(
2 ✷ − M 2)M +
,
(4.4.10)
we obtain the conditions on A:
(Q+ − M+A)Q+ = (Q+ − M+A)M+ = 0 .
(4.4.11)
4.4. Gauge fixing
71
The solution to these equations is
A = M+−1Q+ .
(4.4.12)
Performing the shift (4.4.9), the gauge-fixed lagrangian takes the form
∂ L = 1φ† ˆKφ + 2ψ†(1 − δ
∂c
2
M 3,−2T )Q+φ + ψ†M +ψ
,
(4.4.13a)
ˆ
K = ✷ − M2 − 2Q+M+−1Q+ ,
(4.4.13b)
where T is the “isospin,” as in (4.2.11), for Mαβ. The BRST transformations can
now be written as
δδM3,−2T φ = ǫδM3,−2T Q+φ , δ(1 − δM3,−2T )φ = −ǫM+ψ ,
δψ = ǫδ
ˆ
M 3,2T (Q+ψ − 1 Kφ)
.
(4.4.14)
2
The BRST transformation of ψ is pure gauge, and can be dropped. (In some of
the manipulations we have used the fact that Q, Q+, and ✷ − M2 are symmetric,
i.e., even under integration by parts, while M+ is antisymmetric, and Q and Q are
antihermitian while M+ and ✷ − M2 are hermitian. In a coordinate representa-
tion, particularly for c, all symmetry generators, such as Q, Q+, and M+, would be
antisymmetric, since the fields would be real.)
We can now throw away the BRST-invariant BRST auxiliary fields ψ, but we
must also separate the ghost fields in φ from the physical ones. According to the
usual BRST procedure, the physical modes of a theory are those which are both
BRST-invariant and have vanishing ghost number (as well as satisfy the field equa-
tions). In particular, physical fields may transform into ghosts (corresponding to
gauge transformations, since the gauge pieces are unphysical), but never transform
into BRST auxiliary fields. Therefore, from (4.4.14) we must require that the phys-
ical fields have M3 = −2T to avoid transforming into BRST auxiliary fields, but we
also require vanishing ghost number M3 = 0. Hence, the physical fields (located in
φ) are selected by requiring the simple condition of vanishing isospin T = 0. If we project out the ghosts with the projection operator δT0 and use the identity (4.2.8), we obtain a lagrangian containing only physical fields:
L1 = 1φ†(
2
✷ − M2 + Q2)δT0φ .
(4.4.15)
Its gauge invariance is obtained from the BRST transformations by replacing the
ghosts (the part of φ appearing on the right-hand side of the transformation law)
72
4. GENERAL GAUGE THEORIES
with the gauge parameter (the reverse of the usual BRST quantization procedure),
and we add gauge transformations to gauge away the part of φ with T = 0:
δφ = −i1
M
2 QαΛα + 1
2
αβ Λαβ
,
(4.4.16)
where we have obtained the Q− term from closure of Q+ with Mαβ. The invariance
of (4.4.15) under (4.4.16) can be verified using the above identities. This action and invariance are just the original ones of the OSp(1,1|2) gauge-invariant formalism (or the IGL(1) one, after eliminating the NL auxiliary fields). The gauge-fixing functions for the Λ transformations are also given by the BRST transformations: They are the
transformations of ghosts into physical fields:
F GF = Q+δT0φ = pa(M+aδT0φ) + M(M+mδT0φ) .
(4.4.17)
(The first term is the usual Lorentz-gauge gauge-fixing function for massless fields, the second term the usual addition for Stueckelberg/Higgs fields.) The gauge-fixed
lagrangian (physical fields only) is thus
L
†
GF = L1 − 1 F
M−F
φ†(
4
GF
GF = 1
2
✷ − M2)δT0φ ,
(4.4.18)
in agreement with (4.4.3).
In summary, we see that this first-quantized gauge-fixing procedure is identical
to the second-quantized one with regard to (1) the physical gauge fields, their gauge transformations, and the gauge-invariant action, (2) the BRST transformations of the physical fields, (3) the closure of the BRST algebra, (4) the BRST invariance of the gauge-fixed action, and (5) the invertibility of the kinetic operator after elimination of the NL fields. (1) implies that the two theories are physically the same, (2) and (3) imply that the BRST operators are the same, up to additional modes as in sect.
4.3, (4) implies that both are correctly gauge fixed (but perhaps in different gauges), and (5) implies that all gauge invariances have been fixed, including those of ghosts.
Concerning the extra modes, from the ✷ −M2 form of the gauge-fixed kinetic operator we see that they are exactly the ones necessary to give good high-energy behavior of the propagator, and that we have chosen a generalized Fermi-Feynman gauge. Also,
note the fact that the c = 0 (or xα = 0) part of the field contains exactly the right set of ghost fields, as was manifest by the arguments of sect. 2.6, whereas in the usual second-quantized formalism one begins with just the physical fields manifest, and the ghosts and their ghosts, etc., must be found by a step-by-step procedure. Thus we
see that the OSp from the light cone not only gives a straightforward way for deriving
4.4. Gauge fixing
73
general free gauge-invariant actions, but also gives a method for gauge fixing which is equivalent to, but more direct than, the usual methods.
We now consider gauge fixing to the light cone. In this gauge the gauge theory
reduces back to the original light-cone theory from which it was heuristically obtained by adding 2+2 dimensions in sects. 3.4, 4.1. This proves a general “no-ghost theorem,” that the OSp(1,1|2) (and IGL(1)) gauge theory is equivalent on-shell to the
corresponding light-cone theory, for any Poincaré representation (including strings as a special case).
Consider an arbitrary bosonic gauge field theory, with action (4.1.6). (Fermions
will be considered in the following section.) Without loss of generality, we can choose M2 = 0, since the massive action can be obtained by dimensional reduction. The
light-cone gauge is then described by the gauge-fixed field equations
p2φ = 0
(4.4.19a)
subject to the gauge conditions, in the Lorentz frame p
+
a = δa p+,
Mαβφ = 0 ,
Qαφ = Mα−p+φ = 0 ,
(4.4.19b)
with the residual part of the gauge invariance
δφ = −i12QαΛα ∼ Mα−Λα ,
(4.4.19c)
where ± now refer to the usual “longitudinal” Lorentz indices. (The light-cone gauge is thus a further gauge-fixing of the Landau gauge, which uses only (4.4.19ab).)
(4.4.19bc) imply that the only surviving fields are singlets of the new OSp(1,1|2)
algebra generated by Mαβ, Mα±, M−+ (with longitudinal Lorentz indices ±): i.e.,
those which satisfy MABφ = 0 and can’t be gauged away by δφ = MBAΛAB.
We therefore need to consider the subgroup SO(D−2)⊗OSp(1,1|2) of OSp(D−1,1|2)
(the spin group obtained by adding 2+2 dimensions to the original SO(D−2)), and
determine which parts of an irreducible OSp(D−1,12) representation are OSp(1,1|2)
singlets. This is done most simply by considering the corresponding Young tableaux
(which is also the most convenient method for adding 2+2 dimensions to the original representation of the light-cone SO(D−2)). This means considering tensor products of the vector (defining) representation with various graded symmetrizations and antisymmetrizations, and (graded) tracelessness conditions on all the indices. The
obvious OSp(1,1|2) singlet is given by allowing all the vector indices to take only
74
4. GENERAL GAUGE THEORIES
SO(D−2) values. However, the resulting SO(D−2) representation is reducible, since
it is not SO(D−2)-traceless. The OSp(D−1,1|2)-tracelessness condition equates its
SO(D−2)-traces to OSp(1,1|2)-traces of representations which differ only by replacing the traced SO(D−2) indices with traced OSp(1,1|2) ones. However, OSp(1,1|2) (or
OSp(2N|2N) more generally) has the unusual property that its traces are not true
singlets. The simplest example [4.11] (and one which we’ll show is sufficient to treat the general case) is the graviton of (4.1.15). Considering just the OSp(1,1|2) values of the indices, there are 2 states which are singlets under the bosonic subgroup GL(2) generated by Mαβ, M−+, namely (+ −) , |α |α . However, of these two states, one
linear combination is pure gauge and one is pure auxiliary:
φ1 = A A
→
MAB φ1 = 0 ,
but
φ
α
1
= −1M
2
±
(∓
α)
,
φ2 = (+ −) − α α
→
φ2 = MAB ΛBA
,
but
M±α φ2 = −2 (± α) = 0 .
(4.4.20)
This result is due basically to the fact that the graded trace can’t be separated out in the usual way with the metric because of the identity
ηABη
A
BA = ηAA = (−1)AδA
= 2 − 2 = 0 .
(4.4.21)
Similarly, the reducible OSp(1,1|2) representation which consists of the unsymmetrized direct product of an arbitrary number of vector representations will contain no singlets, since any one trace reduces to the case just considered, and thus the representations which result from graded symmetrizations and antisymmetrizations will also
contain none. Thus, no SO(D−2)-traces of the original OSp(D−1,1|2) representation
need be considered, since they are equated to OSp(1,1|2)-nonsinglet traces by the
OSp(D−1,1|2)-tracelessness condition. Hence, the only surviving SO(D−2) repre-
sentation is the original irreducible light-cone one, obtained by restricting all vector indices to their SO(D−2) values and imposing SO(D−2)-tracelessness.
These methods apply directly to open strings. The modification for closed strings
will be discussed in sect. 11.1.
4.5. Fermions
75
4.5. Fermions
The results of this section are based on the OSp(1,1|2) algebra of sect. 3.5.
The action and invariances are again given by (4.1.1), with the modified JAB of
sect. 3.5, and (4.1.3) is unchanged except for Mαβ → Mαβ. We also allow for the inclu-sion of a matrix factor in the Hilbert-space metric to maintain the (pseudo)hermiticity of the spin operators (e.g., γ0 for a Dirac spinor, so Φ† → ¯
Φ = Φ†γ0). Under the
action of the δ functions, we can make the replacement
p 2
2
b
+ J −α
→ − 3(p2 + M2) + (M p
4
α
b + M αmM )2
− 1 ˜γα(M bp
1 (p2 + M2)
,
(4.5.1)
2
α
b + M αmM ) ˜
γ− + ˜γ+ 2
where p2 ≡ papa. Under the action of the same δ functions, the gauge transformation generated by J−α is replaced with
δΦ = J α
1
− Λα
→
(Mαbpb + MαmM) − 1 ˜γα ˜γ
(p2 + M2)
Λ
2
− + ˜
γ+ 2
α
.
(4.5.2)
Choosing Λα = ˜γαΛ, the ˜γ− part of this gauge transformation can be used to choose the partial gauge
˜
γ+Φ = 0 .
(4.5.3)
The action then becomes
S =
dDx ¯
φ δ(M
2
b
αβ )i˜
γα(Mα pb + MαmM) φ ,
(4.5.4)
where we have reduced Φ to the half represented by φ by using (4.5.3). (The ˜
γ± can
be represented as 2×2 matrices.) The remaining part of (4.5.2), together with the
Jαβ transformation, can be written in terms of ˜γ±-independent parameters as
δφ = (Mαbp
b
b + M αmM )(Aα + 1 ˜
γ
(p2 + M2) + (M p
2 α ˜
γβAβ) + −14
α
b + M αmM )2 B
+ 1M αβΛ
2
αβ
.
(4.5.5)
One way to get general irreducible spinor representations of orthogonal groups
(except for chirality conditions) is to take the direct product of a Dirac spinor with an irreducible tensor representation, and then constrain it by setting to zero the result of multiplying by a γ matrix and contracting vector indices. Since the OSp representations used here are obtained by dimensional continuation, this means we use the same constraints, with the vector indices i running over all commuting and anticommuting
76
4. GENERAL GAUGE THEORIES
values (including m, if we choose to define Mim by dimensional reduction from one
extra dimension). The OSp spin operators can then be written as
Mij = ˇ
M ij + 1 [γ
4
i, γj }
,
(4.5.6)
where ˇ
M are the spin operators for some tensor representation and γi are the OSp
Dirac matrices, satisfying the OSp Clifford algebra
{γi, γj] = 2ηij .
(4.5.7)
We choose similar relations between γ’s and ˜
γ’s:
{γi, ˜γB] = 0 .
(4.5.8)
Then, noting that 1(γα +i˜
γα) and 1(γ
2
2
α − i˜
γα) satisfy the same commutation relations
as creation and annihilation operators, respectively, we define
γ
β
α = aα + a†α
,
˜
γα = i aα − a†α
;
aα, a†β = δα
.
(4.5.9)
We also find
M αβ = ˇ
M αβ + a†(αaβ) .
(4.5.10)
This means that an arbitrary representation ψ(α···β) of the part of the Sp(2) generated by ˇ
M αβ that is also a singlet under the full Sp(2) generated by M αβ can be written
as
ψ(α···β) = a†α · · · a†βψ ,
aαψ = 0 .
(4.5.11)
In particular, for a Dirac spinor ˇ
M = 0, so the action (4.5.4) becomes simply (see
also (4.1.40))
S =
dDx ¯
φ(/
p + /
M)φ ,
(4.5.12)
where /
p ≡ γapa, /
M ≡ γmM (γm is like γ5), all dependence on γα and ˜γα has
been eliminated, and the gauge transformation (4.5.5) vanishes. (The transformation φ → eiγmπ/4φ takes /p + /
M → /p + iM.)
In the case of the gravitino, we start with φ = |i φi, where |i is a basis for
the representation of ˇ
M (only). φ must satisfy not only M αβφ = 0 but also the
irreducibility condition
γiφi = 0
→
φα = 1γ
2 αγaφa
,
aαφa = 0 .
(4.5.13)
4.5. Fermions
77
(a†α or i˜γα could be used in place of γα in the solution for φα.) Then using (4.1.8) for ˇ
M on φ, straightforward algebra gives the action
S =
dDx ¯
φa(ηab/
p − p(aγb) + γa/pγb)φb
=
dDx ¯
φaγabcpbφc ,
(4.5.14)
where γabc = (1/3!)γ[aγbγc], giving the usual gravitino action for arbitrary D. The gauge invariance remaining in (4.5.5) after using (4.5.13-14), and making suitable
redefinitions, reduces to the usual
δφa = ∂aλ .
(4.5.15)
We now derive an alternative form of the fermionic action which corresponds to
actions given in the literature for fermionic strings. Instead of explicitly solving the constraint M αβ = 0 as in (4.5.11), we use the Mαβ gauge invariance of (4.5.5) to
“rotate” the aα†’s. For example, writing aα = (a+, a−), we can rotate them so they
all point in the “+” direction: Then we need consider only φ’s of the form φ(a+†) |0 .
(The + value of α should not be confused with the + index on p
2
+.) The δ(M αβ ) then
picks out the piece of the form (4.5.11). (It “smears” over directions in Sp(2). This use of aα is similar to the “harmonic coordinates” of harmonic superspace [4.12].) We can also pick
φ = e−ia+†a−†φ(a+†) |0
,
(4.5.16a)
since the exponential (after δ(M) projection) just redefines some components by
shifting by components of lower ˇ
M -spin. In this gauge, writing γα = (s, u), ˜
γα =
(˜
s, ˜
u), we can rewrite φ as
˜0 = e−ia+†a−† |0
↔
˜
s ˜0 = u ˜0 = 0 ,
φ = φ(s) ˜0
↔
˜
sφ = 0 .
(4.5.16b)
By using an appropriate (indefinite-metric) coherent-state formalism, we can choose s to be a coordinate (and u = −2i∂/∂s). We next make the replacement
∂
δ(M
2
αβ )
→
ds µ(s) δ(2 ˇ
M+ + s2)δ
ˇ
M 3 + s
(4.5.17)
∂s
after pushing it to the right in (4.5.4) so it hits the φ, where we have just replaced projection onto M
2
αβ
= 0 with M + = M 3 = 0 (which implies M − = 0). The first δ
function factor is a Dirac δ function, while the second is a Kronecker δ. µ(s) is an
78
4. GENERAL GAUGE THEORIES
appropriate measure factor; instead of determining it by an explicit use of coherent states, we fix it by comparison with the simplest case of the Dirac spinor: Using
ds ǫ(s)δ(s2 + r2)(a + bs) = b ,
(4.5.18a)
we find
µ(s) ∼ ǫ(s) .
(4.5.18b)
Then only the ˜
u term of (4.5.4) contributes in this gauge, and we obtain (the ˜
u itself
having already been absorbed into the measure (4.5.18b))
∂
S = −2
dDx ds ǫ(s) ¯
φ Q+δ(2 ˇ
M+ + s2)δ
ˇ
M 3 + s
φ .
(4.5.19)
∂s