40
3. GENERAL BRST
imaginary in coordinate space). However, we will avoid changing notation or Fourier transforming the fields, in order to simplify comparison to the OSp(1,1|2) formalism.
The effect of (3.4.3a) on a field satisfying Φ = ΩΦ* is that it now satisfies
Φ = (−1)c∂/∂c+M3ΩΦ*
(3.4.4)
due to the i in p+ = i∂+.
These results can be extended to interacting field theory, and we use Yang-Mills
as an example [2.3]. Lorentz-covariantizing the light-cone result (2.1.7,2.4.11), we find p− =
−1F ijF
(p
4
ji + 1
2
+A−)2
,
J−α =
ix−(pαAi)(p+Ai) + ixα −1F ijF
(p
4
ji + 1
2
+A−)2 − Aαp+A−
,
1
A− = −
[∇i, p
p 2
+Ai}
.
(3.4.5)
+
When working in the IGL(1) formalism, it’s extremely useful to introduce a
Lorentz covariant type of second-quantized bracket [3.14]. This bracket can be pos-
tulated independently, or derived by covariantization of the light-cone canonical commutator, plus truncation of the ˜
c, x+, and x− coordinates. The latter derivation will
prove useful for the derivation of IGL(1) from OSp(1,1|2). Upon covariantization of the canonical light-cone commutator (2.4.5), the arguments of the fields and of the δ-function on the right-hand side are extended accordingly. We now have to truncate.
The truncation of x+ is automatic: Since the original commutator was an equal-time
one, there is no x+ δ-function on the right-hand side, and it therefore suffices to delete the x+ arguments of the fields. At this stage, in addition to x− dependence, the fields depend on both c and ˜
c and the right-hand side contains both δ-functions.
(This commutator may be useful for OSp approaches to field theory.) We now wish
to eliminate the ˜
c dependence. This cannot be done by straightforward truncation,
since expansion of the field in this anticommuting coordinate shows that one cannot eliminate consistently the fields in the ˜
c sector. We therefore proceed formally and
just delete the ˜
c argument from the fields and the corresponding δ-function, obtaining
1
[Φ†(1), Φ(2)]c = −
δ(x
2p
2− − x1−)δD(x2 − x1)δ(c2 − c1)
,
(3.4.6)
+2
which is a bracket with unusual statistics because of the anticommuting δ-function on the right-hand side. The transformation (3.4.3a) is performed next; its nonunitarity
3.4. From the light cone
41
causes the p+ dependence of (3.4.6) to disappear, enabling one to delete the x−
argument from the fields and the corresponding δ-function to find (using (c∂/∂c)c = c)
[Φ†(1), Φ(2)]c = −1δD(x
2
2 − x1)δ(c2 − c1)
.
(3.4.7)
This is the covariant bracket. The arguments of the fields are (xa, c), namely, the usual D bosonic coordinates of covariant theories and the single anticommuting coordinate of the IGL(1) formalism. The corresponding δ-functions appear on the right-hand
side. (3.4.6,7) are defined for commuting (scalar) fields, but generalize straightforwardly: For example, for Yang-Mills, where Ai includes both commuting (Aa) and
anticommuting (A
†
α) fields, [Ai , Aj ] has an extra factor of ηij . It might be possible
to define the bracket by a commutator [A, B]c = A ∗ B − B ∗ A. Classically it can be defined by a Poisson bracket:
δ
†
δ
[A†, B]c = 1
dz
A
B
,
(3.4.8)
2
δΦ(z)
δΦ(z)
where z are all the coordinates of Φ (in this case, xa and c). For A = B = Φ, the
result of equation (3.4.7) is reproduced. The above equation implies that the bracket is a derivation:
[A, BC]c = [A, B]cC + (−1)(A+1)BB[A, C]c ,
(3.4.9)
where the A’s and B’s in the exponent of the (−1) are 0 if the corresponding quantity is bosonic and 1 if it’s fermionic. This differs from the usual graded Leibnitz rule by a (−1)B due to the anticommutativity of the dz in the front of (3.4.8), which also
gives the bracket the opposite of the usual statistics: We can write (−1)[A,B]c =
(−1)A+B+1 to indicate that the bracket of 2 bosonic operators is fermionic, etc., a direct consequence of the anticommutativity of the total δ-function in (3.4.7). One can also verify that this bracket satisfies the other properties of a (generalized) Lie bracket:
[A, B]c = (−1)AB[B, A]c ,
(−1)A(C+1)[A, [B, C]c]c + (−1)B(A+1)[B, [C, A]c]c + (−1)C(B+1)[C, [A, B]c]c = 0 .
(3.4.10)
Thus the bracket has the opposite of the usual graded symmetry, being antisymmetric for objects of odd statistics and symmetric otherwise. This property follows from
the hermiticity condition (3.4.4): (−1)c∂/∂c gives (−1)−(∂/∂c)c = −(−1)c∂/∂c upon
integration by parts, which gives the effect of using an antisymmetric metric. The
Jacobi identity has the same extra signs as in (3.4.9). These properties are sufficient to perform the manipulations analogous to those used in the light cone.
42
3. GENERAL BRST
Before applying this bracket, we make some general considerations concerning
the derivation of interacting IGL(1) from OSp(1,1|2). We start with the original
untransformed generators J3 and J c
−
= Q. The first step is to restrict our attention
to just the fields at ˜
c = 0. Killing all the fields at linear order in ˜
c is consistent
with the transformation laws, since the transformations of the latter fields include no terms which involve only ˜
c = 0 fields. Since the linear-in-˜
c fields are canonically
conjugate to the ˜
c = 0 fields, the only terms in the generators which could spoil this
property would themselves have to depend on only ˜
c = 0 fields, which, because of the
d˜
c (= ∂/∂˜
c) integration, would require explicit ˜
c-dependence. However, from (2.4.15),
since ηcc = Ccc = 0, we see that J c
−
anticommutes with pc, and thus has no explicit
˜
c-dependence at either the free or interacting levels. (The only explicit coordinate dependence in Q is from a c term.)
The procedure of restricting to ˜
c = 0 fields can then be implemented very simply
by dropping all pc(= −∂/∂˜c)’s in the generators. As a consequence, we also lose all explicit x
c
− terms in Q. (This follows from [J − , p+] = −pc.) Since ˜
c and ∂/∂˜
c now
occur nowhere explicitly, we can also kill all implicit dependence on ˜
c: All fields are
evaluated at ˜
c = 0, the d˜
c is removed from the integral in the generators, and the
δ(˜
c2 − ˜c1) is removed from the canonical commutator, producing (3.4.6). In the case
of Yang-Mills fields Ai = (Aa, Aα) = (Aa, Ac, A˜c), the BRST generator at this point is given by
Q =
ic[−1F ijF
(p
4
ji + 1
2
+A−)2] − Acp+A−
,
(3.4.11)
where the integrals are now over just xa, x−, and c, and some of the field strengths simplify:
F cc = 2(Ac)2
,
other F ic = [∇i, Ac} .
(3.4.12)
Before performing the transformations which eliminate p+ dependence, it’s now
convenient to expand the fields over c as
Aa = Aa + cχa
,
Ac = iC + cB
,
A˜c = iC + cD
,
(3.4.13)
where the fields on the right-hand sides are xα-independent. (The i’s have been
chosen in accordance with (3.4.4) to make the final fields real.) We next perform the dc integration, and then perform as the first transformation the first-quantized one
3.4. From the light cone
43
(3.4.3a), Φ → p −J3
+
Φ (using the first-quantized J3 = c∂/∂c + M3), which gives
1
2
1
2
−iQ =
1 F 2
2B +
i[
p 2C,
˜
C
4
ab
− 12
∇a, p
p
+Aa] − {C, ˜
C} −
+
+
p+
p+
1
2
− 2D + ˜
C2 − 2p 2
˜
+
C
C2
p+
1
+ 2χa − i[∇a, ˜
C] − 2i p
˜
+Aa,
C
[∇
p
a, C ]
.
(3.4.14)
+
This transformation also replaces (3.4.6) with (3.4.7), with an extra factor of ηij for [A †
i , Aj ], but still with the x− δ-function. Expanding the bracket over the c’s,
[χa, Ab]c = 1 η
,
[B, ˜
C]
,
(3.4.15)
2 ab
,
[D, C]c = 12
c = − 12
where we have left off all the δ-function factors (now in commuting coordinates only).
Note that, by (3.4.10), all these brackets are symmetric.
We might also define a second-quantized
∂
J3 =
Aip+ c
+ M3 A
∂c
i
,
(3.4.16a)
but this form automatically keeps just the antihermitian part of the first-quantized operator c∂/∂c = 1[c, ∂/∂c] + 1: Doing the c integration and transformation (3.4.3a), 2
2
J3 =
χaAa − ˜
CB − 3CD .
(3.4.16b)
As a result, the terms in Q of different orders in the fields have different second-quantized ghost number. Therefore, we use only the first-quantized ghost operator
(or second-quantize it in functional form).
As can be seen in the above equations, despite the rescaling of the fields by
suitable powers of p+ there remains a fairly complicated dependence on p+. There is no explicit x− dependence anywhere but, of course, the fields have x− as an argument.
It would seem that there should be a simple prescription to get rid of the p+’s in the transformations. Setting p+ = constant does not work, since it violates the Leibnitz rule for derivatives (p+φ = aφ implies that p+φ2 = 2aφ2 and not aφ2). Even setting
p+φi = λiφi does not work. An attempt that comes very close is the following: Give
the fields some specific x− dependence in such a way that the p+ factors can be
evaluated and that afterwards such dependence can be canceled between the right-
hand side and left-hand side of the transformations. In the above case it seems that
44
3. GENERAL BRST
the only possibility is to set every field proportional to (x−)0 but then it is hard to define 1/p+ and p+. One then tries setting each field proportional to (x−)ǫ and
then let ǫ → 0 at the end. In fact this prescription gives the correct answer for the quadratic terms of the Yang-Mills BRST transformations. Unfortunately it does not
give the correct cubic terms.
It might be possible to eliminate p+-dependence simply by applying J−+ = 0 as a
constraint. However, this would require resolving some ambiguities in the evaluation of the nonlocal (in x−) operator p+ in the interaction vertices.
We therefore remove the explicit p+-dependence by use of an explicit transforma-
tion. In the Yang-Mills case, this transformation can be completely determined by
choosing it to be the one which redefines the auxiliary field B in a way which eliminates interaction terms in Q involving it, thus making B + i1 p
2
· A BRST-invariant.
The resulting transformation [3.14] redefines only the BRST auxiliary fields:
Q → eL∆Q ,
L
2
AB ≡ [A, B]c
,
L∆ = 0 ,
1
1
2
∆ =
˜
C
i[Aa, p
˜
C
(p 2C) ,
(3.4.17)
p
+Aa] − ˜
C2C + 2
+
+
p+
simply redefines the auxiliary fields to absorb the awkward interaction terms in
(3.4.14). (We can also eliminate the free terms added to B and χ by adding a
term
˜
CipaAa to ∆ to make the first term ˜
C(1/p+)i[∇a, p+Aa].) We then find for
the transformed BRST operator
−iQ =
1 F 2
(2B + ip
4
ab − 12
· A)2 − 2DC2 + (2χa − ipaC)[∇a, C] .
(3.4.18)
The resulting transformations are then
QAa = − i[∇a, C] ,
Qχa = i1[
p
p
2 ∇b, Fba] + i{C, χa − i 1
2 aC} + pa(B + i 1
2
· A) ,
QC = − 2i(B + i1p
2
· A) ,
QD = − i ∇a, χa − i1p
2 aC
+ i[C, D] ,
QC = iC2
,
QB = − 1pa [
2
∇a, C] .
(3.4.19)
Since all the p+’s have been eliminated, we can now drop all x− dependence from the fields, integration, and δ-functions. On the fields Aa, C, C, B of the usual BRST2
3.5. Fermions
45
formalism, this result agrees with the corresponding transformations (3.2.8), where this B = 1 B of (3.2.11). By working with the second-quantized operator form of Q
2
(and of the redefinition ∆), we have automatically obtained a form which makes QΦ
integrable in Φ, or equivalently makes the vertices which follow from this operator cyclic in all the fields (or symmetric, if one takes group-theory indices into account).
The significance of this property will be described in the next chapter.
This extended-light-cone form of the OSp(1,1|2) algebra will be used to derive
free gauge-invariant actions in the next chapter. The specific form of the generators for the case of the free open string will be given in sect. 8.2, and the generalization to the free closed string in sect. 11.1. A partial analysis of the interacting string along these lines will be given in sect. 12.1.
3.5. Fermions
These results can be extended to fermions [3.15]. This requires a slight modi-
fication of the formalism, since the Sp(2) representations resulting from the above analysis for spinors don’t include singlets. This modification is analogous to the addition of the B∂/∂C terms to Q in (3.2.6a). We can think of the OSp(1,1|2) generators of (3.4.2) as “orbital” generators, and add “spin” generators which themselves generate OSp(1,1|2). In particular, since we are here considering spinors, we choose the spin generators to be those for the simplest spinor representation, the graded generalization of a Dirac spinor, whose generators can be expressed in terms of graded Dirac “matrices”:
{˜γA, ˜γB] = 2ηAB , S
′
AB = 1 [˜
γ
= J
4
A, ˜
γB} , JAB
AB + SAB
,
(3.5.1)
where { , ] is the opposite of [ , }. These ˜γ matrices are not to be confused with the “ordinary” γ matrices which appear in Mij from the dimensional continuation
of the true spin operators. The ˜
γA, like γi, are hermitian. (The hermiticity of γi
in the light-cone formalism follows from (γi)2 = 1 for each i and the fact that all states in the light-cone formalism have nonnegative norm, since they’re physical.)
The choice of whether the ˜
γ’s (and also the graded γi’s) commute or anticommute
with other operators (which could be arbitrarily changed by a Klein transformation) follows from the index structure as usual (bosonic for indices ±, fermionic for α).
(Thus, as usual, the ordinary γ matrices γa commute with other operators, although
they anticommute with each other.)
46
3. GENERAL BRST
In order to put the OSp(1,1|2) generators in a form more similar to (3.4.2), we
need to perform unitary transformations which eliminate the new terms in J−+ and
J+α (while not affecting Jαβ, although changing J−α). In general, the appropriate
transformations J
′
AB = U J AB U −1 to eliminate such terms are:
ln U = −(ln p+)S−+
(3.5.2a)
to first eliminate the S−+ term from J−+, and then
ln U = S α
+ pα
(3.5.2b)
to do the same for J+α. The general result is
J−+ = −ix−p+ + k ,
J+α = ixαp+ ,
Jαβ = −ix(αpβ) + Mαβ ,
1
J
1
β
−α = −ix−pα +
−ix (pbp
p
p
α 2
b + M 2 + pβ pβ ) + M α
β + kpα + Qα
; (3.5.3a)
+
Mαβ = Mαβ + Sαβ ,
Q
b
1
α = (M α pb + M αmM ) + S−α + S+α (pbp
2
b + M 2)
.
(3.5.3b)
(3.5.3a) is the same as (3.4.2a), but with Mαβ and Qα replaced by Mαβ and Qα. In
this case the last term in Qα is
S
1
1
−α + S+α (pbp
˜
γ
(pbp
2
b + M 2) = − 12 α ˜
γ− + ˜γ+ 2
b + M 2)
.
(3.5.4)
We can again choose k = 0.
This algebra will be used to derive free gauge-invariant actions for fermions in
sect. 4.5. The generalization to fermionic strings follows from the representation of the Poincaré algebra given in sect. 7.2.
3.6. More dimensions
In the previous section we saw that fermions could be treated in a way similar to
bosons by including an OSp(1,1|2) Clifford algebra. In the case of the Dirac spinor, there is already an OSp(D−1,1|2) Clifford algebra (or OSp(D,1|2) in the massive
case) obtained by adding 2+2 dimensions to the light-cone γ-matrices, in terms of
which Mij (and therefore the OSp(1,1|2) algebra) is defined. Including the addi-
tional γ-matrices makes the spinor a representation of an OSp(D,2|4) Clifford algebra (OSp(D+1,2|4) for massive), and is thus equivalent to adding 4+4 dimensions to the
3.6. More dimensions
47
original light-cone spinor instead of 2+2, ignoring the extra spacetime coordinates.
This suggests another way of treating fermions which allows bosons to be treated
identically, and should thus allow a straightforward generalization to supersymmetric theories [3.16].
We proceed similarly to the 2+2 case: Begin by adding 4+4 dimensions to the
light-cone Poincaré algebra (2.3.5). Truncate the resulting IOSp(D+1,3|4) algebra to ISO(D−1,1)⊗IOSp(2,2|4). IOSp(2,2|4) contains (in particular) 2 inequivalent truncations to IOSp(1,1|2), which can be described by (defining-representation direct-
product) factorization of the OSp(2,2|4) metric into the OSp(1,1|2) metric times the metric of either SO(2) (U(1)) or SO(1,1) (GL(1)):
ηAB = ηABη
,
âˆb
A = Aâ
→
ˆ
J
bâ
AB = η
J
,
ǫ ∆ = ηBAJ
,
(3.6.1)
Aâ,Bˆ
b
âˆb
Aâ,Bˆ
b
where ∆ is the generator of the U(1) or GL(1) and η
= I or σ
âˆ
b
1. These 2 OSp(1,1|2)’s
are Wick rotations of each other. We’ll treat the 2 cases separately.
The GL(1) case corresponds to first taking the GL(2|2) (=SL(2|2)⊗GL(1)) sub-
group of OSp(2,2|4) (as SU(N)⊂SO(2N), or GL(1|1)⊂OSp(1,1|2)), keeping also half
of the inhomogeneous generators to get IGL(2|2). Then taking the OSp(1,1|2) sub-
group of the SL(2|2) (in the same way as SO(N)⊂SU(N)), we get IOSp(1,1|2)⊗GL(1),
which is like the Poincaré group in (1,1|2) dimensions plus dilatations. (There is also an SL(1|2)=OSp(1,1|2) subgroup of SL(2