(11.2.21)
The only terms in Λα which contribute to the transformation of the Sp(2) singlets are Λα = (λcp · a†1 + λcbg†1 + λcgb†1)f†1α + (λfp · a†1 + λfbg†1 + λfgb†1)c†1α
+ λcf †2α + λf c†2α 0
.
(11.2.22)
The gauge transformations of the components are then
1
δhab = ∂(aλcpb) − ηab √ λc ,
2
√
δha =
2λcpa + ∂aλc ,
δBa = ∂aλcb + 2Kλcpa ,
δGa = −1λcp
2
a + λf pa + ∂aλcg
,
1
δh = √ λc − 1λcb + λfb + 2Kλcg ,
2
2
δB = 2Kλcb ,
δG = −1λ
2
cg + λf g
,
212
11. BRST FIELD THEORY
√
δB =
2λcb + 2Kλc ,
√
δG = −1λ
2λ
2
c + λf +
cg
,
δη+ = λcb ,
√
δη− =
2λf − 1λ
2
f b − ∂ · λfp + 2Kλfg
,
√
δη0 =
2λc − 1λ
λ
∂
4
cb + 1
2
f b − 12
· λcp + Kλcg .
(11.2.23)
We then gauge away
Ga = 0
→
λfpa = 1λcp
2
a − ∂aλcg
,
G = 0
→
λfg = 1λ
2
cg
,
η+ = 0
→
λcb = 0 ,
1
1
1
η− = 0
→
λf = √ λfb + √ ∂ · λcp − √ (✷ + K)λcg ,
2 2
2 2
2
√
η0 = 0
→
λfb = −2 2λc + ∂ · λcp − 2Kλcg .
(11.2.24)
The transformation laws of the remaining fields are
δhab = ∂(aλb) − ηabλ ,
√
δha =
2(λa + ∂aλ) ,
δh = −3λ + ∂ · λ ,
δBa = −(✷ − 2)λa ,
√
δB = − 2(✷ − 2)λ ,
δB = 0 ,
1
δG = √ (−3λ + ∂ · λ) ,
(11.2.25)
2
√
where λa = λcpa and λ = λc/ 2. λcg drops out of the transformation law as a result
of the gauge invariance for gauge invariance
δφ = QαΛα
→
δΛα = QβΛ(αβ) .
(11.2.26)
The lagrangian can then be computed from (4.1.6) and (11.2.17) to be
L = 1hab(
ha(
h(
4
✷ − 2)hab + 12
✷ − 2)ha − 12 ✷ − 2)h
√
− 1B 2
B2 + 2B( 2G
2
a − 12
− h)
√
3
1
+ Ba(−∂bhab + ∂ah + 2ha) + B(−∂ · h + √ h − √ haa) . (11.2.27)
2
2 2
11.2. Components
213
The lagrangian of (11.2.19) is equivalent to that obtained from the IGL(1) action,
whereas the lagrangian of (11.2.27) is like the IGL(1) one but contains in addition
√
the 2 gauge-invariant auxiliary fields B and
2G − h. Both reduce to the OSp(1,1|2)
lagrangians after elimination of auxiliary fields.
The results for the closed string are similar. Here we consider just the massless
level of the nonoriented closed string (which is symmetric under interchange of + and
− modes). The Sp(2) and ∆N invariant components are
φ = (haba†+aa†−a + Bag†(+a†−)a + Gab†(+a†−)a + hg†(+b†−) + Bg†+g†− + Gb†+b†−
+ η
α
α
α
+f †+ f †−α + η−c†+ c†−α + η0f †(+ c†−)α) |0
,
(11.2.28)
where all oscillators are from the first mode (n = 1), and hab is symmetric. The gauge parameters are
Λα = (λcp · a† + λcbg† + λcgb†)(+f†−)α + (λfp · a† + λfbg† + λfgb†)(+c†−)α |0
.
(11.2.29)
The component transformations are then
δhab = ∂(aλcpb) ,
δBa = ∂aλcb − 12✷ λcpa ,
δGa = 2λfpa − λcpa + ∂aλcg ,
δh = 2λfb − λcb − 12✷ λcg ,
δB = −✷ λcb ,
δG = 4λfg − 2λcg ,
δη+ = 2λcb ,
δη− = −λfb − ∂ · λfp − 12✷ λfg ,
δη0 = λfb − 1λ
∂
2
cb − 12
· λcp − 14✷ λcg .
(11.2.30)
We choose the gauges
Ga = 0
→
λfpa = 1λcp
∂
2
a − 12 aλcg
,
G = 0
→
λfg = 1λ
2
cg
,
η+ = 0
→
λcb = 0 ,
η0 = 0
→
λfb = 1∂
2
· λcp + 14✷ λcg .
(11.2.31)
214
11. BRST FIELD THEORY
The remaining fields transform as
δhab = ∂(aλb) ,
δBa = −12✷ λa ,
δh = ∂ · λ ,
δB = 0 ,
δη− = −∂ · λ ,
(11.2.32)
where λa = λcpa, and λcg drops out. Finally, the lagrangian is
L = 1hab
2
2
✷ hab − h✷ h − 4Ba − 8B(η− + h) − 4Ba(∂bhab − ∂ah) .
(11.2.33)
Again we find the auxiliary fields B and η− + h in addition to the usual nonminimal ones. After elimination of auxiliary fields, this lagrangian reduces to that of (11.2.13) (for the nonoriented sector, up to normalization of the fields).
Exercises
(1) Derive the gauge-invariant actions (IGL(1) and OSp(1,1|2)) for free open strings.
Do the same for the Neveu-Schwarz string. Derive the OSp(1,1|2) action for the
Ramond string.
(2) Find all the Sp(2)-singlet fields at the levels indicated in (11.2.2). Separate them into sets corresponding to irreducible representations of the Poincaré group (including their Stueckelberg and auxiliary fields).
(3) Derive the action for the massless level of the open string using the bosonized ghosts of sect. 8.1.
(4) Derive the action for the next mass level of the closed string after those in
(11.2.13). Do the same for (11.2.33).
(5) Rederive the actions of sect. 11.2 for levels which include spin 2 by first decompos-ing the corresponding light-cone representations into irreducible representations,
and then using the Hilbert-space constructions of sect. 4.1 for each irreducible
representation.
12.1. Introduction
215
12. GAUGE-INVARIANT
INTERACTIONS
12.1. Introduction
The gauge-invariant forms of the interacting actions for string field theories are far from understood. Interacting closed string field theory does not yet exist. (Although a proposal has been made [12.1], it is not even at the point where component actions can be examined.) An open-string formulation exists [4.8] (see the following section), but it does not seem to relate to the light-cone formulation (chapt. 10), and has more complicated vertices. Furthermore, all these formulations are in the IGL(1) formalism, so relation to particles is less direct because of the need to eliminate BRST auxiliary fields. Most importantly, the concept of conformal invariance is not clear in these formulations. If a formulation could be found which incorporated the world-sheet
metric as coordinates, as the free theory of sect. 8.3, it might be possible to restore conformal transformations as a larger gauge invariance which allowed the derivation of the other formulations as (partial) gauge choices.
In this section we will mostly discuss the status of the derivation of an interacting gauge-covariant string theory from the light cone, with interactions similar to those of the light-cone string field theory. The derivation follows the corresponding derivation for Yang-Mills described in sects. 3.4 and 4.2 [3.14], but the important step (3.4.17) of eliminating p+ dependence has not yet been performed.
As performed for Yang-Mills in sect. 3.4, the transformation (3.4.3a) with Φ →
U−1Φ is the first step in deriving an IGL(1) formalism for the interacting string from the light cone. Since the transformation is nonunitary, the factor of p+ in (2.4.9) is canceled. In (2.4.7), using integration by parts, (3.4.3a) induces the transformation of the vertex function
1
V(n)
→
U(1) · · · U(n)V(n) ,
(12.1.1)
p+1 · · · p+n
216
12. GAUGE-INVARIANT INTERACTIONS
where we work in momentum space with respect to p+. The lowest-order interacting
contribution to Q then follows from applying this transformation to the OSp-extended form of the light-cone vertex (10.6):
p
J (3) c
+r
c
−1
−
= −2ig
X (σ
X˜c] , (12.1.2)
p
r
int) δ
p+ ∆[Xa] ∆[p+Xc] ∆[p+
+1p+2p+3
effectively giving conformal weight -1 to Xc and conformal weight 1 to X˜c. A δ-
functional that matches a coordinate must also match the σ-derivative of the coordinate, and with no zero-modes one must have
σ
σ
σ
∆ p −1
−1
−2
+
X˜c
= ∆ ∂
X˜c
= ∆ p
X˜c′
. (12.1.3)
p
σ p+
+
+
˜
c=0
p+
p+
(Even the normalization is unambiguous, since without zero-modes ∆ can be normal-
ized to 1 between ground states.) We now recognize Xc and X˜c′ to be just the usual Faddeev-Popov ghost C(σ) of τ -reparametrizations and Faddeev-Popov anti ghost
δ/δC(σ) of σ-reparametrizations (as in (8.1.13)), of conformal weights -1 and 2, respectively, which is equivalent to the relation (8.2.2,3) (as seen by using (7.1.7b)).
Finally, we can (functionally) Fourier transform the antighost so that it is replaced with the canonically conjugate ghost. Our final vertex function is therefore
p
J (3) c
+r
− = −2ig
C
p
r(σint)δ
p+ ∆[Xa] ∆[p+C] ∆[p+C] ,
(12.1.4a)
+1p+2p+3
or in terms of momenta
δ
δ
J (3) c
−1
−2
−2
−
= −2igp+rCr(σint)δ
p+ ∆[p+ P a] ∆ p+
∆ p
.
δC
+
δC (12.1.4b)
The extra p+’s disappear due to Fourier transformation of the zero-modes c:
1
p˜c
dc e−cp˜cf (p
.
(12.1.5)
p
+c) =
dc e−cp˜c/p+f (c) = ˜
f
+
p+
Equivalently, the exponent of U by (3.4.3a) is c∂/∂c + M3, so the zero-mode part
just scales c, but c∂/∂c = 1 − (∂/∂c)c, so besides scaling ∂/∂c there is an extra
factor of p+ for each zero-mode, canceling those in (12.1.1,2). There is no effect on the normalization with respect to nonzero-modes because of the above-mentioned
normalization in the definition of ∆ with respect to the creation and annihilation
operators. A similar analysis applies to closed strings [12.2].
Hata, Itoh, Kugo, Kunitomo, and Ogawa [12.3] proposed an interacting BRST
operator equivalent to this one, and corresponding gauge-fixed and gauge-invariant
12.2. Midpoint interaction
217
actions, but with p+ treated as an extra coordinate as in [2.7]. (A similar earlier attempt appeared in [4.9,7.5], with p+ fixed, as a consequence of which the BRST
algebra didn’t close to all orders. Similar attempts appeared in [12.4].) However,
as explained in [2.7], such a formalism requires also an additional anticommuting
coordinate in order for the loops to work (as easily checked for the planar 1-loop
graph with external tachyons [2.7]), and can lead to problems with infrared behavior
[4.4].
The usual four-point vertex of Yang-Mills (and even-higher-point vertices of grav-
ity for the closed string) will be obtained only after field redefinitions of the massive fields. This corresponds to the fact that it shows up in the zero-slope limit of the S-matrix only after massive propagators have been included and reduced to points.
In terms of the Lagrangian, for arbitrary massive fields µ and massless fields ν, the terms, for example,
L = 1µ[
2
✷ + M2 + M2U(ν)]µ + M2µV (ν)
(12.1.6a)
(where U(ν) and V (ν) represent some interaction terms) become, in the limit M2 →
∞,
L = M2[1 (1 + U)µ2 + V µ] .
(12.1.6b)
2
The corresponding field redefinition is
V
µ = ˜
µ −
,
(12.1.6c)
1 + U
which modifies the Lagrangian to
V 2
L = 1 ˜
µ(
M2
+
2
✷ + M2 + M2U)˜µ + 12
O(M0) .
(12.1.6d)
1 + U
The redefined massive fields ˜
µ no longer contribute in the zero-slope limit, and can
be dropped from the Lagrangian before taking the limit. However, the redefinition
has introduced the new interaction term 1M2 V 2 into the ν-part of the Lagrangian.
2
1+U
12.2. Midpoint interaction
Witten has proposed an extension of the IGL(1) gauge-invariant open bosonic
string action to the interacting case [4.8]. Although there may be certain limitations with his construction, it shares certain general properties with the light-cone (and covariantized light-cone) formalism, and thus we expect these properties will be common to any future approaches. The construction is based on the use of a vertex which 218
12. GAUGE-INVARIANT INTERACTIONS
consists mainly of δ-functionals, as in the light-cone formalism. Although the geometry of the infinitesimal surface corresponding to these δ-functionals differs from that of the light-cone case, they have certain algebraic features in common. In particular, by considering a structure for which the δ-functional (times certain vertex factors) is identified with the product operation of a certain algebra, the associativity of the product follows from the usual properties of δ-functionals. This is sufficient to define an interacting, nilpotent BRST operator (or Lorentz generators with [J−i, J−j] = 0), which in turn gives an interacting gauge-invariant (or Lorentz-invariant) action.
The string fields are elements of an algebra: a vector space with an outer product ∗.
We write an explicit vector index on the string field Φi, where i = Z(σ) is the co-
ordinates (X, C, C for the covariant formalism and XT , x± for the light cone), and excludes group-theory indices. Then the product can be written in terms of a rank-3
matrix
(Φ ∗ Ψ)
jk
i = f i
ΦkΨj .
(12.2.1)
In order to construct actions, and because of the relation of a field to a first-quantized wave function, we require, in addition to the operations necessary to define an algebra, a Hilbert-space inner product
Φ|Ψ =
DZ tr Φ†Ψ = tr Φi†Ψi ,
(12.2.2)
where tr is the group-theory trace. Furthermore, in order to give the hermiticity
condition on the field we require an indefinite, symmetric charge-conjugation matrix Ω on this space:
Φi = ΩijΦj† ,
(ΩΦ)[X(σ), C(σ), C(σ)] ≡ Φ[X(π − σ), C(π − σ), −C(π − σ)] .
(12.2.3a)
Ω is the “twist” of (11.1.1). To allow a reality condition or, combining with (12.2.3a), a symmetry condition for real group representations (for SO(N) or USp(N)), we also
require that indices can be freely raised and lowered:
(ηΦ)i = Ωijδjk(ηΦ)tk ,
(12.2.3b)
where η is the group metric and t is the group-index transpose. In order to perform the usual graphical manipulations implied by duality, the twist must have the usual effect on vertices, and thus on the inner product:
Ω(Φ ∗ Ψ) = (ΩΨ) ∗ (ΩΦ) .
(12.2.4)
12.2. Midpoint interaction
219
Further properties satisfied by the product follow from the nilpotence of the BRST
operator and integrability of the field equations QΦ = 0: Defining
QΦ = Q
j
jk
0Φ + Φ ∗ Φ
→
(QΦ)i = Q0i Φj + f i ΦkΦj
,
(12.2.5)
S =
DZ tr Φ†( 1 Q
Φ
(ΩQ
(Ωf )ijkΦ
2
0Φ + 1
3
∗ Φ) = tr 12
0)ij Φj Φi + 1
3
kΦj Φi
,
(12.2.6)
we find that hermiticity requires
(ΩQ0)ij = ( ¯
Q0Ω)ji ,
(Ωf )ijk = ( ¯
fΩΩ)kji
,
(12.2.7)
integrability requires antisymmetry of Q0 and cyclicity of ∗:
(ΩQ0)ij = −(ΩQ0)ji ,
(Ωf )ijk = (Ωf )jki
(12.2.8)
(where permutation of indices is in the “graded” sense, but we have omitted some
signs: e.g., (ΩQ0)ijΦjΨi = +(ΩQ0)ijΨjΦi when Φ[Z] and Ψ[Z] are anticommuting,
but Φi and Ψi include components of either statistics), and nilpotence requires, besides Q 2
0
= 0, that ∗ is BRST invariant (i.e., Q0 is distributive over ∗) and associative:
Q0(Φ ?