→ 12
V, V = W +θV , show that V is determined explicitly
by W .
(2) Derive (9.1.7b).
(3) Fill in all the steps needed to obtain (9.1.15) from (9.1.6,8). Derive all parts of (9.1.16). Derive (9.1.19).
(4) Evaluate (9.1.20) by using the result (9.1.15) with N → N + 2 (but dropping 2
dτ integrations) and letting k0 = kN+1 = 0.
(5) Derive (9.2.18).
(6) Generalize (9.2.16) to arbitrary z1, z2, zN and derive (9.2.22c) by the method
of (9.2.20,21). Take the infinitesimal form of the Sp(2) transformation and show
that it’s generated by L0, L±1 with the correspondence (8.1.3), where z = eiσ.
(7) Derive (9.2.27). Evaluate Grsmn of (9.2.4) using (9.2.19,26,27).
10. LIGHT-CONE FIELD THEORY
197
10. LIGHT-CONE FIELD THEORY
In this chapter we extend the discussion of sect. 2.1 to the string and consider
interacting contributions to the Poincaré algebra of sect. 7.1 along the lines of the Yang-Mills case treated in sect. 2.3.
For the string [10.1,9.5,10.2], it’s convenient to use a field Φ[Xi(σ), p+, τ ], since p+ is the length of the string. This X for σ ∈ [0, πp+] is related to that in (7.1.7) for p+ = 1 by X(σ, p+) = X(σ/p+, 1). The hermiticity condition on the (open-string)
field is
Φ[Xi(σ), p+, τ ] = Φ†[Xi(πp+ − σ), −p+, τ] .
(10.1a)
The same relation holds for the closed string, but we may replace πp+ − σ with just
−σ, since the closed string has the residual gauge invariance σ → σ + constant,
which is fixed by the constraint (or gauge choice) ∆NΦ = 0. (See (7.1.12). In
loops, this gauge choice can be implemented either by projection operators or by
Faddeev-Popov ghosts.) As described in sect. 5.1, this charge-conjugation condition corresponds to a combination of ordinary complex conjugation (τ reversal) with a
twist (matrix transposition combined with σ reversal). The twist effectively acts as a charge-conjugation matrix in σ space, in the sense that expressions involving tr Φ†Φ
acquire such a factor if reexpressed in terms of just Φ and not Φ† (and (10.1a) looks like a reality condition for a group for which the twist is the group metric). Here Φ is an N×N matrix, and the odd mass levels of the string (including the massless
Yang-Mills sector) are in the adjoint representation of U(N), SO(N), or USp(N) (for even N) [10.3], where in the latter 2 cases the field also satisfies the reality condition ηΦ = (ηΦ)* ,
(10.1b)
where η is the group metric (symmetric for SO(N), antisymmetric for USp(N)). The
fact that the latter cases use the operation of τ reversal separately, or, by combining with (10.1a), the twist separately, means that they describe nonoriented strings: The string field is constrained to be invariant under a twist. The same is true for closed strings (although closed strings have no group theory, so the choice of oriented vs.
198
10. LIGHT-CONE FIELD THEORY
nonoriented is arbitrary, and η = 1 in (10.1b)). The twist operator can be defined
similarly for superstrings, including heterotic strings. For general open strings the twist is most simply written in terms of the hatted operators, on which it acts as
Ô(σ) → Ô(σ − π) (i.e., as eiπN). For general closed strings, it takes Ô(±)(σ) →
Ô(±)(−σ). (For closed strings, σ → σ − π is irrelevant, since ∆N = 0.)
Light-cone superstring fields [10.2] also satisfy the reality condition (in place of (10.1b), generalizing (5.4.32)) that the Fourier transform with respect to Θa is equal to the complex conjugate (which is the analog of a certain condition on covariant
superfields)
DΘ e ΘaΠap+/2ηΦ[Θa] = ηΦ[ ¯
Πa] * .
(10.1c)
Θa has a mode expansion like that of the ghost C or the spinning string’s ˆ
Ψ (of
the fermionic sector). The ground-state of the open superstring is described by the light-cone superfield of (5.4.35), which is a function of the zero-modes of all the above coordinates. Thus, the lowest-mass (massless) sector of the open superstring
is supersymmetric Yang-Mills.
The free action of the bosonic open string is [10.1,9.5]
∞
∞
∂
S0 = −
DXi
dp+
dτ tr Φ†p+ i
+ H Φ ,
−∞
−∞
∂τ
2πα′p
2
+ dσ
δ2
1
πp+ dσ
p + M2
H =
1
+
X ′2
( 1P 2
i
.
2
−α′
i
− 1 =
2
i − 1) =
0
2π
δX 2
i
α′
−πp
2π
2p
+
+(10.2)
The free field equation is therefore just the quantum mechanical Schrödinger equation.
(The p+ integral can also be written as 2 ∞, due to the hermiticity condition. This 0
form also holds for closed strings, with H the sum of 2 open-string ones, as described in sec. 7.1.) Similar remarks apply to superstrings (using (7.3.13)).
As in the first-quantized approach, interactions are described by splitting and
joining of strings, but now the graph gets chopped up into propagators and vertices: 1
3
6
→
2
5
4
7
10. LIGHT-CONE FIELD THEORY
199
The 3-open-string vertex is then just a δ functional setting 1 string equal to 2 others, represented by an infinitesimal strip in the world sheet. The interaction term in
the action is given by (9.2.23) for Z = X. The 3-closed-string vertex is a similar δ
functional for 3 closed strings, which can be represented as the product of 2 open-
string δ functionals, since the closed-string coordinates can be represented as the sum of 2 open-string coordinates (one clockwise and one counterclockwise, but with
the same zero-modes). This vertex generally requires an integration over the σ of
the integration point (since closed strings can join anywhere, not having any ends, corresponding to the gauge invariance σ → σ+constant), but the equivalent operation of projection onto ∆N = 0 can be absorbed into the propagators.
General vertices can be obtained by considering similar slicings of surfaces with
general global topologies [10.2,9.7]. There are 2 of order g, corresponding locally to a splitting or joining:
↔
✬ ✩
✬ ✩
↔
✫ ✪
✫ ✪
The former is the 3-open-string vertex. The existence of the latter is implied by
the former via the nonplanar loop graph (see sect. 9.3). The rest are order g2, and correspond locally to 2 strings touching their middles and switching halves:
❅
❅
❅
❅
❅
↔
❅
❅
❅
✬ ✩
✬ ✩
↔
✫ ✪
✫ ✪
200
10. LIGHT-CONE FIELD THEORY
✬ ✩
✬ ✩
↔
✫ ✪
✫ ✪
✬
✩
✬ ✩
✬
✫
✩
✪
✬
✫ ✩
✪
↔
✫
✪
✫ ✪
✬
✩
✬ ✩
✬
✫
✩
✪
✫ ✪
↔ ✬ ✩
✫
✪
✫ ✪
The type-I (SO(32) open-closed) theory has all these vertices, but the type-IIAB and heterotic theories have only the last one, since they have only closed strings, and they are oriented (clockwise modes are distinguishable from counterclockwise). If the type-I theory is treated as a theory of fundamental open strings (with closed strings as bound states, so ¯
h can be defined), then we have only the first of the order-g
vertices and the first of the order-g2.
The light-cone quantization of the spinning string follows directly from the cor-
responding bosonic formalism by the 1D supersymmetrization described in sect. 7.2.
In particular, in such a formalism the vertices require no factors besides the δ functionals [10.4]. (In converting to a non-superfield formalism, integration of the vertex over θ produces a vertex factor.) However, the projection (7.2.5) must be put in by hand. Also, the fact that boundary conditions can be either of 2 types (for bosons
vs. fermions) must be kept in mind.
The interactions of the light-cone superstring [10.2] are done as for the light-cone bosonic string, but there are extra factors. For example, for the 3-open-string vertex
10. LIGHT-CONE FIELD THEORY
201
(the interacting contribution to p−), we have the usual δ-functionals times
δ
δ
δ
δ
δ
δ
V (p
¯
¯
−) = P L + 1 P ab
+ 1 P
,
(10.3a)
2
δΘa δΘb
24
LC abcd δΘa δΘb δΘc δΘd
evaluated at the splitting point. The interacting contributions to q− are given by the same overlap δ-functionals times the vertex factors
δ
δ
δ
δ
V (q
a
−a) =
,
V (¯
q
) = 1 Cabcd
.
(10.3b)
δΘa
−
6
δΘb δΘc δΘd
(The euphoric notation for q is as in sect. 5.4 for d.) These are evaluated as in (9.2.28), where P and δ/δΘ have weight w = 1. Their form is determined by requiring that the
supersymmetry algebra be maintained. The δ functional part is given as in (9.2.27), but now the ∆ of (9.2.14) instead of just (9.2.15) is
δ
∆ = 1∆( ˇ
P , ˇ
P )
.
(10.4)
2
− ∆ ˇ
Θ′, δˇΘ
As for the bosonic string, the closed-superstring vertex is the product of 2 open-
string ones (including 2 factors of the form (10.3a) for type I or II but just 1 for the heterotic, and integrated over σ). For the general interactions above, all order-g interactions have a single open-string vertex factor, while all order-g2 have 2, since the interactions of each order are locally all the same. The vertex factor is either 1
or (10.3a), depending on whether the corresponding set of modes is bosonic or supersymmetric. When these superstring theories are truncated to their ground states, the factor (10.3a) keeps only the zero-mode contributions, which is the usual light-cone, 3-point vertex for supersymmetric Yang-Mills, and the product of 2 such factors (for closed strings) is the usual vertex for supergravity.
The second-quantized interacting Poincaré algebra for the light-cone string can
be obtained perturbatively. For example,
[p−, J−i] = 0
→
[p(2)−, J(2)−i] = 0 ,
(10.5a)
[p(3)−, J(2)−i] + [p(2)−, J(3)−i] = 0 ,
(10.5b)
etc., where (n) indicates the order in fields (see sect. 2.4). The solution to (10.5a) is known from the free theory. The solution to (10.5b) can be obtained from known
results for the first-quantized theory [1.4,10.5]: The first term represents the invariance of the 3-point interaction of the hamiltonian under free Lorentz transformations.
202
10. LIGHT-CONE FIELD THEORY
The fact that this invariance holds only on-shell is an expected consequence of the fact that the second term in (10.5b) is simply the commutator of the free hamiltonian with the interaction correction to the Lorentz generator. Thus, the algebra of the
complete interacting generators closes off shell as well as on, and the explicit form of J(3)−i