,
1 [C
γδ γ
2π αβ(1), Da(2)] = 2iδ′(2 − 1)γa
bγ[αP bCβ]δ(1)
− 4iδ(2 − 1)(γ
δ
aγ[αD′β]Bγ + P bγba[α C′β]δ)
,
1 [D
β D
2π
a(1), Db(2)] = − δ′′(2 − 1)γabα
β Bα(1) + (2)
+ δ′(2 − 1) −4iP (aDb) + ηab(3D′αBα − DαBα′) (1) − (2)
+ δ(2 − 1) −2i(3P ′[aDb] − P [aD′b])
+2γ
β
αβ
abα (3D′β B′α − D′′βBα) + γabc
(3P ′cC′αβ − P ′′cCαβ)
.
(7.3.8)
(Due to identities like /
P DD ∼ BD ∼ /
P C, there are other forms of some of these
relations.)
BRST quantization can again be performed, and there are an infinite number of
ghosts, as in the particle case. However, a remaining problem is to find the appropriate ground state (and corresponding string field). Considering the results of sect. 5.4, this may require modification of the generators (7.3.7) and BRST operator, perhaps
to include Lorentz generators (acting on the ends of the string?) or separate contributions from the BRST transformations of Yang-Mills field theory (the ground state
of the open superstring, or of a set of modes of one handedness of the corresponding closed strings). On the other hand, the ground state, rather than being Yang-Mills, might be purely gauge degrees of freedom, with Yang-Mills appearing at some excited level, so modification would be unnecessary. The condition Q2 = 0 should reproduce
the conditions D = 10, α0 = 1.
The covariant derivatives and constraints can also be derived from a 2D lagrangian
of the general form (3.1.10), as for the superparticle [7.5]. This classical mechanics Lagrangian imposes weaker constraints than the Green-Schwarz one [7.6] (which sets
Dα = 0 via Gupta-Bleuler), and thus should not impose stronger conditions.
On the other hand, quantization in the light-cone formalism is understood. Spinors
are separated into halves, with the corresponding separation of the γ matrices giving
7.3. Supersymmetric
141
the splitting of vectors into transverse and longitudinal parts, as in (5.4.27). The light-cone gauge is then chosen as
P +(σ) = p+ ,
Θ− = 0 .
(7.3.9)
Other operators are then eliminated by auxiliary field equations:
1
A = 0
→
P
1
2
− = −
P
+ iΘ+′D
p
2
i
+
,
+
1
B = 0
→
D
†
− = √
/
P T D+ .
(7.3.10)
2p+
The remaining coordinates are x±, Xi(σ), and Θ+(σ), and the remaining operators
are
δ
D+ =
+ p
δΘ+
+ΣΘ+
,
P i = P i ,
Ω+ = iΘ+′
.
(7.3.11)
However, instead of imposing C = D = 0 quantum mechanically, we can solve them
classically, in analogy to the particle case. The C+ij(σ) are now local (in σ) SO(8) generators, and can be used to gauge away all but 1 Lorentz component of D+, by
the same method as (5.4.34ab) [5.30,29]. After this, D+ is just the product of this one component times its σ-derivative. Furthermore, D+ is a Virasoro algebra for D+, and can thus be used to gauge away all but the zero-mode [5.30,29] (as the usual one A did for P + in (7.3.9)), after this constraint implies D+ factors in a way analogous to (5.4.34a):
D+ = 0
→
D+ = cξ(σ) .
(7.3.12)
(The proof is identical, since D+ = 0 is equivalent to D+(σ1)D+(σ2) = 0.) We
are thus back to the particle case for D, with a single mode remaining, satisfying
√
the commutation relation c2 = p+ → c = ± p+, so D is completely determined.
Alternatively, as for the particle, we could consider D = 0 as a second-class constraint
[7.6], or impose the condition D+ = 0 (which eliminates all auxiliary string fields), as a Gupta-Bleuler constraint. This requires a further splitting of the spinors, as in (5.4.29), and the Gupta-Bleuler constraint is again a chirality condition, as in (5.4.33).
C = D = 0 are then also satisfied ala Gupta-Bleuler (with appropriate “normal
ordering”). Thus, in a “chiral” representation (as in ordinary supersymmetry) we have a chiral, “on-shell” string superfield, or wave function, Φ[x±, Xi(σ), Θa(σ)], which satisfies a light-cone field equation
∂
i
+ H Φ = 0 ,
∂x+
142
7. LIGHT-CONE QUANTUM MECHANICS
1
dσ
dσ
δ
H = −p
1
2
− = −
P
P
+ iΘa′
.
(7.3.13)
α′
2π
− =
2π
2
i
δΘa
The dimension of spacetime D = 10 and the constant in H (zero) are determined by
considerations similar to those of the bosonic case (Lorentz invariance in the light-cone formalism, or BRST invariance in the yet-to-be-constructed covariant formalism).
Similarly, light-cone expressions for qα can be obtained from the covariant ones:
dσ
δ
dσ
q+ =
− p
Q
2π
δΘ+
+ΣΘ+
≡
2π
+
,
dσ
1
q
†
− =
√
/
P
Q
2π
T
+
.
(7.3.14)
2p+
If the superstring is formulated directly in the light cone, (7.3.14) can be used as the starting point. Q+ and D+ can be considered as independent variables (instead of
Θ+ and δ/δΘ+), defined by their self-conjugate commutation relations (analogous to
those of P ):
1 {Q
2π
+(σ1), Q+(σ2)} = −δ(σ2 − σ1)2p+Σ
,
1 {D
2π
+(σ1), D+(σ2)} = δ(σ2 − σ1)2p+Σ
.
(7.3.15)
However, as described above and for the particle, D+ is unnecessary for describing
physical polarizations, so we need not introduce it. In order to more closely study the closure of the algebra (7.3.14), we introduce more light-cone spinor notation (see sect. 5.3): Working in the Majorana representation Σ = I, we introduce (D − 2)-
dimensional Euclidean γ-matrices as
/
pT → γiµν′pi ,
γ(iµµ′γj)νµ′ = 2δijδµν
,
γ(iµµ′γj)µν′ = 2δijδµ′ν′ ,
(7.3.16)
where not only vector indices i but also spinor indices µ and µ′ can be raised and
lowered by Kronecker δ’s, and primed and unprimed spinor indices are not necessarily related. (However, as for the covariant indices, there may be additional relations
satisfied by the spinors, irrelevant for the present considerations, that differ in different dimensions.) Closure of the supersymmetry algebra (on the momentum in the usual
way (5.4.4), but in light-cone notation, and with the light-cone expressions for pi, p+, and p−) then requires the identity (related to (7.3.16) by “triality”)
γi(µ|µ′γi|ν)ν′ = 2δµνδµ′ν′ .
(7.3.17)
7.3. Supersymmetric
143
This identity is actually (7.3.3b) in light-cone notation, and the equality of the dimensions of the spinor and vector can be derived by tracing (7.3.17) with δµν.
Returning to deriving the light-cone formalism from the covariant one, we can
also obtain the light-cone expressions for the Poincaré generators, which should prove important for covariant quantization, via the OSp(1,1|2) method. As in general, they are completely specified by Mij, MimM, and M2:
′
Mij =
−i ˆ
XiP j + Mij ,
′
M
j
imM
=
i ˆ
Xip+P − + Mi P j ,
′
M2 =
−2p+P − ,
(7.3.18)
where
1
1
−p
2
+P − = 1 P
+ i
Qγ
2
i
D
8p
+Q′ − i
+
,
+
8p+
1
1
Mij = −
Qγ
C
16p
+γij Q +
+ij
,
(7.3.19)
+
16p+
contain all D dependence (as opposed to X and Q dependence) only in the form of
C and D, which can therefore be dropped. (Cf. (5.4.22). γ+ picks out Q+ from Q, as
in (5.4.27).)
We can now consider deriving the BRST algebra by the method of adding 4+4
dimensions to the light-cone (sects. 3.6, 5.5). Unfortunately, adding 4+4 dimensions doesn’t preserve (7.3.17). In fact, from the analysis of sect. 5.3, we see that to preserve the symmetries of the σ-matrices requires increasing the number of commuting
dimensions by a multiple of 8, and the number of time dimensions by a multiple of 4.
This suggests that this formalism may need to be generalized to adding 8+8 dimen-
sions to the light-cone (4 space, 4 time, 8 fermionic). Coincidentally, the light-cone superstring has 8+8 physical (σ-dependent) coordinates, so this would just double
the number of oscillators. Performing the reduction from OSp(4,4|8) to OSp(2,2|4)
to OSp(1,1|2), if one step is chosen to be U(1)-type and the other GL(1)-type, it may be possible to obtain an algebra which has the benefits of both formalisms.
As for the bosonic string, the closed superstring is constructed as the direct
product of 2 open strings (1 for the clockwise modes and 1 for the counterclockwise): The hamiltonian is the sum of 2 open-string ones, and the closed-string ground state is the product of 2 open-string ones. In the case of type I or IIB closed strings, the 2 sets of modes are the same kind, and the former (the bound-state of type I open
144
7. LIGHT-CONE QUANTUM MECHANICS
strings) is nonoriented (in order to be consistent with the N = 1 supersymmetry of
the open string, rather than the N = 2 supersymmetry generated by the 2 sets of
modes of oriented, type II strings). Type IIA closed strings have Θ’s with the opposite chirality between the two sets of modes (i.e., one set has a Θα while the other has a Θα). The ground states of these closed strings are supergravity (N = 1 supergravity for type I and N = 2 for type II). The heterotic string is a closed string for which one set of modes is bosonic (with the usual tachyonic scalar ground state) while the other is supersymmetric (with the usual supersymmetric Yang-Mills ground state).
The lowest-mass physical states, due to the ∆N = 0 restriction, are the product
of the massless sector of each set (since now ∆N = H(+) − H(−) = (N(+) − 1) −
N(−)). The dimension of spacetime for the 2 sets of modes is made consistent by
compactification of some of the 26 dimensions of one and some (or none) of the 10 of the other onto a torus, leaving the same number of noncompactified dimensions (at
least the physical 4) for both sets of modes. These compactified bosonic modes can
also be fermionized (see the next section), giving an equivalent formulation in which the extra dimensions don’t explicitly appear: For example, fermionization of 16 of
the dimensions produces 32 (real) fermionic coordinates, giving an SO(32) internal
symmetry (when the fermions are given the same boundary conditions, all periodic or all antiperiodic). The resulting spectrum for the massless sector of heterotic strings consists of supergravity coupled to supersymmetric Yang-Mills with N = 1 (in 10D
counting) supersymmetry. The vectors gauging the Cartan subalgebra of the full
Yang-Mills group are the obvious ones coming from the toroidal compactification
(i.e., those that would be obtained from the noncompactified theory by just dropping dependence on the compactified coordinates), while the rest correspond to “soliton”
modes of the compactified coordinates for which the string winds around the torus. As for the dimension and ground-state mass, quantum consistency restricts the allowed
compactifications, and in particular the toroidal compactifications are restricted to those which, in the case of compactification to D = 10, give Yang-Mills group SO(32) or E8⊗E8. (These groups give anomaly-free 10D theories in their massless sectors.
There is also an SO(16)×SO(16) 10D-compactification which can be considered to
have broken N=1 supersymmetry. There are other 10D-compactifications which have
tachyons.)
Some aspects of the interacting theory will be described in chapts. 9 and 10.
Exercises
145
Exercises
(1) Use (7.1.7) as the classical solution for ˆ
Xi, and set ani = 0 for n = 1 and i = 1.
Find Xa(σ, τ ). In the center-of-mass frame, find the energy and spin, and relate
them.
(2) Do (1) in the conformal gauge by using (7.1.7) for ˆ
Xa for a = 0, 1 (same n), and
applying the constraint (6.2.5). Compare results.
(3) Prove the light-cone Poincaré algebra closes only for D = 26, and determines the constant in (7.1.8a).
(4) Find explicit expressions for all the states at the 4 lowest mass levels of the open bosonic string. For the massive levels, combine SO(D−2) representations into
SO(D−1) ones. Do the same for the 4 lowest (nontrivial) mass levels of the
closed string.
(5) Derive (7.1.15), including the expressions in terms of σ-integrals. What happens to the part of this integral symmetric in ij for Mij?
(6) Derive (7.2.8).
(7) Derive (7.3.17), both from (7.3.3b) and closure of (7.3.14). Show that it implies D − 2 = 1, 2, 4, 8.
(8) Show that (7.3.5) satisfies (7.3.3). Show that (7.3.6) gives a supersymmetry
algebra, and that the operators of (7.3.5) are invariant. Check (7.3.8) till you
drop.
(9) Find all the states in the spinning string at the tachyonic, massless, and first massive levels. Show that, using the truncation of sect. 7.2, there are equal
numbers of bosons and fermions at each level. Construct the same states using
the X and Q oscillators of sect. 7.3.
146
8. BRST QUANTUM MECHANICS
8. BRST QUANTUM MECHANICS
8.1. IGL(1)
We first describe the form of the BRST algebra obtained by first-quantization
of the bosonic string by the method of sect. 3.2, using the constraints found in the conformal (temporal) gauge in sect. 6.2.
The residual gauge invariance in the covariant gauge is conformal transformations
(modified by the constraint that they preserve the position of the boundaries). After quantization in the Schrödinger picture (where the coordinates have no τ dependence), the Virasoro operators [8.1]
G(σ) = −i 1P 2(σ)
2
− 1
,
(8.1.1)
with P as in (7.1.3) but for all Lorentz components, generate only these transfor-
mations (instead of the complete set of 2D general coordinate transformations they
generated when left as arbitrary off-shell functions of σ and τ in the classical mechanics). Using the hamiltonian form of BRST quantization, we first find the classical
commutation relations (Poisson brackets, neglecting the normal-ordering constant in (8.1.1))
[G(σ1), G(σ2)] = 2πδ′(σ2 − σ1)[G(σ1) + G(σ2)]
= 2π [δ(σ2 − σ1)G′(σ2) + 2δ′(σ2 − σ1)G(σ2)] ,
(8.1.2a)
or in mode form
iG(σ) =
Lne−inσ
→
[Lm, Ln] = (n − m)Lm+n .
(8.1.2b)
These commutation relations, rewritten as
dσ1
dσ
dσ
λ
2 λ
λ
′(σ)G(σ) ,
(8.1.3a)
2π
1(σ1)G(σ1),
2π
2(σ2)G(σ2)
=
2π [2(σ)λ1]
8.1. IGL(1)
147
correspond to the 1D general coordinate transformations (in “zeroth-quantized” no-
tation)
∂
∂
∂
λ
′
2(σ)
, λ
= λ
,
(8.1.3b)
∂σ
1(σ) ∂σ
[2λ1] ∂σ
giving the structure constants
f (σ1, σ2; σ3) = 2πδ′(σ2 − σ1) [δ(σ1 − σ3) + δ(σ2 − σ3)] ,
(8.1.4a)
or
←
→
∂
λ j
i
k
1 λ2 f ij
↔
λ1
λ
∂σ 2
.
(8.1.4b)
More generally, we have operators whose commutation relations
[G(σ1), O(σ2)] = 2π [δ(σ2 − σ1)O′(σ2) + wδ′(σ2 − σ1)O(σ2)]
(8.1.5a)
represent the transformation properties of a 1D tensor of (covariant) rank w, or a
scalar density of weight w:
λG, O = λO′ + wλ′O .
(8.1.5b)
Equivalently, in terms of 2D conformal transformations, it has scale weight w. (Re-
member that conformal transformations in D = 2 are equivalent to 1D general co-
ordinate transformations on σ±: See (6.2.7).) In particular, we see from (8.1.2a)
that G itself is a 2nd-rank-covariant (as opposed to contravariant) tensor: It is the energy-momentum tensor of the mechanics action. (It was derived by varying that
action with respect to the metric.) The finite form of these transformations follows from exponentiating the Lie algebra represented in (8.1.3): (8.1.5) can then also be rewritten as the usual coordinate transformations
∂σ′ w O′(σ′) = O(σ) ,
(8.1.6a)
∂σ
where the primes here stand for the transformed quantities (not σ-derivatives) or as (dσ′)wO′(σ′) = (dσ)wO(σ) ,
(8.1.6b)
indicating their tensor structure. In particular, a covariant vector (w = 1) can be integrated to give an invariant. P is such a vector (and the momentum p the corresponding conformal invariant), which is why G has twice its weight (by (8.1.1)).
Before performing the BRST quantization of this algebra, we relate it to the
light-cone quantization of the previous chapter. The constraints (8.1.1) can be solved
148
8. BRST QUANTUM MECHANICS
in a Gupta-Bleuler fashion in light-cone notation. The difference between that and
actual light-cone quantization is that in the light-cone quantization P − is totally eliminated at the classical level, whereas in the light-cone notation for the covariant-gauge quantization the constraint is used to determine the dependence on the ±
oscillators in terms of the transverse oscillators. One way to do this would be to start with a state constructed from just transverse oscillators (as in light-cone quantization) and add in terms involving longitudinal oscillators until the constraints are satisfied (or actually half of them, ala Gupta-Bleuler). A simpler way is to start at the classical level in an arbitrary conformal gauge with transverse oscillators, and then conformally transform them to the light-cone gauge to see what a transverse oscillator (in the
physical sense, not the light-cone-index sense) looks like. We thus wish to consider dσ′P ′i(σ′) = dσP i(σ) ,
1
σ′ =
ˆ
X
p
+(σ) = σ + oscillator-terms
.
(8.1.7)
+
(Without loss of generality, we can work at x+ = 0. Equivalently, we can explicitly subtract x+ from ˆ
X+ everywhere the latter appears in this derivation.) If we consider
the same transformation on P
′
+ (using ∂ ˆ
X/∂σ ≡ P ), we find P + = p+, the light-cone
gauge. (8.1.7) can be rewritten as
1
P ′(σ
ˆ
1) =
dσ2 δ σ1 −
X
p
+(σ2)
P (σ2) .
(8.1.8)
+
(8.1.7) follows upon replacing σ
′
1 with σ1
and integrating out the δ-function (with
the Jacobian giving the conformal weight factor). A more convenient form for quan-
tization comes from the mode expansion: Multiplying by einσ1 and integrating,
dσ
α ′
n =
ein ˆ
X+(σ)/p+ P (σ) .
(8.1.9)
2π
These (“DDF”) operators [8.2] (with normal ordering, as usual, upon quantization,
and with transverse Lorentz index i, and n > 0) can be used to create all physi-
cal states. Due to their definition in terms of a conformal transformation from an
arbitrary conformal gauge to a completely fixed (light-cone) gauge, they are auto-
matically conformally invariant: i.e., they commute with G. (This can be verified to remain true after quantization.) Consequently, states constructed from them satisfy the Gupta-Bleuler constraints, since the conformal generators push past these operators to hit the vacuum. Thus, these ope