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We have mentioned above that the non-gauged action of the supersymmetric string is invariant under diffeomorphisms and (super-)Weyl transformations. Therefore, it is not

3

A detailed discussion of spinors in higher dimensions and the charge conjugation matrix follows in Chapter 3 and Appendix A.

2.1 Conformal Field Theory on the World-Sheet 17

possible to naively plug this action into the path integral. One must only integrate over configurations of the world-sheet metric and the gravitino which are not related by these symmetry transformations. Otherwise this leads to a massive over-counting. Usually, this is settled by introducing a Fadeev–Popov determinant to the partition function. As can be shown this is equivalent to adding a ghost action to (2.6) which becomes in superconformal gauge Sgh = 1 4π Z d2z b∂c¯ + ¯b ∂¯c+β∂γ¯ + ¯β ∂¯γ. (2.25) The fields b, c are anticommuting ghost fields, which are necessary for the quantization of the bosonic action (2.5), while the commuting super-ghost fields β, γ are required in addition for the supersymmetric action (2.6). The equations of motion, derived from the action above, identify these fields as chiral or anti-chiral, respectively:

¯

∂b= ¯∂c= 0, ∂β¯ = ¯∂γ = 0. (2.26) We restrict our discussion to the chiral fields in the following and quickly summarize their conformal properties. The energy momentum tensors

Tb,c(z) =−2b(z)∂c(z)−∂b(z)c(z), Tβ,γ(z) =₋3

2β(z)∂γ(z)− 1

2∂β(z)γ(z), (2.27) imply that the ghost fields have the conformal weights

h(c) =₋1, h(b) = 2, h(γ) =₋1

2, h(β) = 3

2. (2.28)

The central charges of the ghost and superghost CFTs can be obtained from the OPEs of the energy momentum tensors. One finds that cb,c _{=−26 and c}β,γ _{= +11. If the central}

charge of the matter system (2.11) is also taken into account, the total central charge vanishes for D= 10:

cX,ψ_+cb,c_+cβ,γ ₌ 3

2D−26 + 11 = 0. (2.29)

As previously described the quantum theory does not suffer in this case from a supercon- formal anomaly. The cb- andγβ-propagators, derived from the action (2.25), demand that the ghost fields satisfy the OPEs

c(z)b(w)_∼ 1

z−w, γ(z)β(w)∼ 1

z−w. (2.30)

As in the case of the matter fields we can perform a Laurent expansion of the ghost and superghost fields. We obtain for the former

c(z) =X n cnz−n+1, b(z) = X n bnz−n−2, (2.31)

18 2. Scattering in String Theory

where the modes have to satisfy the anticommutation relations

{bm, cn}=δm+n, {bm, bn}={cm, cn}= 0 (2.32)

due to the OPE ofc(z) andb(w). The operator-state correspondence implies that vacuum state of the ghost system is annihilated by all bn, n > −2 and cn, n > 1, but not by the

mode c1:

lim

z→0c(z)|0ib,c =c1|0ib,c ≡ |1ib,c 6= 0. (2.33)

Due to [L0, c1] = −1, where L0 is the zero mode of the energy momentum tensor Tb,c(z),

the state_|1_ib,c is the state with lowest energy and therefore the proper ground state of the

ghost system. It is also annihilated by c1 because of {c1, c1}= 0.

We now discuss the superghost fields. These are associated to the fermions ψm _and

hence satisfy the same periodicity conditions. Therefore, the mode expansion also yields an NS and an R sector, γ(z) =X r γrz−r+1/2, β(z) = X r βrz−r−3/2, r ∈ ( Z₊1 2 : NS sector, Z _{: R sector.} (2.34)

The modes must satisfy the commutation relations

[γr, βs] =δr+s, [βr, βs] = [γr, γs] = 0 (2.35)

because the superghost fields are of bosonic type. As in the case above, the shift in the mode expansion implies that the vacuum in the NS sector is not a highest weight state. In fact, due to [L0, γ1/2] = −1/2γ1/2, the vacuum can be lowered to arbitrary negative

energies because γ1/2 does not square to zero. In the R sector an operator analogous to the

spin field is needed creating a branch cut and interpolating between the different boundary conditions. As shown in [36] by bosonizing the superghost system, the proper ground states for the two sectors are

e−φ(0)|0iβ,γ ≡ q=−1 2 β,γ : NS sector, e−φ(0)/2_|0_iβ,γ ≡ q=₋1_β,γ : R sector. (2.36) These states are annihilated by γ1/2 and γ1 as required.

The action (2.25) is invariant under two chiral U(1) symmetries generated by the cur- rents

jb,c =−b(z)c(z), jβ,γ =−β(z)γ(z). (2.37) The OPE of these currents with the respective energy momentum tensor Tb,c _orTb,j

T(z)j(w)_∼ Q (z₋w)3 + j(w) (z₋w)2 + ∂j(w) z₋w (2.38)

2.1 Conformal Field Theory on the World-Sheet 19

exhibits an anomaly. The charge Q takes the value −3 for the ghost and +2 for the superghost system. The anomalous conservation law of the currents reads

¯

∂j(z) = 1 4Q

√

hR , (2.39)

where h is the determinant of the world-sheet metric and R the corresponding curvature scalar. It can be shown that the anomaly arises from the presence of (super-)ghost zero modes. It is possible to calculate their number from (2.39) using the Riemann-Roch theo- rem:

Nc−Nb = 3−3g , Nγ−Nβ = 2−2g . (2.40)

This has profound consequences. For string scattering at g loops the string world-sheet is a Riemann surface of genus g. The vertex operators creating string states have to be inserted with the right superghost factors in order to cancel the superghost background charge of 2−2g. Furthermore, at tree-level, i.e. g = 0, the presence of three ghost zero modes follows from the three globally defined diffeomorphisms on the sphere. In order to cancel this residual gauge freedom three vertex operators positions can be fixed in the calculation of the amplitude.