The superconformal ghosts

The vertex operator Θ_sfor an R state |s is

Θ_s∼= exp

 i^

a

s_aH^a



. (10.3.29)

This operator, which produces a branch cut in ψ^µ, is sometimes called a spin field. For closed string states, this is combined with the appropriate antiholomorphic vertex operator, built from ˜H^a.

The general bc CFT, renaming ψ → b and ψ → c, is obtained by modifying the energy-momentum tensor of the λ = ¹₂ theory to

T_B^(λ)= T_B^(1/2)− (λ −¹₂)∂(bc) . (10.3.30) The equivalences (10.3.13) give the corresponding bosonic operator

T_B^(λ) ∼=T_B^H− i(λ − ¹₂)∂²H . (10.3.31) This is the same as the linear dilaton CFT, with V =−i(λ − ¹₂). With this correspondence between V and λ, the linear dilaton and bc theories are equivalent,

b ∼=e^iH , c ∼=e^−iH . (10.3.32) As a check, the central charges agree,

c = 1− 3(2λ − 1)²= 1 + 12V² . (10.3.33) So do the dimensions of the fields (10.3.32), λ for b and 1−λ for c, agreeing with k²/2 + ikV for e^ikH. The nontensor behaviors of the currents bc and i∂H are also the same. Since the inner product for the reparameterization ghosts makes b and c Hermitean, the bosonic field H must be anti-Hermitean in this application. The bosonization of the ghosts is usually written in terms of a Hermitean field with the opposite sign OPE,

H → iρ ; c ∼=e^ρ , b ∼=e^−ρ . (10.3.34)

10.4 The superconformal ghosts

To build the BRST current we will need, in addition to the anticommuting b and c ghosts of the bosonic string, commuting ghost fields β and γ of weight (³₂, 0) and (−¹₂, 0), and the corresponding antiholomorphic fields.

The action for this SCFT was given in eq. (10.1.17) and the currents T_B and TF in eq. (10.1.21). The ghosts β and γ must have the same periodicity (10.2.4) as the generator TF with which they are associated.

This is necessary to make the BRST current periodic, so that it can be

integrated to give the BRST charge. Thus, and similarly for the antiholomorphic fields. The (anti)commutators are

[γr, β_s] = δr,−s , {bm, c_n} = δn,−m . (10.4.2) Define the ground states|0 _NS,R by

β_r|0 _NS = 0 , r≥ ¹₂ , γ_r|0 _NS = 0 , r≥ ¹₂ (10.4.3a) β_r|0 _R = 0 , r≥ 0 , γr|0 _R= 0 , r ≥ 1 , (10.4.3b) b_m|0 _NS,R = 0 , m≥ 0 , cm|0 _NS,R= 0 , m≥ 1 . (10.4.3c) We have grouped β0 with the lowering operators and γ0 with the raising ones, in parallel with the bosonic case. The spectrum is built as usual by acting on the ground states with the raising operators. The generators are

The normal ordering constant is determined by the usual methods to be R: a^g =−5

8 , NS: a^g=−1

2 . (10.4.5)

Vertex operators

We focus here on the βγ CFT, as the bc parts of the vertex operators are already understood. Let us start by considering the state corresponding to the unit operator. From the Laurent expansions (10.4.1) it is in the NS sector and satisfies

βr|1 = 0 , r ≥ −1

2 , γr|1 = 0 , r ≥ 3

2 . (10.4.6) This is not the same as the ground state |0 _NS: the mode γ_1/2 annihilates

|0 NS while its conjugate β_−1/2 annihilates |1 . We found this also for the bc ghosts with c₁ and b₋₁. Since anticommuting modes generate just two states, we had the simple relation|0 = c1|1 (focusing on the holomorphic side). For commuting oscillators things are not so simple: there is no state

10.4 The superconformal ghosts 17 in the Fock space built on |1 by acting with γ1/2 that has the properties of|0 _NS. The definition of the state |0 _NS translates into

γ(z)δ(γ(0)) = O(z) , β(z)δ(γ(0)) = O(z⁻¹) , (10.4.7) for the corresponding operator δ(γ). The notation δ(γ) reflects the fact that the field γ has a simple zero at the vertex operator. Recall that for the bc ghosts the NS ground state maps to the operator c, which is the anticommuting analog of a delta function. One can show that an insertion of δ(γ) in the path integral has the property (10.4.7).

To give an explicit description of this operator it is again convenient to bosonize. Of course β and γ are already bosonic, but bosonization here refers to a rewriting of the theory in a way that is similar to, but a bit more intricate than, the bosonization of the anticommuting bc theory.

Start with the current βγ. The operator product βγ(z) βγ(0)∼ −1

z² (10.4.8)

is the same as that of ∂φ, where φ(z)φ(0) ∼ − ln z is a holomorphic scalar. Holomorphicity then implies that this equivalence extends to all correlation functions,

βγ(z) ∼=∂φ(z) . (10.4.9)

The OPE of the current with β and γ then suggests

β(z)∼=^? e^−φ(z) , γ(z)∼=^? e^φ(z). (10.4.10) For the bc system we would be finished: this approach leads to the same bosonization as before. For the βγ system, however, the sign of the current–current OPE and therefore of the φφ OPE is changed. The would-be bosonization (10.4.10) gives the wrong OPEs: it would imply

β(z)β(0)= O(z^{? −1}) , β(z)γ(0)= O(z^{? 1}) , γ(z)γ(0)= O(z^{? −1}) , (10.4.11) whereas the correct OPE is

β(z)β(0) = O(z⁰) , β(z)γ(0) = O(z⁻¹) , γ(z)γ(0) = O(z⁰) . (10.4.12) To repair this, additional factors are added,

β(z) ∼=e^−φ(z)∂ξ(z) , γ ∼=e^φ(z)η(z) . (10.4.13) In order not to spoil the OPE with the current (10.4.9), the new fields η(z) and ξ(z) must be nonsingular with respect to φ, which means that the ηξ theory is a new CFT, decoupled from the φ CFT. Further, the equivalence (10.4.13) will hold — all OPEs will be correct — if η and ξ satisfy

η(z)ξ(0)∼ 1

z , η(z)η(0) = O(z) , ∂ξ(z)∂ξ(0) = O(z) . (10.4.14)

This identifies the ηξ theory as a holomorphic CFT of the bc type: the OPE of like fields has a zero due to the anticommutativity.

It remains to study the energy-momentum tensor. We temporarily con-sider the general βγ system, with β having weight λ^. The OPE

T (z)βγ(0) = 1− 2λ^

z³ + . . . (10.4.15) determines the φ energy-momentum tensor,

T_B^φ=−1

2∂φ∂φ +1

2(1− 2λ^)∂²φ . (10.4.16) The exponentials in the bosonization (10.4.13) thus have weights λ^−1 and

−λ^ respectively, as compared with the weights λ^ and 1− λ^ of β and γ.

This fixes the weights of η and ξ as 1 and 0: this is a λ = 1 bc system, with

T_B^ηξ =−η∂ξ (10.4.17)

and

T_B^βγ ∼=T_B^φ+ T_B^ηξ . (10.4.18) As a check, the central charges are 3(2λ^− 1)²+ 1 for T_B^φ and−2 for TB^ηξ, adding to the 3(2λ^− 1)²− 1 of the βγ CFT. The need for extra degrees of freedom is not surprising. The βγ theory has a greater density of states than the bc theory because the modes of a commuting field can be excited any number of times. One can check that the total partition functions agree, in the appropriate sectors.

If need be one can go further and represent the ηξ theory in terms of a free boson, conventionally χ with χ(z)χ(0)∼ ln z, as in the previous section. Thus

η ∼=e^−χ , ξ ∼=e^χ , (10.4.19a) β ∼=e^−φ+χ∂χ , γ ∼=e^φ−χ . (10.4.19b) The energy-momentum tensor is then

T_B =−1

2∂φ∂φ +1

2∂χ∂χ + 1

2(1− 2λ^)∂²φ + 1

2∂²χ . (10.4.20) For the string, the relevant value is λ^ = ³₂. The properties (10.4.7) of δ(γ) determine the bosonization,

δ(γ) ∼=e^−φ , h = 1

2 . (10.4.21)

The fermionic parts of the tachyon and massless NS vertex operators are then

e^−φ , e^−φe^±iHa (10.4.22) respectively. For λ^= ³₂, the exponential e^lφ has weight −¹₂l²− l.

10.4 The superconformal ghosts 19 The operator Σ corresponding to|0 _R satisfies

β(z)Σ(0) = O(z^−1/2) , γ(z)Σ(0) = O(z^1/2) . (10.4.23) This determines

Σ = e^−φ/2 , h = 3

8 . (10.4.24)

Adding the contribution −1 of the bc ghosts, the weight of e^−φ/2 and of e^−φ agree with the values (10.4.5). The R ground state vertex operators are then

Vs= e^−φ/2Θ_s , (10.4.25) with the spin field Θ_s having been defined in eq. (10.3.29).

We need to extend the definition of world-sheet fermion number F to be odd for β and γ. The ultimate reason is that it anticommutes with the supercurrent TF and we will need it to commute with the BRST operator, which contains terms such as γTF. The natural definition for F is then that it be the charge associated with the current (10.4.9), which is l for e^lφ. Again, it is conserved by the OPE. This accounts for the ghost contributions in eq. (10.2.24). Note that this definition is based on spin rather than statistics, since the ghosts have the wrong spin-statistics relation; it would therefore be more appropriate to call F the world-sheet spinor number.

For completeness we give a general expression for the cocycle for exponentials of free fields, though we emphasize that for most purposes the details are not necessary. In general one has operators

exp(ikL· HL+ ikR· HR) , (10.4.26) with the holomorphic and antiholomorphic scalars not necessarily equal in number. The momenta k take values in some lattice Γ. The naive operator product has the phase of z^−k◦k, and for all pairs in Γ, k◦ k^ must be an integer. The notation is as in section 8.4, k◦ k^ = k_L· k_L^ − k_R· k^_R. When k ◦ k^ is an odd integer the vertex operators anticommute rather than commute. A correctly defined vertex operator is

C_k(α0)^◦◦exp(ikL· HL+ ikR· HR)^◦◦ (10.4.27) with the cocycle Ck defined as follows. Take a set of basis vectors kα for Γ; that is, Γ consists of the integer linear combinations nαk_α. Similarly write the vector of zero-mode operators in this basis, α0= α0αkα, Then for k = n_αk_α,

C_k(α0) = exp

 πi^

α>β

n_αα_0βk_α◦ kβ



. (10.4.28)

This generalizes the simple case (8.2.22). The reader can check that vertex operators with even k◦ k now commute with all vertex operators, and those with odd k◦ k anticommute among themselves. Note that a cocycle has no effect on the commutativity of a vertex operator with itself, so an exponential must be bosonic if k◦ k is even and fermionic if k ◦ k is odd.