— DRAFT — IASSNS-HEP-97/72

Lecture 14

IX. Superstring Perturbation Theory

Eric D’Hoker

We shall now develop a systematic treatment of superstring perturbation theory in the RNS formulation. Local N = 1 supersymmetry (SUSY) on the worldsheet plays a fundamental rôle, but this symmetry is not manifest in the component formalism, set up in §VII. The N = 1 superfield formalism, formulated in the language of N = 1 supergeometry, treats diffeomorphisms and local supersymmetry transformations on an equal footing, and renders both manifest. Their combined action makes up the group of super-diffeomorphisms of the worldsheet super-manifold, and local supersymmetry invariants can be constructed from super-diffeomorphism tensors. We first present the different ingredients needed and then apply these methods to the RNS superstrings. A general reference for this subject is E. D’Hhoker and D. H. Phong, “The Geometry of String Perturbation Theory”, Rev. Mod. Phys. 60 (1988), p. 917.

A) N = 1 supergeometry (also called N = 1, D = 2 supergravity)

We consider a supermanifold Σ of dimension (2|2), and use local coordinates ξ^{M} = (ξ^{m}, θ^{μ}), m = 1, 2, μ = 1, 2 with the grading

ξ^{M}ξ^{N} = (−)^{MN}ξ^{N}ξ^{M}  (9.A.1)

where (−)^{MN} equals 1 unless M = μ, N = ν, in which case it equals −1. We introduce a local orthonormal frame

E^{A} ≡ dξ^{M}E_{M}^{A}  A = (a, α) a = z, z̄; α = +, − ,  (9.A.2)

where it is understood that a has the same grading as m, while α has the same grading as μ. We denote the inverse frame by E_{A}^{M}, and we have E_{A}^{M} E_{M}^{B} = δ_{A}^{B} and E_{M}^{A} E_{A}^{N} = δ_{M}^{N}.

IX.1

Next, we consider frame rotations by a group of U(1) gauge transformations, under which the fields E^{z}, Ez̄, E^{+}, E^{−} have weights −1, +1, −{1 \atop 2}, +{1 \atop 2}, respectively. A general superfield V has U(1) weight −n if it transforms as (E^{z})^{⊗n}. We introduce a U(1) gauge field or connection

Ω = dξ^{M}Ω_{M}  (9.A.3)

with associated covariant derivatives

𝒟_{A}^{(n)}V ≡ E_{A}^{M}(∂_{M} + inΩ_{M})V .  (9.A.4)

Here ∂_{M} ≡ ∂/∂ξ^{M} and i = \sqrt{−1} . Torsion T_{AB}^{C} and curvature R_{AB} superfield tensors are defined by the graded commutators

[𝒟_{A}, 𝒟_{B}]V = T_{AB}^{C}𝒟_{C}V + in R_{AB}V .  (9.A.5)

The data of N = 1 supergeometry is the frame E^{A} and the U(1) connection Ω, together with the following constraints on the torsion and curvature, arising in [𝒟_{α}, 𝒟_{β}]:

T_{αβ}^{γ} = 0; T_{αβ}^{c} = 2γ_{αβ}^{c}; R_{++} = R_{−−} = 0  (9.A.6)

Here, γ^{c} are the 2-dimensional Dirac matrices, satisfying {γ^{a}, γ^{b}} = −η^{ab}, with a convenient representation given by γ_{++}^{z} = γz̄ ; −− = 1, all other components = 0. The constraints of (9.A.6) are equivalent to the following set of constraints

\begin{matrix}𝒟_{+}^{2} = 𝒟_{z}, 𝒟_{−}^{2} = 𝒟z̄ \\ \{𝒟_{+}, 𝒟_{−}\}V = in R_{+−}V , \end{matrix}  (9.A.7)

where V is any superfield of weight n. The structure relations of the remaining covariant derivatives are defined by using the (graded) Jacobi identities. The only free quantity in this supergeometry is the curvature superfield R_{+−} = R_{−+} ≡ R.

The symmetry transformations on the data E^{A} and Ω that preserve the constraints (9.A.6) or equivalently (9.A.7) are

IX.2 1) sDiff(Σ): super-diffeomorphisms ξ^{M} → ξ′^{M}(ξ^{N})

contains Diff(Σ), local N = 1 susy, and further algebraic transformatons.

2) a) sWeyl(Σ): generated by a superfield Σ, contains Weyl(Σ) and further algebraic trans-

formations.

b) sU(1): generated by a superfield L, contains U(1) gauge transformations and

further algebraic transformations.

It is convenient to combine transformations a) and b) into complexified Weyl transformations, given in terms of a complex superfield Λ, defined by

Λ ≡ Σ − iL, Λ^{∗} ≡ Σ + iL .  (9.A.8)

The superderivatives transform under sWeyl × sU(1) as

\begin{aligned}𝒟_{+}^{(n)} & = e^{nΛ−{1 \atop 2}Λ^{∗}} 𝒟̂_{+}^{(n)}e^{−nΛ} \\ 𝒟_{−}^{(n)} & = e^{−nΛ^{∗}−{1 \atop 2}Λ}𝒟̂_{−}^{(n)}e^{nΛ^{∗}} \end{aligned}  (9.A.9)

and the curvature transforms as

R_{+−} = e^{−Σ} (R̂_{+−} − 2i𝒟̂_{+}𝒟̂_{−}Σ) .  (9.A.10)

From (9.A.9), the transformation rules for E_{±} and Ω_{±} may be deduced, and from these combined with the torsion constraints (9.A.6), the transformation laws of E_{z}, Ez̄, Ω_{z} and Ωz̄ follow.

N = 1 supergeometry admits an almost complex structure

J_{M}^{N} = E_{M}^{a}ϵ_{a}^{b}E_{b}^{N} + E_{M}^{α}γ̄_{α}^{β}E_{β}^{N}  (9.A.11)

where ϵ^{2} = γ̄^{2} = −1. In fact, J is a complex structure provided the constraints given in (9.A.6) are satisfied. (This may be viewed as one of the motivations for choosing those constraints.) We shall denote local complex coordinates by (ξ, ξ̄, θ, θ̄) with ξ = 1/\sqrt{2} (ξ^{1} + iξ^{2}), θ = 1/\sqrt{2} (θ^{1} + iθ^{2}), and ξ̄ and θ̄ their complex conjugates.

IX.3

By a combined sDiff, sWeyl and sU(1) transformation, any N = 1 supergeometry may be mapped on a locally flat (Euclidean) supergeometry, characterized by the superderivatives

\begin{aligned}𝒟_{+} & = \frac{∂}{∂θ} + θ \frac{∂}{∂ξ} \\ 𝒟_{−} & = \frac{∂}{∂θ̄} + θ̄\frac{∂}{∂ξ̄} \end{aligned}  \begin{aligned}𝒟_{z} & = 𝒟_{+}^{2} = \frac{∂}{∂ξ} \\ 𝒟z̄ & = 𝒟_{−}^{2} = \frac{∂}{∂ξ̄} . \end{aligned}  (9.A.12)

The residual symmetry group of sDiff × sU(1) × sWeyl that leaves this supergeometry invariant is the group of superconformal transformations, which consists of superanalytic diffeomorphisms, ξ → ξ′(ξ, θ), ξ̄ → ξ̄′(ξ̄, θ̄), θ → θ′(ξ, θ); θ̄ → θ̄′(ξ̄, θ̄), combined with sWeyl and sU(1) transformations.

Wess-Zumino gauge is obtained by fixing a gauge for the action of the algebraic parts of sDiff(Σ), sU(1) and sWeyl. The remaining independent fields are e_{m}^{a}, χ_{m}^{α} and an auxiliary field A, and we have

\begin{aligned}E_{m}^{a} & = e_{m}^{a} + θγ^{a}χ_{m} − {1 \atop 2}θθ̄e_{m}^{a}A \\ E_{m}^{α} & = −{1 \atop 2}χ_{m}^{α} − \frac{i}{2}θ^{β}(γ^{a})_{β}^{α}e_{m}^{a}A + ⋯ \\ E_{μ}^{a} & = (γ^{a})_{μ}^{β}θ_{β} \\ E_{μ}^{α} & = δ_{μ}^{α}(1 + \frac{i}{4}θθ̄A) \end{aligned}  (9.A.13)

This presentation of the supergeometry is so-called in components.

Integration over Σ is performed with the help of a super-measure dμ_{E}, defined as follows:

dμ_{E} ≡ s det E_{M}^{A}dξ dξ̄dθ dθ̄  (9.A.14)

where the superdeterminant (also sometimes referred to as the Berezinian) is defined by

s det E_{M}^{A} ≡ det(E_{m}^{a} − E_{M}^{α}(E_{μ}^{α})^{−1}E_{μ}^{a}) · (det E_{μ}^{α})^{−1}  (9.A.15)

Of course, dμ_{E} is sDiff invariant. Inner products on superfields Ψ_{1} and Φ_{2} of weight n are defined by

⟨Φ_{1}, Φ_{2}⟩ = ∫_{Σ} dμ_{E} Φ_{1}^{∗}Φ_{2} .  (9.A.16)

IX.4 B) Functional Integral Representation of Transition Amplitudes

The RNS worldsheet action, introduced in §VII in components, admits a simple expression in superfield form. We introduce a string coordinate superfield of sU(1) weight 0; X = Σ → ℝ^{10}

X^{μ} = x^{μ} + θ^{μ}ψ_{+}^{μ} + θ̄ψ_{−}^{μ} + θθ̄F^{μ}  μ = 0, 1, …, 9 .  (9.B.1)

The action

S_{X} = \frac{1}{4π} ∫_{Σ} dμ_{E}𝒟_{−}X · 𝒟_{+}X  (9.B.2)

reduces to the RNS action,^{∗} except for the fact that it has an entire term of the form F^{2}. This term is non-dynamical and the F field may be integrated out without any consequences.

The next action we need is that of the super (B, C) system: let B and C be superfields of sU(1) weight n and {1 \atop 2} − n, respectively. The (B, C) action is then

S_{BC} = \frac{1}{2π} ∫_{Σ} dμ_{E}(B𝒟_{−}^{(n)}C + B̄𝒟_{+}^{(−n)}C̄) .  (9.B.4)

To make contact with the (b, c) action, we decompose the superfields B, C in components

\begin{aligned}B & = β + θb + θ̄b′ + θθ̄β′ \\ C & = c + θγ + θ̄γ′ + θθ̄c′ \end{aligned}  (9.B.5)

The fields b′, β′, γ′ and c′ are auxiliary, just as F was an auxiliary field in S_{X}, and we shall drop their contribution to S_{BC}. The U(1) weight assignments in Wess-Zumino gauge are now obtained by using the fact that θ (resp. θ̄) have weight −\begin{matrix}1 \\ 2 \end{matrix} (resp. \begin{matrix}1 \\ 2 \end{matrix}). Hence, the weights are, for n = −1:

field  β b  γ  c ; U(1) weight  3/2 2 −1/2 −1  (9.B.6)
S_{X} = \frac{1}{4π} ∫_{Σ} d^{2}z(∂_{z}x · ∂z̄x − ψ_{+} · ∂z̄ψ_{+} − ψ_{−} · ∂_{z}ψ_{−} − F^{2})  (9.B.3)

IX.5 We recognize the (b, c) system of anti-commuting ghost fields, and we now also find a (β, γ) system of commuting ghost fields (called superghost) which has precisely the correct weight assignments to be the ghost system for local supersymmetry. The (βγ) fields are worldsheet spinors, and their spin structure must be the same as that of ψ_{+}^{μ}, in order for (βγ) to be N = 1 local supersymmetry ghosts. The BC action may be worked out in components and becomes in W Z gauge:

∗ In locally flat coordinates, this action equals in components so that the sign of the ψ-kinetic term is opposite to the convention we used previously. Notice that this reverses the sign of the ψ stress tensor. Apologies! (Conventions are to be uniformized in the final write-up.)

S_{BC} = \frac{1}{2π} ∫_{Σ} d^{2}z[b∇_{(−1)}^{z}c + β∇_{(−1/2)}^{z}γ − χz̄^{+}Sz̄ ; + + c.c.]  (9.B.7)

where the BC supercurrent is

S_{z+} ≡ +{1 \atop 2} bγ − \frac{3}{2} β∇_{z}c − ∇_{z}βc  (9.B.8)

This current is the N = 1 superpartner of the stress tensor and it is conserved; in fact it is analytic.

The combined X, (B, C) system for D = 10 components of X is superconformal invariant, and so the super-conformal field theory associated with this system depends upon N = 1 supergeometry only through its supermoduli. Recall that supermoduli space is defined by

sℳ_{h} ≡ \frac{\{E_{M}^{A}, Ω_{M} satisfying constraints of (9.A.6)\}}{sDiff × sWeyl × sU(1)}  (9.B.9)

and is a space of dimension

dim sℳ_{h} = \{ \begin{matrix}(0|0) & h = 0 \\ \{ & (1|0)  h = 1 even spin structure (1|1)  h = 1 odd spin structure \\ (3h − 3|2h − 2) & n ≥ 2 \end{matrix}  (9.B.10)

The holomorphic cotangent space to sℳ_{h} is spanned by super-holomorphic 3/2 differentials Φ_{K}, which satisfy

𝒟_{−}^{(3/2}Φ_{K} = 0 Φ_{K} odd graded K = 1, …, dim sℳ_{h} (9.B.11)

Super-Beltrami differentials are defined as the duals to Φ_{K}, and may be parametrized by the super-moduli variations of E_{M}^{A}:

μ_{K} ≡ E_{−}^{M} \frac{∂E_{M}^{z}}{∂m_{K}}  K = 1, …, dim sℳ_{h} .  (9.B.12)

IX.6 It is instructive to work out the μ_{K} in Wess-Zumino gauge:

μ_{K} = θ̄(ez̄^{n} \frac{∂e_{n}^{z}}{∂m_{K}} − θ \frac{∂χz̄^{+}}{∂m_{K}} ) .  (9.B.13)

The first term on the r.h.s. is recognized as the ordinary Beltrami differential, for m_{K} an even modulus, while the second term corresponds to a novel odd Beltrami differential for m_{K} an odd modulus.

We are now ready to state a preliminary Definition of transition amplitudes for RNS superstring amplitudes. This preliminary definition is the natural generalization of the “summation over surfaces” definition adopted for bosonic string amplitudes in §I. It is preliminary in the sense that it treats left and right chirality worldsheet spinors with the same spin structure. In the final definition of transition amplitudes, these spin structures will have to be summed over independently for left and right movers, and this will be presented in §IX.F, after the Chiral Splitting Theorem has been established. The preliminary definition will be useful, however, in order to introduce superghosts, and it is needed in order to set up the Chiral Splitting Theorem. Transition amplitudes are defined by

A_{h} = ∫ DE_{M}^{A}DΩ_{M}δ (constraints) ∫ DX^{μ}V_{1} … V_{N}e^{−S_{X}}  (9.B.14)

for certain vertex operators V_{i}, to be specified shortly.

In the critical dimension, D = 10, and for super-conformally invariant vertex operators V_{i}, we have the following Theorem

\begin{aligned}A_{h} = ∫_{sℳ_{h}} ∏_{K} dm_{K} ∫ & DX^{u} ∫ D(BB̄) ∫ D(CC̄)V_{1} … V_{N}e^{−S_{X}−S_{BC}} \\ & · ∏_{K} δ(⟨μ_{κ}, B⟩)δ(⟨μ̄_{κ}, B̄⟩) \end{aligned}  (9.B.15)

The proof of this result is much more involved than in the case or bosonic strings, because the inner products ⟨Φ_{1}, Φ_{2}⟩ are no longer positive on superfields, and extra care must be used in dealing with the 𝒟_{+} and 𝒟_{−} operators. We shall not prove it here; we refer the reader to Rev. Mod. Phys. 60 (1988) p. 917.

IX.7 C) Superconformal Field Theory (some basics)

A conformal field theory with an additional N = 1 local supersymmetry invariance is automatically invariant under superconformal transformations, and is an N = 1 superconformal field theory. Each conformal field has a superpartner (which is also a conformal field), and together both fields fit into a superconformal field; X^{μ}, B, C are examples of such fields. Conserved currents are other examples of conformal fields that generalize to supercurrents. Conservation of a supercurrent J is given by

𝒟_{−}J = 0  (9.C.1)

In local super-complex coordinates, this means that

J = J(z, θ) = J_{1/2}(z) + θJ_{1}(z)  (9.C.2)

with J_{1/2} and J_{1} holomorphic in z. The associated charge is defined by

Q_{J} = ∮ dz dθ J = ∮ dz J_{1}(z)  (9.C.3)

Important examples of supercurrents include the stress tensor, which is built up out of the ordinary stress tensor T_{zz} and the (worldsheet) supercurrent S_{z+}:

T (z, θ) ≡ S_{z+}(z) + θT_{zz}(z) .  (9.C.4)

For the X-system, T is given by

\begin{aligned}T^{(X)} & = −{1 \atop 2}𝒟_{+}X · 𝒟_{+}^{2}X \\ & = −{1 \atop 2} ψ_{+} · ∂_{z}x + θ (−{1 \atop 2} ∂_{z}x · ∂_{z}x − {1 \atop 2} ∂_{z}ψ_{+} · ψ_{+}) \end{aligned}  (9.C.5)

We recognize indeed the super-current and the xψ stress tensor. For the (B, C) system, T is given by

\begin{aligned}T^{(BC)} & = −C𝒟_{+}^{2}B + \frac{1}{2}𝒟_{+}C𝒟_{+}B − \frac{3}{2}𝒟_{+}^{2}CB \\ & = −c∂_{z}β + \frac{1}{2}γb − \frac{3}{2}∂_{z}cβ + θ(T^{(bc)} + T^{(βγ)}) \end{aligned}  (9.C.6)

IX.8

The OPE of T with superfields of weight (h, 0) is determined by Ward identities, and we have the asymptotic expansions

\begin{aligned}T (z, θ)Φ(w, ζ) ∼ n\frac{θ − ζ}{(z − w − θζ)^{2}} Φ(w, ζ) & + {1 \atop 2} \frac{1}{z − w − θζ} 𝒟_{+}Φ(w, ζ) \\ & + \frac{θ − ζ}{z − w − θζ} 𝒟_{+}^{2}Φ(w, ζ) \end{aligned}  (9.C.7)

For example 𝒟_{+}X, B and C have weights n = 1/2, 3/2, −1 respectively. The super-Virasoro algebra is similarly expressed. Denote s ≡ z − w − θζ

T (z, θ)T (w, ζ) ∼ \frac{\frac{1}{4} ĉ}{s^{3}} + \frac{3}{2} \frac{θ − ζ}{s^{2}} T (w, ζ) + {1 \atop 2} \frac{1}{s}𝒟_{+}T (w, ζ) + \frac{θ − ζ}{s} 𝒟_{+}^{2}T (w, ζ)  (9.C.8)

where ĉ = \frac{2}{3}c represents the central charge.

D) BRST Quantization

The combined action S_{X} + S_{BC} is BRST invariant. For completeness, we record the BRST transformations on the fields

δX^{μ} = λC𝒟_{+}^{2}X^{μ} − {1 \atop 2} λ 𝒟_{+}C𝒟_{+}X^{μ} + complex conjugate
δC = λC𝒟_{+}^{2}C − \frac{1}{4}λ𝒟_{+}C𝒟_{+}C
δB = −λT

(9.D.1)

where λ is a constant Grassmann-valued parameter and T is the stress tensor superfield, defined in C). The associated BRST supercurrent is

J_{BRST} = C(T^{(X)} + {1 \atop 2}T^{(BC)}) − \frac{3}{4}𝒟_{+}(C𝒟_{+}CB)  (9.D.2)

The BRST charge

Q_{BRST} = ∮ dz dθ J_{BRST}  (9.D.3)

is conserved and nilpotent precisely in D = 10, in analogy with the bosonic string. Vertex operators for physical states commute with Q_{BRST}, while vertex operators of spurious states are Q_{BRST}-commutators of some operator. We shall not carry out BRST quantization any further here.

IX.9 E) Vertex operators for physical states

We distinguish two kinds of vertex operators: those for the NS sector, creating spacetime bosons, and those for the R sector, creating space-time fermions (also sometimes called the fermion vertex operator). For closed strings, this produces four different kinds of vertex operators: NS-NS and R-R creating bosons, and NS-R and R-NS creating fermions. The Rvertex cannot be constructed directly out of the fields X^{μ}, since it must produce a branch cut starting at its insertion points.

(1) The NS vertex operator For simplicity, we specialize to the closed RNS string, and consider NS-NS vertex

operators first; the left-moving part may be constructed using the chiral splitting theorem for the RNS string, to be derived in the next section, §F. We begin by constructing RNS vertex operators for NS-NS states without ghost dependence.

Vertex operators in the NS-NS sector are given in a fashion completely analogous to the bosonic string

V (ε, k) = ∫_{Σ} dμ_{E}P_{n}(ε, 𝒟_{+}X, 𝒟_{−}X, 𝒟_{+}^{2}X, …)e^{ik·X}  (9.E.1)

Here, P_{n} is a polynomial involving a sum over terms, each of which has a total number n of 𝒟_{+} derivatives. By sU(1) invariance of V, P_{n} must then also have a total number n of 𝒟_{−} derivatives. (On non-flat N = 1 supergeometries, the curvature superfield R_{+−} may also enter, just as the Gaussian curvature R_{g} entered for the bosonic string in the dilaton vertex. R_{+−} effectively counts for one 𝒟_{+} and for one 𝒟_{−}.) By construction, V (ε, k) is invariant under sDiff transformations. It remains to guarantee that V (ε, k) is sWeyl invariant.

Under constant sWeyl transformations, 𝒟_{+} and 𝒟_{−} have weight 1/2, dμ_{E} has weight −1 and the exponential has weight k^{2}. Given that #𝒟_{+} = n_{𝒟} = #𝒟_{−}, we have

M^{2} = −k^{2} = n_{𝒟} − 1  (9.E.2)

The GSO projection forces n to be odd only, since the number of NS fields applied to the ground state in each sector (left and right moving) must be odd. Thus, we have n_{𝒟} = 2n+1,

IX.10 with n = 0, 1, 2, … and the masses of NS-NS states are given by

M^{2} = 2n  n = 0, 1, 2, …  (9.E.3)

Non-constant sWeyl transformations will require in addition that certain conditions be fulfilled on the polarization tensors ε, such as transversality.

The massless string NS-NS vertex operator is

V (ε, k) = ε_{μν}(k) ∫_{Σ} dμ_{E}𝒟_{+}X^{μ}𝒟_{−}X^{ν}e^{ik·X}  (9.E.4)

sWeyl invariance requires ε_{μν}(k)k^{μ} = ε_{μν}(k)k^{ν} = 0, and we recover the vertex operator for dilaton, graviton and B_{μν} fields. It is useful to work out this vertex operator in components, dropping the auxiliary field F, and other terms that vanish by the field equations, such as ∂z̄ψ_{+}. (The full justification for dropping terms that vanish by the field equations may be given with the help of the Chiral Splitting Theorem in the next section, §F.) Restricting to left-movers only, we use the fact that

X^{μ} = X_{L}^{μ} + X_{R}^{μ}  (9.E.5)

and obtain the left-moving NS part of the vertex operator (9.E.?) as follows:

∫ dθ 𝒟_{+}X_{L}^{μ}e^{ik·X_{L}} = (∂_{z}x_{L}^{μ} − iψ_{+}^{μ}ψ_{+}^{ν}k_{ν})e^{ik·x_{L}}  (9.E.6)

This vertex contains the bosonic vertex operator, as well as an operator that allows for an application of an even number of NS fermions only.

(2) The R vertex operator Ramond states transform under spinor representations of the (double cover of the)

Lorentz group, Spin(1, 9). This group has two inequivalent Majorana-Weyl spinors S_{α} and S′^{α}, each of dimension 16 so that we let α = 1, …, 16. To describe the representations S_{α} and S′^{α}, we introduce the Clifford algebra of Spin(1, 9)

{Γ^{μ}, Γ^{ν}} = 2η^{μν}I_{(32)}  (9.E.7)

IX.11 where I_{(32)} is the identity matrix of dimension 32. The chirality matrix Γ̄ is defined by

Γ̄ ≡ Γ^{0}Γ^{1} ⋯ Γ^{9}

and satisfies

Γ̄^{2} = I_{(32)} and {Γ^{μ}, Γ̄} = 0  (9.E.8)

Weyl spinors of left (resp. right) chirality are eigenvectors of Γ̄ with eigenvalues +1 (resp. −1). The charge conjugation matrix C is defined by the relation

(Γ^{μ})^{T} = −CΓ^{μ}C^{−1}  (9.E.9)

and anti-commutes with Γ̄:

{Γ̄, C} = 0 .  (9.E.10)

The charge conjugate ψ^{c} of ψ is defined by

ψ^{c} ≡ Cψ̄^{T}  ψ̄ = ψ^{†}Γ^{0}  (9.E.11)

A Majorana spinor is a spinor whose charge conjugate is equal to itself, ψ^{c} = ψ. It is consistent (and convenient) to choose C = Γ^{0}, so that a Majorana spinor is real: ψ^{c} = ψ^{∗} = ψ. We shall choose a basis in which

Γ̄ = ( \begin{matrix}I_{(16)} & 0 \\ 0 & −I_{(16)} \end{matrix} ) ,  C = Γ^{0} = ( \begin{matrix}0 & I_{(16)} \\ −I_{(16)} & 0 \end{matrix} )  (9.E.12)

A 32 component Majorana spinor ψ now decomposes into two 16 component Majorana-Weyl spinors S_{α} and S′^{α}. It will be convenient to work with Γ-matrices in this basis:

ψ = ( \begin{matrix}S_{α} \\ S′^{α} \end{matrix} )  Γ^{μ} = ( \begin{matrix}0 & (γ^{μ})_{αβ} \\ (γ^{μ})^{αβ} & 0 \end{matrix} )  (9.E.13)

with α, β = 1, …, 16. It then follows from (9.E.9), (9.E.12) that γ_{αβ}^{μ} = γ_{βα}^{μ} and γ^{μαβ} = γ^{μβα}. The generator of Spin(1, 9) transformations in this representation is {1 \atop 2} Γ^{μν}, with

\begin{aligned}Γ^{μν} & ≡ \begin{matrix}1 \\ 2 \end{matrix} [Γ^{μ}, Γ^{ν}] = ( \begin{matrix}γ^{μν} & 0 \\ 0 & γ^{μν} \end{matrix} ) \\ γ^{μν} & ≡ \frac{1}{2} [γ^{μ}, γ^{ν}] . \end{aligned}  (9.E.14)

IX.12 Clearly, Γ^{μν} commutes with Γ̄, and {1 \atop 2} γ^{μν} are the representation generators of Spin(1, 9) in one of the Weyl representations.

The SO(1, 9) transformation properties in the Ramond sector are governed by the SO(1, 9) currents (we omit the contribution from the x-field)

j^{μν}(z) = ψ_{+}^{μ} ψ_{+}^{ν}(z)  (9.E.15)

Clearly, the field ψ_{+}^{μ} itself transforms under the vector representation of SO(1, 9):

j^{μν}(z)ψ_{+}^{ρ}(w) ∼ \frac{1}{z − w} (η^{μρ}ψ_{+}^{ν}(w) − η^{νρ}ψ_{+}^{μ}(w)) .  (9.E.16)

We seek to construct fields S_{α}(z) and S′^{α}(z) such that transform as spinors under Spin(1, 9) with representation matrices γ^{μν}:

j^{μν}(z)S_{α}(w) ∼ \frac{1}{z − w} (−\begin{matrix}1 \\ 2 \end{matrix}) (γ^{μν})_{α}^{β}S_{β}(w)  (9.E.17a)
j^{μν}(z)S′^{α}(w) ∼ \frac{1}{z − w} (−\begin{matrix}1 \\ 2 \end{matrix}) (γ^{μν})^{α}_{β}S′^{β}(w) .  (9.E.17b)

From these OPE’s, and from the fact that j^{μν} is bilinear in ψ_{+}^{μ}, we may infer the OPE of ψ_{+} with S:

ψ_{+}^{μ}(z)S_{α}(w) ∼ \frac{1}{(z − w)^{\frac{1}{2}}} \frac{1}{\sqrt{2}} (γ^{μ})_{αβ}S′^{α}(w)  (9.E.18a)
ψ_{+}^{μ}(z)S′^{α}(w) ∼ \frac{1}{(z − w)^{\frac{1}{2}}} \frac{1}{\sqrt{2}} (γ^{μ})^{αβ}S_{β}(w) .  (9.E.18b)

The relative normalization between S and S′ is arbitrary, and has been fixed above on symmetry grounds. The conformal weights of S_{α}(z) and S′^{α}(z) are computed from their OPE with the stress tensor T^{(ψ)} = −1 – 2 ∂_{z}ψ_{+} · ψ_{+}:

\begin{aligned}T^{(ψ)}(z)S_{α}(w) & ∼ (−\begin{matrix}1 \\ 2 \end{matrix}) (−\begin{matrix}1 \\ 2 \end{matrix}) \frac{1}{\sqrt{2}} \frac{1}{(z − w)^{3/2}} γ_{αβ}^{μ} ψ_{+}^{μ}(z)S′^{β}(w) \\ & ∼ \frac{1}{16}(γ^{μ}γ_{μ})_{α}^{β}S_{β}(w)\frac{1}{(z − w)^{2}} . \end{aligned}  (9.E.19)

IX.13 Since γ^{μ} γ_{μ} = 10 I_{(16)}, we find that the conformal weight of S_{α} (and similarly of S′^{α}) is 5/8.

To obtain the spin fields S_{α} and S′^{α}, we make use first of the action of the currents j^{μν} in the Cartan subalgebra of Spin(1, 9) only. These components mutually commute and may be bosonized in terms of 5 independent free scalar Bose fields:

j^{2a,2a+1}(z) = −i∂_{z}ϕ^{a}(z)  a = 0, 1, 2, 3, 4 .  (9.E.20)

The R-fields ψ_{+}^{μ} are exponentials of ϕ^{a} with ±1 weights:

\frac{1}{\sqrt{2}} (ψ_{+}^{2a} ± iψ_{+}^{2a+1}) = e^{±iϕ^{a}} .  (9.E.21)

The spin fields are exponentials of ϕ^{a} with ± {1 \atop 2} weights:

S_{α}(z) = e^{iα·ϕ} .  (9.E.22)

where we identify the space-time indices with the spinor representation weights, given by α = (± {1 \atop 2}, ± {1 \atop 2}, ± {1 \atop 2}, ± {1 \atop 2}, ± \begin{matrix}1 \\ 2 \end{matrix}). The spinors S_{α} are obtained for α with an even number of − signs, while S′^{α} is obtained with an odd number of − signs. The conformal dimension of this field is easily verified to be {1 \atop 2} α^{2} = 5/8, with the help of the standard formula for the conformal weight of an exponential of a c = 1 conformal field.

One may complete the analysis by showing that the currents j^{μν} corresponding to roots may also be constructed in terms of exponentials (multiplied by certain cocycle factors) and that j^{μν} indeed satisfies the correct OPE with S_{α} and S′^{α}.

With the above normalization, one also obtains the OPE’s of S with S and with S′:

\begin{aligned}S′^{α}(z)S_{β}(w) & ∼ \frac{δ_{β}^{α}}{(z − w)^{5/4}} + \frac{1}{(z − w)^{1/4}} 1 – 2 γ^{μν}j_{μν}(w) \\ S_{α}(z)S_{β}(w) & ∼ \frac{1}{(z − w)^{3/4}} γ_{αβ}^{μ} ψ_{μ} \end{aligned}  (9.E.23)

(Notice that the construction of spinor fields that we have just given for Spin(1, 9) may be carried over in a straightforward way to Spin(2n). In particular, this type of construction is needed to obtain the spinor of SO(16) necessary to build the full current algebra of

IX.14 E_{8} × E_{8} in the heterotic string. There, the spin fields have precisely conformal dimension (1, 0), since the weight vectors all have length α^{2} = 2.)

It remains to construct the Ramond vertex operator. The spin field by itself cannot produce a vertex operator for massless fermions, since its conformal weight is 5/8 ≠ 1. The resolution of this issue is found in a contribution to the R vertex operator from the (β, γ) and possibly (b, c) ghost system. The relevant conformal primary field is obtained as follows. (We consider only the left-moving part of the vertex operator.)

W_{−}(z, u, k) = u^{α}(k)𝒪(z)S_{α}(z)e^{ik·x_{L}(z)}  (9.E.24)

Here, u^{α} is a spinor analogue of the polarization tensors ε that entered the construction of NS states and vertex operators. The operator 𝒪(z) depends on the β and γ ghosts, and is a primary field of weight 3/8, so that W_{−} is of weight 1 for massless states (with k^{2} = 0). 𝒪(z) is constructed in such a way that W_{−} is BRST invariant; it is an analogue of the spin field, but now for the superghost system.

The (β, γ) ghost system may be represented in terms of two free boson fields ϕ and χ, such that

γ = e^{ϕ−χ}  β = e^{−ϕ+χ}∂_{z}χ  (9.E.25)

with

\begin{aligned}ϕ(z) ϕ(w) & ∼ & − ln(z − w) & c_{ϕ} = 13 \\ χ(z)χ(w) & ∼ & ln(z − w) & c_{χ} = −2 \end{aligned}  (9.E.26)

The ghost number current is j_{z} = −∂_{z}ϕ. The operator 𝒪(z) is again an exponential with weight {1 \atop 2} :

𝒪(z) = e^{−{1 \atop 2}ϕ(z)} .  (9.E.27)

Because the central charge of the ϕ system is 13, the dimension of 𝒪(z) is 3/8, instead of 1/8, as would be the case for a field ϕ of central charge 1.

The above construction of W_{−} implies, though, that W_{−} has ghost number −1/2! But overall ghost number is given in terms of the index theorem, and this would suggest that the number of Ramond vertex operators that can be inserted on any Riemann surface is determined in terms of the topology of the surface, which is clearly absurd.

IX.15

The resolution of this problem lies in the subtleties of the quantization of the (β, γ) system, which we shall not discuss here. The final result is that the quantization of the (β, γ) system yields vertex operators in different possible pictures, labeled by their ghost number. One shows, for example, that there is an equivalenT vertex operator for W_{−}, which had ghost number −1/2. This operator is W_{+} with ghost number +{1 \atop 2} and is given by

W_{+}(z, u, k) = e^{ϕ/2}u^{α}(γ_{μ})_{αβ}S′^{β}(z)(∂_{z}x^{μ} − iψ_{+}^{μ}k · ψ_{+})e^{ik·x}  (9.E.28)

Vertex operators for R states may now be inserted on any worldsheets in any arbitrary (even) numbers.

F) The Chiral Splitting Theorem

In critical RNS string theory, we have powerful results on the superholomorphicity properties of correlation functions for (factorized) vertex operators, at fixed internal momenta. This result is contained in the Chiral Splitting Theorem for the RNS string.

We begin by defining factorized vertex operators. A vertex operator

V_{i} = ∫ d^{2}z_{i} d^{2}θ_{i} W_{i}(z_{i}, θ_{i}, z̄_{i}, θ̄_{i})

is factorized provided W_{i} is the product of holomorphic and anti-holomorphic factors

W_{i}(z_{i}, θ_{i}, z̄_{i}, θ̄_{i}) = W_{i}(z_{i}, θ_{i})W̃_{i}(z̄_{i}, θ̄_{i})

Any vertex operator may be written as a linear combination of factorized vertex operators. For example, for NS-NS massless states, we have

W_{i}(z_{i}, θ_{i}; z̄_{i}, θ̄_{i}) = (ξ_{iμ}𝒟_{+}X^{μ}e^{ik·X_{L}}) (ξ̄_{iμ̄}𝒟_{−}Xμ̄e^{ik·X_{R}})  (9.F.1)

Next, let A_{I} and B_{I}, I = 1, …, h be any canonical basis of homology 1-cycles with canonical intersection form. We define h independent internal loop momenta p_{I} ∈ ℝ^{10}, associated with each A_{I}-cycle, by

p_{I} = ∮_{A_{I}} dz dθ 𝒟_{+}X  (9.F.2)

IX.16 Let the spin structures for left and right chirality spinors be equal, and equal to ν.

The correlation function of the unintegrated vertex operators W_{i} at fixed internal loop momenta p_{I} and fixed supermoduli are defined by^{∗}

\begin{matrix}⟨W_{1} … W_{N}⟩_{E} (p_{I}) \\ ≡ ∫ DX ∫ DBDC W_{1} … W_{N} ∏_{I=1}^{h} δ (p_{I}^{μ} − ∫_{A_{I}} dz dθ 𝒟_{+}X^{μ}) \\ ∏_{K=1}^{dim sℳ_{h}} | ⟨μ_{K}, B⟩ |^{2}e^{−S_{X}−S_{BC}} \end{matrix}  (9.F.3)

(I) The first part of the Chiral Splitting Theorem states that the RNS amplitudes factorize at fixed internal loop momenta:

⟨W_{1} … W_{N}⟩_{E} (p_{I}) = δ(k)𝒞_{ν}^{F} 𝒞̄_{ν}^{F}  (9.F.4)

Here, 𝒞_{ν}^{F} = 𝒞_{ν} F̄(z_{i}, θ_{i}; ζ_{i},; m_{i}; p_{I}, k_{i}) is a complex analytic function of supermoduli m_{K}, K = 1, …, dim sℳ_{h}, of the insertion points (z_{i}, θ_{i}), and of the left-moving factors ζ_{iμ} of the polarization tensors. 𝒞̄_{ν}^{F} is the complex conjugate of 𝒞_{ν}^{F}, with the same spin structure

𝒞̄_{ν}^{F} (z̄_{i}, θ̄_{i}; ζ̄_{i}; m̄_{K}; p_{I}, k_{i}) = 𝒞_{ν}^{F} (z_{i}, θ_{i}; ζ_{i}; m_{K} p_{I}, k_{i})^{∗}  (9.F.5)

The functions C_{ν}^{F} may be evaluated explicitly using Green functions and determinants, using the prime form and the Szegö kernel introduced in §V. (II) The second part of the Chiral Splitting Theorem prescribes the transition amplitudes for Type II and heterotic strings. We shall denote the analogous complex functions 𝒞^{B} = 𝒞^{B}(z_{i}; m_{k}; p_{I}, k_{i}) for the bosonic string, compactified on the 16-dim lattices of E_{8} × E_{8} or Spin(32)/Z_{2}. \underline{Type II A, B}:

A_{h} = δ(k) ∑_{ν,ν̄} Q_{νν̄} ∫_{ℝ^{10h}} d^{10}p_{I} ∫_{sℳ_{h}} dm_{K} dm̄_{K} ∏_{i=1}^{N} ∫_{Σ} d^{2}z_{i} d^{2}θ_{i} C_{ν}^{F} C̄F ; ν̄  (9.F.6)

IX.17 Heterotic E_{8} × E_{8}, Spin(32)/Z_{2}

∗ It is understood that for genus h = 0 (resp. h = 1) three (resp. 1) of the vertex operators should not be integrated over, and should be multiplied by a factor of δ((C, ψ)), where ψ is a superconformal Killing vector.

A_{h} = δ(k) ∑_{ν} Q_{ν} ∫_{ℝ^{10 h}} d^{10}p_{I} ∫_{sℳ_{h}} dm_{K} ∫_{ℳ_{h}} dm̄_{k} ∏_{i=1}^{N} ∫_{Σ} dz_{i} dθ_{i} ∫_{Σ} dz̄_{i} C_{ν}^{F} C̄^{B} (9.F.7)

Here Q_{νν̄} and Q_{ν} are spin structure dependent weight factors which realize the GSO projection. If carried out in the supermoduli picture, ν and ν̄ should run only over even or odd. But, if the integrals are understood in components, then one should sum over all individual spin structures in each class as well. In general Q_{νν̄} = ±1, and Q_{ν} = ±1.

G) Tree-Level Amplitudes for NS-NS states

To tree level, the worldsheet has the topology of the sphere, or stereographically projected plane, and there are neither moduli, nor odd moduli. Thus, we let g = 2|dz|^{2} and χ = 0. We concentrate on the scattering of massless NS-NS states, which can be represented by the vertex operator

V (ε, k) = ε_{μν}(k) ∫_{Σ} d^{2}z d^{2}θ 𝒟_{+}X^{μ}𝒟_{−}X^{ν}e^{ik·x} .  (9.G.1)

By factorizing ε_{μν}(k) = ζ_{μ}(k)ζ̄_{ν}(k), we may start from a more convenient vertex operator

V (ζ, ζ̄; k) = ∫_{Σ} d^{2}z d^{2}θ e^{ik·X+ξ·𝒟_{+}X+ξ̄·𝒟_{−}X}  (9.G.2)

and restrict to its contributions linear in ζ and linear in ζ̄. It is understood that ζ and ζ̄ are Grassmann parameters. The X-propagator is easily calculated:

⟨X^{μ}(z, θ)X^{ν}(z′, θ′)⟩ = − ln |z − z′ − θθ′|^{2} η^{μν}  (9.G.3)

The determinants of x, ψ, b, c and β, γ fields all combine into an overall constant, which we denote by 𝔷.

The chiral splitting theorem is easily used here: there are no internal loop momenta, so that the correlation function of the unintegrated vertex operators

W (z_{i}, θ_{i}; z̄_{i}, θ̄_{i}) = W_{L}(z_{i}, θ_{i})W_{R}(z̄_{i}, θ̄_{i})  (9.G.4)

IX.18 will yield a product of a function C^{F}, complex analytic in z_{i}, θ_{i} times its complex conjugate. Here, the chiral vertex operators are

W_{L}(z_{i}, θ_{i}) = e^{ik_{i}·X_{L}(z_{i},θ_{i})+ζ_{i}·𝒟_{+}X_{L}(z_{i},θ_{i})}  (9.G.5)

and analogously for W_{R}, where the chiral fields X_{L} are contracted with the effective propagator

⟨X_{L}^{μ}(z, θ)X_{L}^{ν}(z′, θ′)⟩ = − ln(z − z′ − θθ′)η^{μν} .  (9.G.6)

One readily finds

\begin{matrix}⟨W_{L}(z_{1}, θ_{1}) ⋯ W_{L}(z_{N}, θ_{N})⟩ = 𝒞^{F} (z_{i}, θ_{i}) \\ 𝒞^{F} = exp ∑_{i≠j}^{N} \{ik_{i} · ζ_{j} \frac{θ_{ij}}{z_{ij}} + {1 \atop 2} ζ_{i} · ζ_{j} \frac{1}{z_{ij}} + 1 – 2 k_{i} · k_{j} ln z_{ij}\} \end{matrix}  (9.G.7)

where we have used the standard notation

\{ \begin{matrix}θ_{ij} & = θ_{i} − θ_{j} ; \\ z_{ij} & = z_{i} − z_{j} − θ_{i}θ_{j} . \end{matrix}  (9.G.8)

At tree level, we have invariance under superconformal automorphisms in OSp(1, 2); the integration measure must be properly treated to account for this invariance. This group of transformations is defiend as follows; consider the matrices

K = ( \begin{matrix}0 & +1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )  T = ( \begin{matrix}a & b & α \\ c & d & β \\ γ & δ & A \end{matrix} )  (9.G.9)
z → \frac{az + b + αθ}{cz + d + βθ}  θ → \frac{γz + δ + Aθ}{cz + d + βθ}  (9.G.10)

where Latin entries are commuting and Greek entries are anticommuting. To obtain a superconformal transformation T, the line element dz = dz + θ dθ must be transformed into itself up to a conformal scaling. Equivalently the form

z_{12} = z_{1} − z_{2} − θ_{1}θ_{2} = \frac{v_{1}w_{2} − v_{2}w_{1} − ψ_{1}ψ_{2}}{w_{1}w_{2}}  (9.G.11)

IX.19 should transform into itself up to a conformal scaling. This is uniquely achieved when the orthosymplectic form is left invariant under T:

T^{T} KT = D  (9.G.12)

The weight under which the difference transforms is easily derived, and we have

T: z_{12} → z̄_{12} = \frac{z_{12}}{(cz_{1} + d + βθ_{1})(cz_{2} + d + βθ_{2})}  (9.G.13)

Similarly the line element transforms as

dz → dz̄ = \frac{dz}{(cz + d + βθ)^{2}}  (9.G.14)

and the volume element as

dz ∧ dθ → \frac{dz∧dθ}{(cz + d + βθ)} .  (9.G.15)

Elements in OSp(1, 2) are in unique correspondence with a triplet of points in the superplane (z_{1}, θ_{1}), (z_{2}, θ_{2}), (z_{3}, θ_{3}) obeying one single (Grassmann valued) constraint. The counting works out because OSp(1, 2) has 3 commuting and 2 anticommuting parameters. The constraint is an OSp(1, 2) invariant Grassmann valued function dependent on three points, given by

Δ = \frac{z_{12}θ_{3} + z_{31}θ_{2} + z_{23}θ_{1} + θ_{1}θ_{2}θ_{3}}{(z_{12}z_{23}z_{31})^{1/2}}  (9.G.16)

The natural value for Δ is of course 0, which implies that one θ is dependent. With this value for Δ, it is easy to see that there is a unique correspondence between triplets of points satisfying Δ = 0, and elements of OSp(1, 2), so that the latter may be accordingly parametrized. The invariant volume element induced on OSp(1, 2) is then

dμ = \frac{dz_{1}dz_{2}dz_{3}dθ_{1}dθ_{2}dθ_{3}}{(z_{12}z_{23}z_{31})^{1/2}} Δ  (9.G.17)

As for the bosonic string, the volume of the superconformal transformation group must be extracted from the amplitudes. As a result of OSp(1, 2) invariance, the 0, 1, 2 point functions vanish.

IX.20

We now work out the three point amplitude. In evaluating 𝒞^{F}, one retains terms proportional to ζ_{1}ζ_{2}ζ_{3}; however, the term with three θ’s vanishes because θ_{12}θ_{23}θ_{31} = 0. Defining the tensors

K_{μ_{1}μ_{2}μ_{3}} = η_{μ_{1}μ_{2}}k_{1μ_{3}} + η_{μ_{2}μ_{3}}k_{2μ_{1}} + η_{μ_{3}μ_{1}}k_{3μ_{2}}  (9.G.18)

we find for the three point function

⟨V (ε_{1}, k_{1})V (ε_{2}k_{2})V (ε_{3}k_{3})⟩ = 4(2π)^{10}δ(k)ε_{1}^{μ_{1}μ̄_{1}}ε_{2}^{μ_{2}μ̄_{2}}ε_{3}^{μ_{3}μ̄_{3}}K_{μ_{1}μ_{2}μ_{3}}K_{μ̄_{1}μ̄_{2}μ̄_{3}} (9.G.19)

As a result of transversality of the polarization tensors, the 3 point function for massless external particles also vanishes.

To compute the four point amplitude, we shall fix superconformal invariance as follows: z = z_{1}, z_{2} = 0, z_{3} = 1, z_{4} = ∞, θ_{1}, θ_{2}, θ_{3} = θ_{4} = 0, and then have

\begin{aligned}exp 𝒢_{4}^{ζ} & = \{ζ_{1} · ζ_{2}ζ_{3} · ζ_{4} \frac{1}{z_{12}z_{34}} + ζ_{1} · ζ_{3}ζ_{2} · ζ_{4} \frac{1}{z_{13}z_{24}} + ζ_{1} · ζ_{4}ζ_{2} · ζ_{3} \frac{1}{z_{14}z_{23}} \} \\ & + \{ζ_{1} · ζ_{2} (k_{1} · ζ_{3}k_{2} · ζ_{4} \frac{θ_{2}θ_{1}}{z_{12}z_{13}z_{24}} + k_{1} · ζ_{4}k_{2} · ζ_{3} \frac{θ_{2}θ_{1}}{z_{12}z_{14}z_{23}} ) + perm.\} \end{aligned} (9.G.20)

In principle, one should now multiply this whole expression by the one involving the ζ̄’s, perform the integrals over z and θ and regroup terms, clearly a feudal task. The calculation is enormously simplified by the factorization properties of the Veneziano-integrals. Recall that we have the ordinary integrals

∫ \frac{d^{2}z}{π} z^{A}z̄Ã(1 − z)^{B}(1 − z̄)_{B} = \frac{Γ(−1 − Ã − B̃}{Γ(−Ã)Γ(−B̃)} \frac{Γ(1 + A)Γ(1 + B)}{Γ(A + B + 2)}  (9.G.21)

provided A − Ã and B − B̃ are integers, which is always the case in string theory. Using the reciprocity formula for Γ-functions and the fact that A − Ã and B − B̃ are integers, this expression is actually symmetric under (A, B) ↔ (Ã, B̃), as one might expect from complex conjugation. More importantly, the answer factorizes into a product of factors each only dependent either on the parameters for the z or z̄ coordinates. This product property implies that one must only consider say the z coordinates to find the full amplitude, which

IX.21 by the same token will also completely factorize as a function of ζ’s and ζ̄’s. An analogous formula is derived for the super-integrals we need:

\begin{aligned}∫ \frac{d^{2}z_{1}}{π} d^{2}θ_{2}[θ_{1}θ_{2}]^{a} & [θ̄_{1}θ̄_{2}]ãz_{12}^{A} z̄Ã ; 12(1 − z_{1})^{B}(1 − z̄_{1})_{B} \\ & = (−2i)^{1−a}(+2i)^{1−ã} \frac{Γ(−ã − Ã − B̃)}{Γ(−Ã)Γ(−B̃)} \frac{Γ(1 + A)Γ(1 + B)}{Γ(A + B + 1 + a)} \end{aligned} (9.G.22)

Here a and ã are either 0 or 1, and the integrals are symmetric under (aAB) ↔ (ãÃB̃). With the help of (9.G.22), it is now straightforward to evaluate the four point function

\begin{matrix}⟨V (ε_{1}, k_{1}) & V (ε_{2}, k_{2})V (ε_{3}, k_{3})V (ε_{4}, k_{4})⟩ \\ = (2π)^{10}δ(k)g^{4} ∫ d^{2}z_{1}d^{2}θ_{2}|z_{12}|^{−s}|z_{1} − 1|^{−u}e^{𝒢_{4}^{ζ}+𝒢ζ̄ ; 4} \\ = π(2π)^{10}δ(k)g^{4} \frac{Γ(−s/2)Γ(−t/2)Γ(−u/2)}{Γ (1 + \frac{s}{2}) Γ (1 + \frac{t}{2}) γ (+\frac{u}{2})}ε^{1¯1}ε^{2¯2}ε^{3¯3}ε^{4¯4}K_{1234}K¯1¯2¯3¯4 \end{matrix} (9.G.23)

Using the abbreviation i for μ_{i} to save some writing, we have

\begin{aligned}K_{1234} & = (stη_{13}η_{24} − suη_{14}η_{23} − tuη_{12}η_{34}) \\ & − s(k_{1}^{4}k_{3}^{2}η_{24} + k_{2}^{3}k_{4}^{1}η_{13} − k_{1}^{3}k_{4}^{2}η_{23} − k_{2}^{4}k_{3}^{1}η_{14}) \\ & + t(k_{2}^{1}k_{4}^{3}η_{13} + k_{3}^{4}k_{1}^{2}η_{24} − k_{2}^{4}k_{1}^{3}η_{34} − k_{3}^{1}k_{4}^{2}η_{12}) \\ & − u(k_{1}^{2}k_{4}^{3}η_{23} + k_{3}^{4}k_{2}^{1}η_{14} − k_{1}^{4}k_{2}^{3}η_{34} − k_{3}^{2}k_{4}^{1}η_{12}) \end{aligned}  (9.G.24)

H) One loop Amplitudes for NS-NS states in Type II A, B

We begin by using part (I) of the Chiral Splitting Theorem, and we compute the Chiral Amplitudes 𝒞_{ν}^{F}. The spin structure for left and right movers is the same here.

For even spin structures ν, there are no odd moduli, and there are no Dirac zero modes. The metric on the torus is taken to be g = 2|dz|^{2}, and the torus is represented as usual by a parallelogram with sides 0, 1, τ, 1 + τ, with τ ∈ H in the upper half complex plane H. The S_{X} and S_{BC} actions are free actions in components:

\begin{aligned}S_{X} + S_{BC} = \frac{1}{4π} ∫_{Σ} d^{2}z & (∂_{z}x · ∂z̄x − ψ_{+} · ∂z̄ψ_{+} − ψ_{−} · ∂_{z}ψ_{−} \\ & + 2b∂z̄c + 2b̄∂_{z}c̄ + 2β∂z̄γ + 2β̄∂_{z}γ̄) . \end{aligned}  (9.H.1)

IX.22 The ghost and anti-ghost insertions are reduced to

bb̄ cc̄ .  (9.H.2)

No anti-superghost insertions occur for even spin structures, since there are no zero modes there. In the NS-NS sector, vertex operators do not involve ghosts, so we may integrate out b, c, b̄, c̄, β, γ, β̄ and γ̄. We shall also retain the determinants of x and ψ_{±} integrations. Thus

δ(k)𝒞_{ν}𝒞_{ν} = M_{ν} M̄_{ν} ℱ_{ν} ℱ̄_{ν}  (9.H.3)

where M_{ν} M̄_{ν} arises from determinants and ℱ_{ν} ℱ̄_{ν} arises from correlation functions of vertex operators. We find

M_{ν} M̄_{ν} = (\frac{det′ Δ_{0}}{(Im τ)^{2}} )^{−5} ^{(^{det / D_{+}} )}_{ν}^{5} ^{(^{det / D_{−}} )}_{ν}^{5}

(\frac{det′ Δ_{−1}}{(Im τ)^{2}} )^{+1} (det Δ_{−1/2}^{−})_{ν}^{−1}

(9.H.4)

As usual, on the torus, these determinants are related to one another, and we find

\frac{det′ Δ_{0}}{(Im τ)^{2}} = \frac{det′ Δ_{−1}}{(Im τ)^{2}} = |η(τ)|^{4}
(det / D_{+} / D_{−})_{ν} = (det Δ_{1/2}^{−})_{ν} = |\frac{ϑ[ν](0|τ)}{η(τ)}|^{2}

(9.H.5)

Putting all together:

M_{ν} = \frac{ϑ[ν](0|τ)^{4}}{η(τ)^{12}}  M̄_{ν} = \frac{\overline{ϑ[ν](0|τ)^{4}}}{\overline{η(τ)}^{12}}  (9.H.6)

up to an overall phase which is independent of τ, but which may depend upon ν. Clearly such a phase can be absorbed into the spin structure weights Q_{νν̄}. The vertex contributions ℱ_{ν} are easily evaluated:

\begin{aligned}ℱ_{ν}(z_{i}, θ_{i}, ζ, k, p) & = exp\{iπp^{2}τ + 2πp · ∑_{i}(−ζ_{i}θ_{i} + ik_{i}z_{i}) \\ & − {1 \atop 2} ∑_{i≠j}(k_{i} · k_{j}G_{ν}(z_{i}, θ_{i}; z_{j}, θ_{j}) + ζ_{i} · ζ_{j}𝒟_{+}^{i}𝒟_{+}^{j}G_{ν} \\ & + 2ik_{i} · ζ_{j}𝒟_{+}^{j}G_{ν})\} . \end{aligned}  (9.H.7)

IX.23 Here, the correlation function G_{ν} is given by

\begin{aligned}G_{ν}(z_{i}, θ_{i}; z_{j}, θ_{j}) & = ⟨X_{L}(z_{i}, θ_{i})X_{L}(z_{j}, θ_{j})⟩ \\ & = − ln E(z_{i}, z_{j}) + θ_{i}θ_{j}S_{ν}(z_{i}, z_{j}) \end{aligned}  (9.H.8)

where E(z, w) is the prime form, defined in §V, and S_{ν} is the Szegö kernel. For the torus, these functions are particularly simple:

\begin{aligned}E(z, w) & = \frac{ϑ_{1}(z − w|τ)}{ϑ′_{1}(0|τ)} ; \\ S_{ν}(z − w) & = \frac{ϑ[ν](z − w|τ)ϑ′_{1}(0|τ)}{ϑ[ν](0|τ)ϑ_{1}(z − w|τ)} . \end{aligned}  (9.H.9)

For odd spin structures, ν, there is now one complex odd modulus χ and there are 10 Dirac zero modes, one for each space-time direction. The action S_{X} + S_{BC} is the one given on in (9.H.1) but there is now also an insertion of the worldsheet supercurrent involving

χS_{z+} S_{z+} = (ψ_{+} · ∂_{z}x − {1 \atop 2} bγ + \frac{3}{2} β∂_{z}c + ∂_{z}β c) + c.c .  (9.H.10)

Integrating out χ will bring down an insertion of ψ_{+} · ∂_{z}x and of ψ_{−} · ∂z̄x. Combining this with the ghost and anti-ghost insertions, we have

bb̄ cc̄δ(β)δ(β̄)δ(γ)δ(γ̄) ∫ S_{z+} ∫ S_{z̄−} .  (9.H.11)

Now, we specialize to correlation functions of massless NS-NS states. As shown in (9.E.5), these operators involve at most 2 factors of ψ_{+}. To obtain a non-zero contribution from odd spin structures, we need to have at least 5 external vertex operators. For example, the anomaly diagrams with 6 external vertex operators receives contributions solely from odd spin structure. We shall concentrate on amplitudes with 4 or fewer external vertex operators, and odd spin structures will never contribute to those. Notice that this means that for these amplitudes, there is no difference between Type II A, B. For completeness, we record that for ν odd, we have M_{ν} = Mν̄ = 1.

It remains to work out the amplitudes with 0, 1, 2, 3, 4 external vertex operators. We begin with determining the factors Q_{νν̄} of spin structure weights. Following Mumford,

IX.24 we label ϑ-functions as follows:

\begin{aligned}ϑ_{00}(z, τ) & = ϑ_{3}(z, τ) = ϑ(z, τ) \\ ϑ_{01}(z, τ) & = ϑ_{4}(z, τ) = ϑ (z + {1 \atop 2}, τ) \\ ϑ_{10}(z, τ) & = ϑ_{2}(z, τ) = e^{iπτ/4+πiz}ϑ (z + \frac{τ}{2} , τ) \\ ϑ_{11}(z, τ) & = ϑ_{1}(z, τ) = e^{iπτ/4+πiz}ϑ (z + {1 \atop 2} + \frac{τ}{2} , τ) \\ ϑ(z, τ) & ≡ ∑_{n=−∞}^{∞} e^{iπn^{2}τ+2πinz} \end{aligned}  (9.H.12)

Modular transformations of ϑ_{2}, ϑ_{3}, ϑ_{4} obey

\begin{aligned}ϑ_{ab}(0, τ + 1) & = e^{\frac{iπ}{4} a}ϑ_{a(b+a)}(0, τ) \\ ϑ_{ab} (0, −\frac{1}{τ} ) & = (−)^{ab}\sqrt{−i τ} ϑ_{ba}(0, τ) \end{aligned}  (9.H.13)

The Dedekind η-function satisfies

\begin{aligned}η(τ + 1)^{12} & = −η(τ)^{12} \\ η (−\frac{1}{τ} )^{12} & = τ^{6}η(τ)^{12} \end{aligned}  (9.H.14)

Thus

\begin{aligned}M_{ab}(τ + 1) & = −e^{iπa}M_{a(b+a+1)}(τ) \\ M_{ab} (−\frac{1}{τ} ) & = −\frac{1}{τ^{4}} M_{ba}(τ) . \end{aligned}  (9.H.15)

We must now require modular invariance of the amplitude:

∑_{(a,b)} Q_{(ab)ν̄}M_{(ab)}(τ + 1) = ∑_{(a,b)} Q_{(a,b)ν̄}M_{(ab)}(τ)  (9.H.16)

∑_{(a,b)} Q_{(ab)ν̄}M_{(ab)}(−1/τ) = −\frac{1}{τ^{4}} ∑_{(a,b)} Q_{(a,b)ν̄}M_{ab}(τ) .

(The factor of −1/τ^{4} is compensated for by the p-integral.) Clearly from τ → −1/τ, we get

Q_{(01)ν̄} = Q_{(10)ν̄}  (9.H.17)

IX.25 while from τ → τ + 1, we get

Q_{(01)ν̄} = −Q_{(00)ν̄}  (9.H.18)

Setting

Q_{(01)ν̄} = Q_{(10)ν̄} = −Q_{(00)ν̄}  (9.H.19)

we fix the overall phase of the amplitude, which is unobservable anyway. One proceeds similarly for ν̄.

It is now straightforward to compute the 0-point function. (Recall that in the bosonic string this object produced the first divergence we had encountered.) Since there are no vertex operator insertions, we simply have

A_{1}(0 − pt) = Vol(M) ∫_{ℳ_{1}}  \frac{d^{2}τ}{τ_{2}^{4}} ∑_{νν̄} Q_{νν̄}M_{ν}Mν̄  (9.H.20)

It suffices to examine the combination

∑_{ν} Q_{νν̄}M_{ν} = Q_{(01)ν̄}M_{(01)} + Q_{(10)ν̄}M_{(10)} + Q_{(00)ν̄}M_{(00)}  (9.H.21)

Notice that since M_{(11)} = 0 the corresponding contribution has been omitted in (9.H.21). Now, using the expressions for M_{ν}, and the equality for Q’s, we find

∑_{ν} Q_{νν̄} = \frac{Q_{(01)ν̄}}{η(τ)^{12}} [ϑ_{4}(0|τ)^{4} + ϑ_{2}(0|τ)^{4} − ϑ_{3}(0|τ)^{4}] = 0  (9.H.22)

by a famous Jacobi ϑ-function identity. This identity arose once before in matching bosonic and fermionic multiplicities at all mass levels. In fact, this cancellation is the result of spacetime supersymmetry. The 0-point function, and thus the 1-point functions vanish. Recall from the discussion of the bosonic string that the vanishing of the 1-point function means that Minkowski flat space-time is a one-loop solution to the superstring equations!

Notice that our argument for cancellation was effected only in the left-moving sector; thus, the same argument holds true in heterotic string theory as well.

The above summation identity on ϑ-functions is only one of a series, usually called the Riemann identities on theta functions. It is most useful here to express two more of

IX.26 these identities in terms of the Dirac propagator (i.e. the Szegö kernel):

\begin{matrix}∑_{ν} Q_{νν̄}M_{ν}S_{ν}(z_{1} − z_{2})S_{ν}(z_{2} − z_{1}) = 0 \\ ∑_{ν} Q_{νν̄}M_{ν}S_{ν}(z_{1} − z_{2})S_{ν}(z_{2} − z_{3})S_{ν}(z_{3} − z_{1}) = 0 \end{matrix}  (9.H.23)

Without giving details here, one similarly shows with the help of these identities that the two- and three-point functions vanish. In physics terms, this means that to one loop order, there is neither mass renormalization (massless particles remain massless) nor coupling constant renormalization. These results again arise from space-time supersymmetry, and are usually referred to as non-renormalization theorems.

I) One-loop amplitudes in the heterotic string

Let us consider the free fermion representation of the heterotic strings, starting from 32 Majorana-Weyl fermions λ^{α}, represented by 32 Euclidean Weyl spinors λ_{−}^{α}(z̄). For simplicity, consider amplitudes with external states that are neutral under the gauge group. Then the one loop contribution of the internal degree of freedom will result entirely from the determinantal factors arising from integrating out λ_{−}^{α}. These depend upon the spin structures, just as the ones for space-time degrees of freedom ψ_{−}^{μ}.

For the gauge group Spin(32)/Z_{2}, all λ_{−}^{α} have the same spin structure, and the determinant for given spin structure contributes a factor

M̄ 4 ; ν̄ = \begin{matrix}\overline{ϑ[ν̄](0|τ)} & 16 \\ \overline{η(τ)} & 48 \end{matrix}  (9.I.1)

The only modular invariant we can form is with all coefficients Qν̄ equal for all 3 even spin structures.

For the gauge group E_{8} × E_{8}, λ_{−}^{α} are divided into two classes of 16. For each class, the determinant is

M̄ 2 ; ν̄ = \begin{matrix}\overline{ϑ[ν](0|τ)} & 8 \\ \overline{η(τ)} & 24 \end{matrix} .  (9.I.2)

The only modular invariant is

∑_{ν̄} M̄_{ν}^{2} = \frac{\overline{ϑ_{2}(0|τ)}^{8} + \overline{ϑ_{3}(0|τ)}^{8} + \overline{ϑ_{4}(0|τ)}^{8}}{\overline{η(τ)}^{24}} .  (9.I.3)

IX.27 One can show, using relations on the number of representations of an integer as a sum of squares, that this function equals the ϑ-function for the E_{8} lattice: ϑ_{E_{8}}(0|τ). Furthermore, the partition functions for Spin(32)/Z_{2} and E_{8} × E_{8} are the same in view of

∑_{ν̄} M̄ 4 ; ν̄ = (∑_{ν̄} M̄ 2 ; ν̄)^{2} .  (9.I.4)

We see that no other modular invariant combinations are allowed. You may think that ∑_{ν} Q_{ν}M_{ν} = 0 is allowed, but this would mean that some states enter with negative weight, and this is impossible since there are no fermions here! This argument shows now that E_{8} × E_{8} and Spin(32)Z_{2} are indeed the only possibilities, as we had promised.

J) The NS-NS 4 point function

It remains to discuss the 4-point functions, again for massless NS-NS states. Clearly, the 4-point function cannot vanish, lest the theory be non-interacting. It is evaluated by making use of a Riemann identity on ϑ-functions with 4 Szegö kernels:

∑_{ν} Q_{νν̄}M_{ν}S_{ν}(z_{1} − z_{σ(1)})S_{ν}(z_{2} − z_{σ(2)})S_{ν}(z_{3} − z_{σ(3)})S_{ν}(z_{1} − z_{σ(4)}) = 16π^{4}Q_{(00)ν̄} (9.J.1)

where σ(i) is a permutation of i, without fixed points.

The final result for the 4-point function can be conveniently expressed as follows:

A_{1}(k_{i}, ε_{i}) = δ(k)ε^{1¯1}ε^{2¯2}ε^{3¯3}ε^{4¯4}K^{1234}K^{¯1¯2¯3¯4}A_{1}(s, t)  (9.J.2)

where the single Lorentz-invariant function A_{1}(s, t) is given by

A_{1}(s, t) = \begin{matrix}1 \\ – \\ 2 \end{matrix} ∫_{ℳ_{1}} \frac{d^{2}τ}{τ_{2}^{2}} ∫_{Σ} \frac{d^{2}z_{1}}{τ_{2}} ∫_{Σ} \frac{d^{2}z_{2}}{τ_{2}} ∫_{Σ} \frac{d^{2}z_{3}}{τ_{2}} e^{_{\frac{s}{2}} (G_{12}+G_{34}−G_{13}−G_{24})}

e^{+ \frac{t}{2} (G_{23}+G_{14}−G_{13}−G_{24})} (9.J.3)

where G_{ij} stands for the 1-loop propagator of the x-field:

G_{ij} = G(z_{i} − z_{j})|τ) ≡ − ln |\frac{ϑ_{1}(z_{i} − z_{j}|τ)}{θ′_{1}(0|τ)}|^{2} + \frac{2π}{τ_{2}} Im (z_{i} − z_{j})^{2}  (9.J.4)

and K is the kinematical factor that also arose in the tree-level amplitudes A_{0}(k_{i}, ϵ_{i}) for NS-NS transition amplitudes. Recall that K is polynomial in moments. By translation invariance on the torus, the point z_{4} may be chosen at will, say at z_{4} = 0.

IX.28

It remains to study the function A_{1}(s, t)! We begin by examining the convergence structure of the above functional representation as a function of s and t.

(1) For fixed τ, and z_{i} → z_{j}, i ≠ j, we have

G(z_{i} − z_{j}|τ) → − ln |z_{i} − z_{j}|^{2} → +∞  (9.J.5)

(2) while for z_{i}’s well separated, but Im τ → ∞, we have

G(z_{i} − z_{j}|τ) → −τ_{2}  → −∞ .  (9.J.6)

The latter limit imposes the strictest convergence condition:

Re(s), Re(t), Re(u) ≥ 0  (9.J.7)

But, since we also have s + t + u = 0 for on-shell massless external states, the convergence condition (9.J.7) requires Re(s) = Re(t) = Re(u) = 0. It is easily seen that this condition is sufficient for convergence, since the integral is bounded by the value at s = t = 0. This means that the integral is absolutely convergent for at least some values of the external momenta, in contrast with the case of the bosonic string, where the integral representation is never convergent for any momenta. Yet, the integral representation is defined only for unphysical momentum configurations where s, t, u are purely imaginary.

A second shortcoming of the integral representation is that formally, when s, t, u are close to the physical region corresponding to real values of s and t, A_{1}(s, t) would be real. This would be inconsistent in an interacting theory, where the 1-loop 4-point function cannot be real, since its imaginary part is related by unitarity to the square of the tree level amplitude, which cannot vanish in any intracting theory.

Our task is thus to properly define the amplitude throughout (s, t) ∈ ℂ × ℂ, including around the physical region where (s, t) ∈ ℝ × ℝ. Naturally, one seeks an analytic continuation of the amplitude in both s and t. (Recall that even at tree level, the integral representation A_{0}(s, t) was absolutely convergent only in a small region, and the full amplitude had to be analytically continued in s and t as well.)

IX.29

To appreciate the physical meaning of such an analytic continuation, we compare with the situation in quantum field theory. There, the physical principle of causality is equivalent to the requirement of locality of observable fields, and this property, combined with Lorentz invariance, implies analytic dependence of the Green functions on the external momenta. Often, this analytic behavior is conveniently summarized in the form of a spectral representation. For example, the 2-point function (off-shell) may be expressed as

H(s) = ∫_{0}^{∞} dM^{2} \frac{ρ(M^{2})}{s − M^{2} + iε} ε > 0  (9.J.8)

where the real and positive function ρ(M^{2}) is the density of states at mass M, also called the spectral density. Provided the integral over M^{2} converges, it follows that H(s) is analytic throughout ℂ, except possibly on the positive real axis, where it may have poles and branch cuts.

In string theory, we do not have available at present a formulation of the amplitudes in terms of local field observables (i.e. commuting at space-like separations). It is thus not known à priori whether analyticity of the amplitudes can be derived solely from Lorentz invariance and causality. (Probably, in a good second quantized formulation, one should be able to derive this.) Thus, analyticity of string amplitudes must be established, and the questions that must be answered are

(1) does an analytic continuation exist?

(2) is the analytic continuation unique?

(3) is the analytic continuation physically acceptable?

The present problem of analytic continuation of A_{1}(s, t) is a difficult matter of pure analysis. Yet, all three questions above have been answered in the affirmative, so that one has a full analytically continued amplitude A_{1}(s, t) available, defined throughout (s, t) ∈ ℂ × ℂ, with cuts and poles on ℝ × ℝ, but nowhere else.

What we have achieved here is the construction of a finite and unitary amplitude, at least to one loop order, that defines NS-NS string scattering! We shall not reproduce the

IX.30 details of the analytic continuatin procedure here. We shall limit ourselves to describing the physical singularities that appear in A_{1}(s, t) upon analytic continuation. To do so it is most convenient to represent the different singularities that appear as a function of s and t in terms of QFT Feynman diagrams. The Feynman diagrams indicate the region in moduli space (including the positions of vertex operators z_{i}) that produce the singularities. The singularities essentially arise from combinations of the singular limits of G_{ij} in (9.J.5) and (9.J.6), and fall into five categories, schematically indicated by five different types of Feynman diagrams below. Whenever two external z_{i} and z_{j} approach one another the Feynman diagram representation will show that their external lines merge

This singularity may be treated at the operator level by considering the operator product expansion (OPE) of the two external vertex operators. The internal line then represents all the operators that arise in the OPE. This type of singularity produces a pole in the total momentum traversing the internal line, i.e. in −(k_{i} + k_{j})^{2}, and the pole turns out to be at the value of the mass^{2} of the physical string states, i.e. 2n = M^{2}, n ∈ ℤ^{+}. The five different categories represented by diagrams (a), (b), (c), (d) and (e) arise, respectively, from 4, 3, 2, 1 and 0 external lines coming close together. Within each category, there are additional singularities (represented by a dotted line aross the loop) that arise from the asymptotic behavior in τ, given by (9.J.6).

(a)

IX.31 (b)

(c)  • double poles in s at s = 2n

• branch cuts in s, starting at s = 2n, n ∈ ℤ^{+}.

(d)  • single poles in s at s = 2n

• branch cuts in s, starting at s = 2n, n ∈ ℤ^{+}.

(e)  • no poles

• branch cuts in s, starting at s = 2n, n ∈ ℤ^{+} with spectral density ρ(2n, t)

• ρ(2n, t) has branch cuts in t, starting at t = 2m, m ∈ ℕ^{0}.

It is understood that within each category, we also include permutation of the diagrams shown, for example by adding diagrams on which the external legs have been permuted.

Now, for superstrings, diagrams in categories (a) and (b) vanish identically. This follows from the fact that the on-shell 1- and 2-point functions for massless states vanish identically. The singularities of the remaining diagrams are precisely the ones that one would expect on physical grounds, and their structure confirms the consistency of the string theory up to this order. A complete proof is in E. D’Hoker and D. H. Phong, “The Box Graph in Superstring Theory”, Nucl. Phys. B440 (1995), 24.

IX.32