A Concrete Representation

The results for the RNS correlation functions stated in Chapters 4 and 5 are written down in Lorentz covariant form and are therefore valid for all representations of the Clifford algebra. However for some issues in the calculations, e.g. checking relations between dif-ferent Clebsch–Gordan coefficients and especially for obtaining loop results, an explicit representation of the gamma matrices is helpful.

It is well-known that the gamma matrices in D = 2m dimensions, Γ^µ_D, and the Pauli matrices σⁱ can be used to construct Γ^µ_D+2, the Dirac matrices in D+ 2 dimensions. It is easily verified that for k= 0, . . . , D−1

Γ^k_D+2 ≡σ¹ ⊗Γ^k_D, Γ^D_D+2 ≡i σ¹⊗Γ^D_D, Γ^D+1_D+2 ≡i σ²⊗12^m (A.29) satisfy the Clifford algebra (A.1) in D+ 2 dimensions, if Γ^µ_D satisfy the Clifford algebra in D space-time dimensions². The matrix Γ^D_D is the chirality matrix defined as

Γ^D_D ≡(−1)^12(m+1)

DY−1

i=1

Γⁱ_D. (A.31)

We start in D= 2 dimensions with Γ⁰₂ ≡σ¹ =

0 1 1 0

, Γ¹₂ ≡i σ² =

0 1

−1 0

. (A.32)

2The product⊗in (A.29) is the Kronecker product which takes anm×nmatrix A and ap×qmatrix B to themp×nq matrix

A⊗B ≡





a11B · · · a1nB ... . .. ... am1B · · · amnB



. (A.30)

A.4 A Concrete Representation 129

The Dirac matrices forD= 4,6,8,10 can then be constructed by recursively using (A.29).

Our construction is chosen such that we obtain the Γ^µ_D’s in Weyl basis, i.e. Γ^D_D always has the form

Γ^D_D =

12^m−1 0 0 −12^m−1

. (A.33)

The charge conjugation matrix CD is determined from the equations (A.11) and (A.15).

This yields

C2 =i σ², C4 =12⊗ C2, C6 =−i σ²⊗ C4,

C8 =12⊗ C6, C10 =−i σ²⊗ C8. (A.34) As 1 is diagonal and σ² off-diagonal, this results exactly in the diagonal and off-diagonal structure stated in (A.9).

130 A. Gamma Matrices in D Dimensions

APPENDIX B

Relations between Index Terms

In this Appendix we collect relations between different index terms, that enter the cor-relators calculated in Chapters 4 and 5. The numbers of independent Clebsch–Gordan coefficients for each correlator are summarized in Table 3.3. As we have explained be-fore these relations can be derived from Fierz identities given in Appendix A, multiplying known equations with further γ and C matrices, as well as demanding that a tensor in D dimensions, which is anti-symmetric in 2^D/2−1 + 1 Weyl indices of the same type, has to vanish. We have checked the validity of the following relations by (anti-)symmetry ar-guments and furthermore by explicit verification using the representation of the Clifford algebra given in Appendix A.4.

Before we start listing the relations we recap our conventions. The Clifford algebra establishes that

γ_α^µ_β˙¯γ^νβγ˙ +γ_α^ν_β˙γ¯^µβγ˙ =−2δ_α^γη^µν, γ¯^µαβ˙ γ_β^ν_γ˙ + ¯γ^ναβ˙ γ_β^µ_γ˙ =−2δ^α_γ˙^˙η^µν. (B.1) These can be used to interchange Lorentz indices in expressions where chains ofγ matrices appear. Antisymmetric products ofγ matrices are defined with a normalization factor 1/p!

in contrast to [21],

γ^µ1...µp ≡ 1 p!

X

ρ∈Sp

sgn(ρ)

(γ^µρ(1)¯γ^µρ(2). . . γ^µρ(p) :p odd,

γ^µρ(1)¯γ^µρ(2). . .γ¯^µρ(p) :p even. (B.2) On the right hand side of this equation we can interchange the Lorentz indices with the help of (B.1). Then the antisymmetric γ-product can be written as an ordered γ-product and further η terms. Let us illustrate this for the simplest case p= 2:

γ^µν = 1

2(γ^µγ¯^ν −γ^ν¯γ^µ) =η^µν+γ^µγ¯^ν. (B.3) In deriving the following relations the behavior of tensors under interchanging their spinor indices is very important. These can be derived by employing the identities from Table 3.4.

132 B. Relations between Index Terms

Contractions of γ and C matrices in the following should always be understood as matrix multiplications. We refrain from explicitly denoting C⁻¹ in index notation. This matrix differs from C in the positioning of its indices and hence can be distinguished in the respective contexts.

B.1 Relations for D = 4

In the following we state relations that allow to reduce the number of index terms for a given correlator in four space-time dimensions. We adopt the widely used notation in four dimensions and denote the blocks of the Γ matrices and the charge conjugation matrix C by σ_α^µ_α˙ and εαβ. The Fierz identity

εαγεβδ =εαβεγδ+εαδεβγ (B.4) is a good starting point for deriving further relations.

Correlator

ψ^µψ^νSαSβ˙SγSδ˙

The relevant index terms for this correlation function up to permutations in the spinor indices are 2η^µνε_αγεβ˙δ˙, σ^µ

αβ˙σ_γ^ν_δ˙, (σ^µσ¯^νε)_αγεβ˙δ˙ and (εσ¯^µσ^ν)β˙δ˙ε_αγ. The latter two terms can be eliminated by multiplying (B.4) with εβ˙α˙σ¯^µαβ˙ εδ˙γ˙σ¯^νγδ˙ and treating the anti-chiral version of (B.4) in the same manner:

(σ^µσ¯^νε)αγεβ˙δ˙ = σ_α^µ_β˙σ^ν_γδ˙−σ_α^µ_δ˙σ^ν_γβ˙, (B.5a) (ε¯σ^µσ^ν)β˙δ˙ε_αγ = σ^µ

αβ˙σ^ν_γδ˙−σ^µ

γβ˙σ_α^ν_δ˙. (B.5b) By symmetrizing in the vector indices µ, ν we arrive at:

2η^µνεαγεβ˙δ˙ = σ_α^µ_δ˙σ^ν_γβ˙ +σ_γ^µ_β˙σ^ν_αδ˙−σ^µ_αβ˙σ_γ^ν_δ˙−σ_γ^µ_δ˙σ_α^ν_β˙. (B.6) Correlator

ψ^µψ^νS_αS_βS_γS_δ

This correlator can be expressed in terms of η^µνεαβεγδ, (σ^µσ¯^νε)αβεγδ and permutations in the spinor indices of these terms. Forming antisymmetric combinations of the latter, where one spinor index of the σ-chains is kept fixed, yields three independent relations:

(σ^µσ¯^νε)_αδε_βγ = −(σ^µσ¯^νε)_αβε_γδ+ (σ^µσ¯^νε)_αγε_βδ, (B.7a) (σ^µσ¯^νε)βδεαγ = −(σ^µσ¯^νε)αβεγδ+ (σ^µσ¯^νε)βγεαδ−2η^µνεαβεγδ, (B.7b) (σ^µσ¯^νε)γδεαβ = −(σ^µσ¯^νε)αγεβδ+ (σ^µσ¯^νε)βγεαδ−2η^µνεαβεγδ. (B.7c) The Fierz identity (B.4) can be used to eliminate η^µνεαγεβδ from the calculations and hence one arrives at five independent index terms. In order to write the correlator in terms of antisymmetric σ-products the following equation is of use:

(σ^µνε)αβεγδ−(σ^µνε)αδεγβ+ (σ^µνε)γδεαβ −(σ^µνε)γβεαδ = 0. (B.8)

B.1 Relations for D= 4 133 Correlator

ψ^µψ^νψ^λS_αS_βS_γS_δ˙

The anti-symmetrization argument can be used also for this correlator to derive relations between different index terms,

(σ^µσ¯^νσ^λ)_βδ˙εαγ = (σ^µσ¯^νσ^λ)_αδ˙εβγ+ (σ^µσ¯^νσ^λ)_γδ˙εαβ, (B.9a) η^νλσ_β^µ_δ˙εαγ = η^νλσ^µ_αδ˙εβγ +η^νλσ_γ^µ_δ˙εαβ. (B.9b) In the second equation it is possible to permute the Lorentz indicesµ, ν, λin order to derive two additional equations. Furthermore, applying (σ^µσ¯^ν)ακε[κβσ_{γ] ˙}^λ_δ = 0 yields identities that mix different σ configurations:

εβγ(σ^µσ¯^νσ^λ)_αδ˙ = (σ^µσ¯^νε)αγσ^λ_βδ˙−(σ^µσ¯^νε)αβσ_γ^λ_δ˙, (B.10a) εαβ(σ^µσ¯^νσ^λ)_γδ˙ = (σ^µσ¯^νε)βγσ_α^λ_δ˙−(σ^µσ¯^νε)αγσ_β^λ_δ˙−2η^µνεαβσ_γ^λ_δ˙. (B.10b) Hence, the triple products (σ^µσ¯^νσ^λ) can be completely eliminated. By permuting the Lorentz indices in (B.10) one obtains four linearly independent relations:

2η^µλσ_α^ν_δ˙εβγ −2η^µνσ_α^λ_δ˙εβγ = (σ^νσ¯^λε)αβσ_γ^µ_δ˙−(σ^ν¯σ^λε)αγσ_β^µ_δ˙

−(σ^µ¯σ^νε)αβσ^λ_γδ˙+ (σ^µσ¯^νε)αγσ^λ_βδ˙, (B.11a) 2η^µλσ_γ^ν_δ˙εαβ−2η^νλσ_γ^µ_δ˙εαβ = (σ^νσ¯^λε)αγσ^µ_βδ˙−(σ^ν¯σ^λε)βγσ_α^µ_δ˙

+(σ^µ¯σ^νε)βγσ^λ_αδ˙−(σ^µσ¯^νε)αγσ_β^λ_δ˙, (B.11b) 2η^νλσ_α^µ_δ˙εβγ = (σ^µ¯σ^νε)αβσ^λ_γδ˙−(σ^µσ¯^νε)αγσ_β^λ_δ˙

+(σ^µ¯σ^λε)_αβσ_γ^ν_δ˙−(σ^µσ¯^λε)_αγσ^ν_βδ˙, (B.11c) 2η^µνσ_γ^λ_δ˙ε_αβ = (σ^µ¯σ^λε)_βγσ_α^ν_δ˙−(σ^µσ¯^λε)_αγσ^ν_βδ˙

+(σ^νσ¯^λε)βγσ_α^µ_δ˙−(σ^ν¯σ^λε)αγσ_β^µ_δ˙. (B.11d) Another relation is necessary in order to reduce the number of Clebsch–Gordan coefficients to ten. This is achieved by multiplying (B.5a) with ¯σ^λβδ˙ εδβ. After a further permutation in the spinor indices one finds:

(σ^µσ¯^νε)αβσ^λ_γδ˙−(σ^µ¯σ^λε)αγσ_β^ν_δ˙+ (σ^νσ¯^λε)βγσ_α^µ_δ˙ = 0. (B.12) Correlator

ψ^µS_αS_βS_γS_δS_ǫS_ζ˙

The starting point for relating different index term for this correlator is

εαβεγδεǫζ−εαβεγζεǫδ+εαδεγζεǫβ −εαδεγβεǫζ+εαζεγβεǫδ−εαζεγδεǫβ = 0, (B.13) which stems from δ_[β^α δ^γ_δδ_ζ]^ǫ = 0. Contracting this equation with ε^ζκσ^µ

κζ˙ yields the only needed relation for putting the correlator into minimal form:

σ_ǫ^µ_ζ˙εαβεγδ+σ_ǫ^µ_ζ˙εαδεβγ +σ^µ_αζ˙εβǫεγδ+σ^µ_αζ˙εβγεδǫ+σ_γ^µ_ζ˙εαβεδǫ−σ^µ_γζ˙εαδεβǫ = 0. (B.14)

134 B. Relations between Index Terms

Correlator

ψ^µψ^νψ^λψ^ρS_αS_β˙S_γS_δ˙

This correlation function can be expressed in terms of permutations in the spinor indices of η^µνσ_α^λ_β˙σ^ρ_γδ˙, (σ^µσ¯^νε)αγ(εσ¯^λσ^ρ)β˙δ˙, (σ^µ¯σ^νσ^λ)_αβ˙σ_γ^ρ_δ˙ and (σ^µσ¯^νσ^λσ¯^ρε)αγεβ˙δ˙. The two expressions containing fourσ matrices can be eliminated with the help of (B.5):

(σ^µσ¯^νσ^λσ¯^ρε)αγεβ˙δ˙ = (σ^µσ¯^νσ^λ)_αβ˙σ_γ^ρ_δ˙−(σ^µσ¯^νσ^λ)_αδ˙σ^ρ_γβ˙, (B.15a) (εσ¯^µσ^νσ¯^λσ^ρ)β˙δ˙εαγ = (σ^νσ¯^λσ^ρ)_γδ˙σ_α^µ_β˙−(σ^νσ¯^λσ^ρ)_αδ˙σ_γ^µ_β˙. (B.15b) Relations between the Clebsch–Gordan coefficients consisting of threeσ matrices are found by decomposing terms of the type (σσ ε) (ε¯ ¯σ σ) using (B.5). Applying these relations either to the first or the second σ-chain results in

(σ^µ¯σ^νε)αγ(εσ¯^λσ^ρ)β˙δ˙ =

((σ^µσ¯^νσ^λ)_αβ˙σ^ρ_γδ˙−(σ^µσ¯^νσ^ρ)_αδ˙σ_γ^λ_β˙, (σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙−(σ^µσ¯^λσ^ρ)_αδ˙σ^ν_γβ˙. (B.16) Now we can write down relations between the terms on the right hand side of (B.16):

(σ^µσ¯^νσ^λ)_αβ˙σ^ρ

γδ˙ = (σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙−(σ^µσ¯^λσ^ρ)_αδ˙σ_γ^ν_β˙+ (σ^µσ¯^νσ^ρ)_αδ˙σ^λ_γβ˙, (B.17a) (σ^µσ¯^νσ^λ)_αδ˙σ^ρ

γβ˙ = (σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙−(σ^µσ¯^λσ^ρ)_αβ˙σ_γ^ν_δ˙+ (σ^µσ¯^νσ^ρ)_αβ˙σ_γ^λ_δ˙, (B.17b) (σ^µσ¯^νσ^λ)_γδ˙σ^ρ_αβ˙ = (σ^νσ¯^λσ^ρ)_αβ˙σ_γ^µ_δ˙−(σ^µσ¯^λσ^ρ)_γβ˙σ_α^ν_δ˙+ (σ^µσ¯^νσ^ρ)_γβ˙σ_α^λ_δ˙, (B.17c) (σ^µσ¯^νσ^λ)_γβ˙σ_α^ρ_δ˙ = (σ^νσ¯^λσ^ρ)_αδ˙σ^µ_γβ˙−(σ^µσ¯^λσ^ρ)_γδ˙σ^ν_αβ˙+ (σ^µσ¯^νσ^ρ)_γδ˙σ_α^λ_β˙. (B.17d) The last three equations were found by permuting the spinor indices in (B.16). However, one can also perform permutations in the Lorentz indices. This yields

2η^µν(σ_α^λ_β˙σ_γ^ρ_δ˙−σ_γ^λ_β˙σ^ρ_αδ˙) = (σ^νσ¯^λσ^ρ)_αδ˙σ^µ_γβ˙−(σ^νσ¯^λσ^ρ)_γδ˙σ_α^µ_β˙

+(σ^µσ¯^λσ^ρ)_αδ˙σ^ν_γβ˙−(σ^µσ¯^λσ^ρ)_γδ˙σ^ν_αβ˙, (B.18a) 2η^µν(σ_α^λ_δ˙σ^ρ

γβ˙−σ_γ^λ_δ˙σ^ρ

αβ˙) = (σ^νσ¯^λσ^ρ)_αβ˙σ^µ

γδ˙−(σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙

+(σ^µσ¯^λσ^ρ)_αβ˙σ_γ^ν_δ˙−(σ^µσ¯^λσ^ρ)_γβ˙σ_α^ν_δ˙, (B.18b)

−2η^λρ(σ^µ

αβ˙σ_γ^ν_δ˙−σ^µ

αδ˙σ_γ^ν_β˙) = (σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙−(σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙

+(σ^µσ¯^λσ^ρ)_αβ˙σ_γ^ν_δ˙−(σ^µσ¯^λσ^ρ)_αδ˙σ^ν_γβ˙, (B.18c)

−2η^λρ(σ_γ^µ_δ˙σ_α^ν_β˙−σ_γ^µ_β˙σ^ν_αδ˙) = (σ^νσ¯^λσ^ρ)_αβ˙σ_γ^µ_δ˙−(σ^νσ¯^λσ^ρ)_αδ˙σ_γ^µ_β˙

+(σ^µσ¯^λσ^ρ)_γδ˙σ^ν_αβ˙−(σ^µσ¯^λσ^ρ)_γβ˙σ_α^ν_δ˙, (B.18d)

B.1 Relations for D= 4 135 and

2 (η^µλσ_α^ν_β˙σ^ρ

γδ˙−η^µλσ^ν_γβ˙σ^ρ

αδ˙−η^νλσ^µ

αβ˙σ^ρ

γδ˙+η^νλσ^µ

γβ˙σ^ρ

αδ˙)

=−(σ^νσ¯^λσ^ρ)_αδ˙σ^µ

γβ˙+ (σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙+ (σ^µσ¯^νσ^ρ)_αδ˙σ^λ_γβ˙−(σ^µσ¯^νσ^ρ)_γδ˙σ_α^λ_β˙, (B.19a) 2 (η^µλσ_α^ν_δ˙σ^ρ

γβ˙ −η^µλσ^ν_γδ˙σ^ρ

αβ˙−η^νλσ^µ

αδ˙σ^ρ

γβ˙+η^νλσ^µ

γδ˙σ^ρ

αβ˙)

=−(σ^νσ¯^λσ^ρ)_αβ˙σ^µ

γδ˙+ (σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙+ (σ^µσ¯^νσ^ρ)_αβ˙σ_γ^λ_δ˙−(σ^µσ¯^νσ^ρ)_γβ˙σ_α^λ_δ˙, (B.19b) 2 (η^νλσ^µ

αβ˙σ^ρ

γδ˙−η^νλσ^µ

αδ˙σ^ρ

γβ˙)−η^νρσ^µ

αβ˙σ_γ^λ_δ˙+η^νρσ^µ

αδ˙σ_γ^λ_β˙)

=−(σ^νσ¯^λσ^ρ)_γδ˙σ_α^µ_β˙+ (σ^νσ¯^λσ^ρ)_γβ˙σ_α^µ_δ˙+ (σ^µσ¯^νσ^ρ)_αβ˙σ_γ^λ_δ˙−(σ^µσ¯^νσ^ρ)_αδ˙σ_γ^λ_β˙, (B.19c) 2 (η^νλσ^µ

γδ˙σ^ρ

αβ˙−η^νλσ^µ

γβ˙σ^ρ

αδ˙−η^νρσ^µ

γδ˙σ^λ_αβ˙+η^νρσ^µ

γβ˙σ_α^λ_δ˙)

=−(σ^νσ¯^λσ^ρ)_αβ˙σ_γ^µ_δ˙+ (σ^νσ¯^λσ^ρ)_αδ˙σ_γ^µ_β˙+ (σ^µσ¯^νσ^ρ)_γδ˙σ^λ_αβ˙−(σ^µσ¯^νσ^ρ)_γβ˙σ_α^λ_δ˙. (B.19d) In addition the following relations hold,

2 (η^µρσ_α^ν_β˙σ_γ^λ_δ˙−η^µρσ_γ^ν_β˙σ_α^λ_δ˙−η^νρσ_α^µ_β˙σ^λ_γδ˙+η^νρσ_γ^µ_β˙σ^λ_αδ˙−η^λρσ^µ_αβ˙σ_γ^ν_δ˙+η^λρσ_γ^µ_β˙σ^ν_αδ˙)

= −(σ^νσ¯^λσ^ρ)_αβ˙σ^µ

γδ˙+ (σ^ν¯σ^λσ^ρ)_αδ˙σ^µ

γβ˙ −(σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙+ (σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙

−(σ^µσ¯^λσ^ρ)_αβ˙σ_γ^ν_δ˙+ (σ^µσ¯^λσ^ρ)_γβ˙σ_α^ν_δ˙+ (σ^µσ¯^νσ^ρ)_αβ˙σ_γ^λ_δ˙−(σ^µσ¯^νσ^ρ)_γβ˙σ^λ_αδ˙, (B.20a) 2 (η^µρσ_α^ν_δ˙σ_γ^λ_β˙−η^µρσ_γ^ν_δ˙σ_α^λ_β˙−η^νρσ^µ

αδ˙σ_γ^λ_β˙+η^νρσ^µ

γδ˙σ_α^λ_β˙ +η^λρσ^µ

αδ˙σ_γ^ν_β˙−η^λρσ^µ

γδ˙σ_α^ν_β˙)

= (σ^νσ¯^λσ^ρ)_αβ˙σ^µ

γδ˙−(σ^νσ¯^λσ^ρ)_αδ˙σ^µ

γβ˙+ (σ^νσ¯^λσ^ρ)_γδ˙σ^µ

αβ˙−(σ^νσ¯^λσ^ρ)_γβ˙σ^µ

αδ˙

−(σ^µσ¯^λσ^ρ)_αδ˙σ^ν_γβ˙+ (σ^µσ¯^λσ^ρ)_γδ˙σ_α^ν_β˙+ (σ^µσ¯^νσ^ρ)_αδ˙σ_γ^λ_β˙−(σ^µσ¯^νσ^ρ)_γδ˙σ_α^λ_β˙, (B.20b) as well as

2 (η^µνσ^λ_αβ˙σ_γ^ρ_δ˙−η^µνσ_γ^λ_δ˙σ_α^ρ_β˙ −η^µλσ^ν_αβ˙σ_γ^ρ_δ˙+η^µλσ_α^ν_δ˙σ_γ^ρ_β˙ +η^µρσ_α^ν_β˙σ_γ^λ_δ˙−η^µρσ_α^ν_δ˙σ_γ^λ_β˙)

= (σ^νσ¯^λσ^ρ)_αδ˙σ_γ^µ_β˙−(σ^µσ¯^λσ^ρ)_γβ˙σ_α^ν_δ˙+ (σ^µσ¯^νσ^ρ)_αβ˙σ_γ^λ_δ˙−(σ^µσ¯^νσ^λ)_αβ˙σ^ρ_γδ˙. (B.21) Using the identities stated above it is possible to arrive at a set of 25 independent index terms for this correlator.

Correlator

ψ^µψ^νψ^λψ^ρS_αS_βS_γS_δ

The index terms η^µνη^λρεαβεγδ, η^µν(σ^λσ¯^ρε)αβεγδ, (σ^µσ¯^νε)αβ(σ^λσ¯^ρε)γδ and the four σ-chains (σ^µσ¯^νσ^λσ¯^ρε)αβεγδ arise for this correlator. Additionally, permutations of these terms in the vector and spinor indices appear. Again, the Fierz identity (B.4) can be used to eliminate terms of the type η η ε_αγε_βδ. Further eliminations can be accomplished by applying (B.7). In particular (B.7b) gives rise to new relations between the terms η(σσ ε)¯ ε:

−2η^µνη^λρεαβεγδ =

(η^µν(σ^λσ¯^ρε)αβεγδ−η^µν(σ^λσ¯^ρε)βγεαδ+η^µν(σ^λσ¯^ρε)βδεαγ, η^λρ(σ^µσ¯^νε)αβεγδ−η^λρ(σ^µσ¯^νε)βγεαδ+η^λρ(σ^µ¯σ^νε)βδεαγ.

(B.22)

136 B. Relations between Index Terms

Tensors consisting of two separate σ-chains satisfy (σ^µσ¯^νε)γβ(σ^λσ¯^ρε)αδ−(σ^µσ¯^νε)αβ(σ^λσ¯^ρε)γδ

= εαγ

h(σ^µσ¯^νσ^λσ¯^ρε)βδ + 2η^µν(σ^λσ¯^ρε)βδ

i, (B.23a)

(σ^µσ¯^νε)αδ(σ^λσ¯^ρε)γβ −(σ^µσ¯^νε)γδ(σ^λ¯σ^ρε)αβ

= εαγ

h

(σ^µσ¯^νσ^λσ¯^ρε)βδ −4η^µλη^νρεβδ + 4η^µρη^νλεβδ

−2η^µλ(σ^νσ¯^ρε)βδ + 2η^µρ(σ^νσ¯^λε)βδ + 2η^λρ(σ^µσ¯^νε)βδ

−2η^νρ(σ^µσ¯^λε)βδ + 2η^νλ(σ^µσ¯^ρε)βδ

i, (B.23b)

which stem from (σ^µσ¯^ν)ακε[κβ(σ^λσ¯^ρε)γ]δ = 0. Applying the same technique to the four-σ terms, namely (σ^µσ¯^νσ^λσ¯^ρε)_α[βε_γδ]= 0, yields the equations

(σ^µσ¯^νσ^λσ¯^ρε)αγεβδ = (σ^µσ¯^νσ^λσ¯^ρε)αβεβδ−(σ^µσ¯^νσ^λσ¯^ρε)αδεγβ, (B.24a) (σ^µσ¯^νσ^λσ¯^ρε)βδεαγ = (σ^µσ¯^νσ^λσ¯^ρε)γδεαβ −(σ^µσ¯^νσ^λσ¯^ρε)γβεαδ. (B.24b) From permutations of the spinor indices in the relations above we find

(σ^µσ¯^νσ^λσ¯^ρε)αβεγδ−(σ^µσ¯^νσ^λ¯σ^ρε)αδεγβ+ (σ^µσ¯^νσ^λσ¯^ρε)γβεαδ−(σ^µσ¯^νσ^λσ¯^ρε)γδεαβ

= −εβδ

h2η^µν(σ^λ¯σ^ρε)αγ−2η^µλ(σ^ν¯σ^ρε)αγ+ 2η^µρ(σ^νσ¯^λε)αγ

+ 2η^λρ(σ^µσ¯^νε)αγ−2η^νρ(σ^µσ¯^λε)αγ+ 2η^νλ(σ^µσ¯^ρε)αγ

+ 4η^µνη^λρεαγ−4η^µλη^νρεαγ+ 4η^µρη^νλεαγ

i

. (B.25)

Note that the spinor indices attached to the four-σ term can be interchanged by (σ^µσ¯^νσ^λσ¯^ρε)βα = −(σ^ρσ¯^λσ^νσ¯^µε)αβ

= −(σ^µ¯σ^νσ^λσ¯^ρε)αβ−2η^µν(σ^λσ¯^ρε)αβ + 2η^µλ(σ^ν¯σ^ρε)αβ−2η^µρ(σ^νσ¯^λε)αβ

−2η^λρ(σ^µσ¯^νε)_αβ+ 2η^νρ(σ^µσ¯^λε)_αβ −2η^νλ(σ^µ¯σ^ρε)_αβ

−4η^µνη^λρεαβ + 4η^µλη^νρεαβ −4η^µρη^νλεαβ, (B.26) where we have successively made use of the Clifford algebra (B.1). Finally, poles inz13z24

in the result of this eight-point function can be removed by the following identities:

−2η^νρ(σ^µσ¯^λε)αγεβδ = (σ^µσ¯^νε)αβ(σ^λσ¯^ρε)γδ−(σ^µσ¯^νε)αδ(σ^λ¯σ^ρε)γβ

+(σ^µσ¯^ρε)αβ(σ^λσ¯^νε)γδ−(σ^µσ¯^ρε)αδ(σ^λσ¯^νε)γβ, (B.27a)

−2η^µλ(σ^νσ¯^ρε)βδεαγ = (σ^µσ¯^νε)αβ(σ^λσ¯^ρε)γδ−(σ^µσ¯^νε)γβ(σ^λσ¯^ρε)αδ

+(σ^λσ¯^νε)_αβ(σ^µσ¯^ρε)_γδ−(σ^λσ¯^νε)_γβ(σ^µσ¯^ρε)_αδ, (B.27b) 4η^µλη^νρεαγεβδ = (σ^µσ¯^νε)αβ(σ^λσ¯^ρε)γδ−(σ^µσ¯^νε)γβ(σ^λσ¯^ρε)αδ

B.2 Relations for D= 6 137

Im Dokument Correlators of Ramond-Neveu-Schwarz Fields in String Theory (Seite 146-155)