Appendix A Spinors in Four Dimensions In this appendix we collect the conventions used for spinors in both Minkowski and Euclidean spaces. In Minkowski space the flat metric has the 0 1 2 3 form ηµν = diag(−1, 1, 1, 1), and the coordinates are labelled (x ,x , x , x ). The analytic continuation into Euclidean space is madethrough the replace- ment x0 = ix4 (and in momentum space, p0 = −ip4) the coordinates in this case being labelled (x1,x2, x3, x4). The Lorentz group in four dimensions, SO(3, 1), is not simply connected and therefore, strictly speaking, has no spinorial representations. To deal with these types of representations one must consider its double covering, the spin group Spin(3, 1), which is isomorphic to SL(2, C). The group SL(2, C) pos- sesses a natural complex two-dimensional representation. Let us denote this representation by S andlet us consider an element ψ ∈ S with components ψα =(ψ1,ψ2) relative to some basis. The action of an element M ∈ SL(2, C) is β (Mψ)α = Mα ψβ. (A.1) This is not the only action of SL(2, C) which one could choose. Instead of M we could have used its complex conjugate M, its inverse transpose (M T)−1,or its inverse adjoint (M †)−1. All of them satisfy the same group multiplication law. These choices would correspond to the complex conjugate representation S, the dual representation S,and the dual complex conjugate representation S. We will use the following conventions for elements of these representations: α α˙ ψα ∈ S, ψα˙ ∈ S, ψ ∈ S, ψ ∈ S. (A.2) These representations are called spinorial representations and are not inde- pendent. The SL(2, C) invariant tensor αβ (and α˙ β˙ ) allows one to relate a representation to its dual. We will use the following conventions for this 204 Spinors in four dimensions 205 tensor: 12 21 21 = = −12 = − = 1, 1˙2˙ 2˙1˙ (A.3) 2˙1˙ = = −1˙2˙ = − = 1. It allows one to raise and lower spinor indices: α αβ β ψ = ψβ, ψα = αβψ , (A.4) α˙ α˙ β˙ β˙ ψ = ψβ˙ , ψα˙ = α˙ β˙ ψ . One can easily verify that these relations are consistent with the assignment ˙ βδ β β˙δ˙ β (A.2). Notice also that in our conventions αβ = δα, α˙ β˙ = δα˙ . The real form of the complex Lie algebras sl(2, C) and su(2)+ × su(2)− are the same. This allows one to use the notation (j1, j2) for the represen- tationsofSL(2, C) where j1 and j2 are the spins of the two su(2)s. The representation S corresponds to (2, 0) whilst S to (0, 2). Notice that the two su(2)’s are not independent but related by complex conjugation. Under complex conjugation the labels of the representations must be interchanged. This allows one to restrict to representations which are fixed by complex con- jugation in the case in which the two labels are the same. The resulting representation is real, and the simplest case, (2, 2), corresponds to the defin- ing representation of the Lorentz group SO(3, 1) or vector representation. The Clebsch–Gordan coefficients which intertwine between the vector representation and the (2, 2) of SL(2, C) involve the Pauli matrices: 0 1 0 −i 10 σ1 = ,σ2 = ,σ3 = , (A.5) 10 i 0 0 −1 and have theform µ 1 2 3 σ = (1,σ ,σ ,σ ) ˙ , (A.6) αβ˙ αβ allowing one to transform a four-vector pµ = (p0,p1,p2, p3) into a bi-spinor: µ p0 + p3 p1 − ip2 σ pµ = . (A.7) p1 + ip2 p0 − p3 The indices for the matrices σµ show how SL(2, C) acts in bi-spinor space. µ µ † If M ∈ SL(2, C), the action of M is given by σ pµ → M(σ pµ)M , which preserves hermiticity and the determinant. The determinant is related to the µ µ 2 2 2 2 Lorentz invariant quantity p pµ = −det(σ pµ) = −p0 +p1 +p2 +p3 (the minus sign in front of the determinant is owed to our choice of metric). Notice that 206 Topological Quantum Field Theory and Four-Manifolds M and −M act in the same way on the vector representation, showing the fact that SL(2, C) is a double cover of SO(3, 1). The dual Clebsch–Gordan coefficients are defined after raising the indices of σµ, ˙ (σµ)αβ˙ = α˙ ββασµ −→ (σµ) = (1, −σ1, −σ2, −σ3)(A.8) αβ˙ and allow to transform a vector into a bi-spinor corresponding to the dual representations. The matrices σµ and σµ satisfy the following set of useful identities: σµσν + σν σµ = − 2ηµν , (A.9) µ αβ˙ − β α˙ (σ ) (σµ)γρ˙ = 2δγ δρ˙ , which can be easily derived after using the following property of the Pauli matrices σi, i = 1, 2, 3: σiσj = δij + iijkσk (A.10) where ijk is the totally antisymmetric tensor with 123 =1. The generators of the Lorentz transformations can be constructed with the help of the matrices σµ and σµ. They take the form: 1 (σµν ) β = σµσν − σν σµ β, α 4 α ˙ (A.11) µν β˙ 1 µ ν ν µ β (σ ) ˙ = σ σ − σ σ ˙ , α 4 α which leads to the simple form: 1 i σ0i = − σi,σij = − ijkσk, 2 2 (A.12) 1 i σ0i = σi, σij = − ijkσk, 2 2 and satisfy the properties: i i σµν = µνρσσ , σµν = − µνρσσ , (A.13) 2 ρσ 2 ρσ 0123 being 0123 =1, = −1. The matrices (A.11) provide spinorial representations of the Lorentz group: M µν → iσµν ,Mµν =→ iσµν , (A.14) satisfying the Lorentz algebra: [M µν ,Mρτ ] = iηµρM ντ − iηµτ M νρ − iηνρM µτ + iηντ M µρ. (A.15) Spinors in four dimensions 207 Under a Lorentz transformation parametrized by a real antisymmetric pa- rameter ωµν spinors transform as: µν β µν β δψα = iωµν (M )α ψβ = −ωµν (σ )α ψβ, (A.16) µν β˙ − µν β˙ δψα˙ = iωµν (M )α˙ ψβ˙ = ωµν (σ )α˙ ψβ˙ . For the scalar product of spinors we will use the following conventions: α α ψχ = ψ χα = −ψαχ = χψ ˙ (A.17) α − α˙ ψχ = ψ χα˙ = ψα˙ χ = χψ. In Euclidean space the relevant group is the orthonormal group SO(4). Inthis case the covering group is Spin(4), which is isomorphic to SU(2)+ ⊗ SU(2)−. The representations are labelled by the spins of the two SU(2)s, which are independent. These spinorial representations are denoted by S+ and S−.The SU(2) invariant tensor can be used to raise and lower in- dices following the same conventions (equations (A.3) and (A.4)) as in the Minkowskian case. The Clebsh-Gordan coefficients which intertwine between the bi-spinorial representations and the defining representation of SO(4), or vector represent- ation, are σµ = (σ1, σ2, σ3,i), σµ =(−σ1, −σ2, −σ3, i). (A.18) Given a four vector (p1,p2, p3,p4) the negative of the determinant of its corre- µ 2 2 2 2 sponding bi-spinor now becomes −det(σ pµ) = p1 +p2 +p3 +p4,asexpected. The matrices σµ and σµ satisfy the identities σµσν + σν σµ = − 2δµν , (A.19) µ αβ˙ − β α˙ (σ ) (σµ)γρ˙ = 2δγ δρ˙ . In order to construct the generators of the rotation group SO(4) one first defines the following matrices: 1 (σµν ) β = σµσν − σν σµ β, α 4 α ˙ (A.20) µν β˙ 1 µ ν ν µ β (σ ) ˙ = σ σ − σ σ ˙ , α 4 α One then easily finds out that the generators M µν = iσµν (or M µν = iσµν ) satisfy the SO(4) algebra: [M µν ,Mρτ ] = iδµρM ντ − iδµτ M νρ − iδνρM µτ + iδντ M µρ. (A.21) 208 Topological Quantum Field Theory and Four-Manifolds The explicit forms of the matrices σµν and σµν are ⎛ ⎞ i 3 i 2 i 1 0 − 2 σ 2 σ 2 σ ⎜ i 3 − i 1 i 2 ⎟ µν ⎜ 2 σ 0 2 σ 2 σ ⎟ σ = ⎝ i 2 i 1 i 3 ⎠ , − 2 σ 2 σ 0 2 σ i 1 i 2 i 3 − 2 σ − 2 σ − 2 σ 0 (A.22) ⎛ ⎞ i 3 i 2 i 1 0 − 2 σ 2 σ − 2 σ ⎜ i 3 − i 1 − i 2 ⎟ µν ⎜ 2 σ 0 2 σ 2 σ ⎟ σ = ⎝ i 2 i 1 i 3 ⎠ . − 2 σ 2 σ 0 − 2 σ i 1 i 2 i 3 2 σ 2 σ 2 σ 0 These matrices act as projectors of antisymmetric tensors and two-forms into their self-dual (SD) and anti-self-dual (ASD) parts. Given an antisymmetric + − tensor Vµν one defines its SD part, Vµν , and ASD part, Vµν , as follows: 1 1 V ± = (V ± V ρσ), (A.23) µν 2 µν 2 µνρσ where µνρσ isthe totally antisymmetric tensor corresponding to thevolume µν µν form (1234 =1). The matrices σ and σ satisfy the relations: 1 1 σµν = − σρσ, σµν = σρσ (A.24) 2 µνρσ 2 µνρσ and therefore they act as projectors into SD and ASD parts, providing the corresponding bi-spinor form: µν γ − µν γ Vαβ =βγ(σ )α Vµν = βγ(σ )α Vµν , (A.25) µν γ˙ + µν γ˙ Vα˙ β˙ =β˙γ˙ (σ ) α˙ Vµν = α˙ γ˙ (σ ) α˙ Vµν . Thetensors Vαβ and Vα˙ β˙ are symmetric and transform as the (3, 0) and the (0, 3) representations of SO(4). The antisymmetric tensor Vµν can bealso written in spinorial form as µ ν Vαα,β˙ β˙ = (σ )αα˙ (σ )ββ˙ Vµν . (A.26) It turns out that the bi-spinor version of the decomposition of the antisym- metric tensor into SD and ASD parts takes the form: + − −→ Vµν = Vµν + Vµν Vαα,β˙ β˙ = α˙ β˙ Vαβ + αβVα˙ β˙ .