N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

A.2. σ-Matrices 73
We use a shorthand notation for σ µν -matrices whose indices have been lowered by
means of the ε-tensor,
They are symmetric in the spinor indices,
σ µν αβ = −(σ µν ε)αβ , ¯σ µν
˙α ˙ β = (ε¯σµν ) ˙α ˙ β
(A.19)
σ µν αβ = σ µν βα , ¯σ µν
˙α ˙ β = ¯σµν ˙ β ˙α . (A.20)
Using the σ µν -matrices, an antisymmetric tensor Fµν can be decomposed into its “selfdual”
and “anti-selfdual” part,
F α ˙α β ˙ β = ε ˙α ˙ β Fαβ + εαβ ¯ F ˙α ˙ β , ˜ Fα ˙α β ˙ β = i ε ˙α ˙ β Fαβ − i εαβ ¯ F ˙α ˙ β
Fαβ = −Fµν σ µν αβ , ¯ F ˙α ˙ β = Fµν ¯σ µν
˙α ˙ β .
(A.21)
There are numerous relations between the quantities defined so far. The ones used
frequently in this thesis shall be listed here.
Identities containing two σ-matrices:
σ µ
α ˙α σ µ β ˙ β = 2εαβε ˙α ˙ β , σ µ
α ˙α ¯σ ˙ ββ
µ = 2 δ β α δ ˙ β
˙α
(A.22)
(σ µ ¯σ ν )α β = η µν δ β α + 2 σ µν α β , (¯σ µ σ ν ) ˙α ˙ β = η µν δ ˙α ˙ β + 2 ¯σ µν ˙α ˙ β (A.23)
Identities containing three σ-matrices:
σ [µ
α ˙α σν]
β ˙ β = εαβ ¯σ µν
˙α ˙ β − ε ˙α ˙ β σµν αβ
ε µνρσ σρσ = 2i σ µν , ε µνρσ ¯σρσ = −2i ¯σ µν
(A.24)
(A.25)
ε µνρσ σ ρ α ˙ασ σ β ˙ β = −2i (εαβ ¯σ µν
˙α ˙ β + ε ˙α ˙ β σµν αβ) . (A.26)
σ µν σ ρ = 1
2 (ηνρ σ µ − η µρ σ ν + iε µνρσ σσ) (A.27)
¯σ µν ¯σ ρ = 1
2 (ηνρ ¯σ µ − η µρ ¯σ ν − iε µνρσ ¯σσ) (A.28)
¯σ µ σ νρ = 1
2 (ηµν ¯σ ρ − η µρ ¯σ ν − iε µνρσ ¯σσ) (A.29)
σ µ ¯σ νρ = 1
2 (ηµν σ ρ − η µρ σ ν + iε µνρσ σσ) (A.30)
σ µν αβ σν γ ˙α = −εγ(β σ µ
α) ˙α , ¯σµν
˙α ˙ β σν α ˙γ = −σ µ
α( ˙α εβ) ˙ ˙γ . (A.31)
Identities containing four σ-matrices:
σ µν σ ρσ = 1
2 (ηµσ σ νρ − η µρ σ νσ + η νρ σ µσ − η νσ σ µρ )
+ 1
4 (ηµσ η νρ − η µρ η νσ + iε µνρσ )
¯σ µν ¯σ ρσ = 1
2 (ηµσ ¯σ νρ − η µρ ¯σ νσ + η νρ ¯σ µσ − η νσ ¯σ µρ )
+ 1
4 (ηµσ η νρ − η µρ η νσ − iε µνρσ )
(A.32)
(A.33)
σ µν α β σν ρ γ δ = 1
2 (δβ γ σ µρ α δ − δ δ ασ µρ γ β ) + 1
4 ηµρ (εαγ ε βδ + δ δ α δ β γ ) (A.34)