N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

40 Chapter 3. The Linear Case
where eqs. (2.26) and (2.27) have been employed. This expression enters the central
charge transformation of ψ i as well as the supersymmetry transformations of W µ and
Gµν, which may be cast into the form
Next we calculate
D i αW µ = i ¯ Zσ µ δz ¯ ψ i + 1
2 Uσµ¯ λ i − D µ ψ i
α
D i αGµν = 2Iσµνδzψ i + Uσµνλ i + iεµνρσ σ ρ D σ ¯ ψ i
α .
DiDjM ij = − 3
I
and insert the result into eq. (BI.1),
G µν ( ˜ Fµν + iFµν) + 2W µ ∂µ ¯ Z + 2i Dµ(L∂ µ ¯ Z)
+ iλ i σ µ Dµ ¯ ψi + iDµψ i σ µ¯ λi + 2i ψ i σ µ ∂µ ¯ λi
+ 2iIλiδzψ i − i
3 (λiDj + ¯ λi ¯ Dj + 3Yij)M ij ,
IDµW µ = i
12 I DiDjM ij − 1
2 I λiδzψ i + c.c. = −W µ ∂µI + DµΛ µ + 1
2 FµνG µν ,
where we introduced the abbreviation
(3.22)
(3.23)
Λµ ≡ L∂µR + 1
2 (ψi σµ ¯ λi + λ i σµ ¯ ψi) . (3.24)
The first Bianchi identity thus reads
µ µ
Dµ IW − Λ = 1
2FµνG µν . (3.25)
To determine the second one, we first apply ¯ D ˙αi to eq. (3.21),
¯D ˙αiDαjM ij = 3
I σµ
α ˙α Gµν∂ ν R + ˜ Gµν∂ ν I + D ν Σµν + ˜ FµνW ν
− 1
2U∂µ|Z| 2 − 1
2δz(Zψ i σµ ¯ λi + ¯ Zλ i σµ ¯ ψi) .
Σµν has been defined in eq. (2.38). When put into eq. (BI.2), it follows that
I Dν ˜ G µν + R DνG µν = 1
6 I Diσ µ ¯ DjM ij − 1
2 U∂µ |Z| 2 − 1
2 δz(Zψ i σµ ¯ λi + ¯ Zλ i σµ ¯ ψi)
= − ˜ G µν ∂νI − G µν ∂νR − DνΣ µν − ˜ F µν Wν ,
(3.26)
and combining the derivatives, we eventually obtain
Dν IG ˜µν µν µν
+ RG + Σ = − 1
2εµνρσFνρWσ . (3.27)
We observe that the Bianchi identities of W µ and Gµν are not independent of each
other but constitute a coupled system of differential equations. We cannot solve them
yet as the covariant derivatives contain the central charge generator δz, whose action
on W µ and Gµν needs to be determined first. Since N ij
α ˙α = 0, eq. (2.36) immediately
gives
δz IG ˜µν µν µν
+ RG + Σ = −ε µνρσ DρWσ , (3.28)