N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

3.2. Transformations and Bianchi Identities 41
whereas the determination of δzW µ is somewhat involved. According to eq. (2.39),
2 µ i
Iδz |Z| W + 2L(Z∂µ Z¯ − Z∂ ¯ µ i Z) + 2 (Zψiσ µ¯ λi − ¯ Zλ i σ µ ψi) ¯ =
= I 2 DνG µν − IR Dν ˜ G µν + 1
6IR Diσ µ DjM ¯ ij
= I 2 DνG µν − G µν R∂νR − R Dν(I ˜ G µν + Σ µν ) + ˜ F µν
1
Wν + 2UR ∂µ |Z| 2
+ 1
2R δz(Zψ i σµ ¯ λi + ¯ Zλ i σµ ¯ ψi) .
The expression in square brackets can be rewritten by means of the Bianchi identity
(3.27),
2 µ i
Iδz |Z| W + 2L(Z∂µ Z¯ − Z∂ ¯ µ i Z) + 2 (Zψiσ µ¯ λi − ¯ Zλ i σ µ ψi) ¯ =
= |Z| 2 DνG µν + 1
2UR ∂µ |Z| 2 + 1
2R δz(Zψ i σµ ¯ λi + ¯ Zλ i σµ ¯ ψi) ,
from which follows
µ µ
δz IW − Λ = DνG µν . (3.29)
One can now easily check that the Bianchi identities, together with the central charge
transformations just obtained, satisfy the integrability condition
D[µDν] = 1
2 Fµνδz ,
which is the covariant analogue of d 2 = 0.
We first solve the constraint on W µ . Let us split the covariant derivative in eq. (3.25)
into the partial derivative and the central charge transformation, which we then replace
using eq. (3.29),
µ µ
∂µ IW − Λ = 1
2 FµνG µν − Aµδz
= 1
2 Fµν − AµDν
= ∂µ(G µν Aν) .
IW µ − Λ µ
G µν
In the last step we employed the identity A[µDν] = A[µ∂ν]. We conclude that the terms
in parentheses equal the dual field strength of an antisymmetric tensor gauge field Bµν,
IW µ = 1
2 εµνρσ (∂νBρσ − Aν ˜ Gρσ) + Λ µ . (3.30)
Note how the first two terms on the right-hand side resemble a covariant derivative,
and indeed we shall find that Bµν transforms into − ˜ Gµν under δz. In the limit Z = i
we recover the relation W µ = H µ just as in the free case.
To solve the Bianchi identity (3.27) we proceed along the same lines,
∂ν IG ˜µν µν µν
+ RG + Σ = −ε µνρσ ( 1
2FνρWσ − AνDρWσ)
Hence there is a vector field Vµ such that
= −ε µνρσ ∂ν(AρWσ) .
I ˜ G µν + RG µν = ε µνρσ (∂ρVσ − AρWσ) − Σ µν . (3.31)