N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

4.2. Transformations and Bianchi Identities 61
where the real composite fields
W µ ≡ |Z| 2 (LW µ − ψ i σ µ ¯ ψi) + i
2 L2 (Z∂ µ ¯ Z − ¯ Z∂ µ Z + iλ i σ µ¯ λi)
+ iL (Zψ i σ µ¯ λi − ¯ Zλ i σ µ ¯ ψi)
(4.24)
Π µν ≡ i(Zψ i σ µν ψi − ¯ Z ¯ ψ i ¯σ µν ¯ ψi) (4.25)
are both bilinear in the components of the vector-tensor multiplet. In view of eq. (2.56)
one might search for a superfield redefintion which simplifies W µ . In the limit Z = i
we would have
LW µ − ψ i σ µ ¯ ψi = LL ′ ˆ W µ − (L ′2 + 1
2 LL′′ ) ˆ ψ i σ µ ˆ¯ ψi ,
so the spinors could be removed indeed (with L ∼ ˆ L 1/3 ). However, this would merely
shift complications from one place to another, as both W µ and W µ occur in the Bianchi
identities and their solutions. Hence, we stick to our original choice (4.13) for the
constraints.
It is quite an effort to derive the first Bianchi identity from eq. (BI.1). Applying ¯ D ˙α i to
eq. (4.16) gives
i
6 Z ¯ Di ¯ DjM ij = iLZ − 2(W µ − iD µ L) ∂µZ − Zλiδzψ i + 2i ∂µλ i σ µ ψi
¯
− 1
i
2 L
Z(Wµ − iDµL) 2 + i
4 ¯ ZG µν (Gµν + i ˜ Gµν) + i
2Z|Z|2 U 2
+ λ i σ µ ψi
¯ (Wµ − iDµL) + iFµν ¯ ψ i ¯σ µν ψi
¯ − Z Dµ(ψ i σ a ψi) ¯ (4.26)
− 2 ψ i σ µ ¯ ψi ∂µZ + ZUψ i λi − iY ij ¯ ψi ¯ ψj − i
4 ¯ Z ¯ Mij ¯ M ij
+ 1
3 Z (ψiDj ¯ M ij − ¯ ψi ¯ DjM ij ) .
This we insert into eq. (BI.1) and multiply the equation with |Z| 2 L, upon which the
covariant divergence of W µ emerges,
DµW µ = − 1
4 (IGµν − R ˜ Gµν)(I ˜ G µν + RG µν ) − 1
+ 1
2FµνΠ µν − i
1
2 4Z(ZMij − 4i λiψj)M ij + (ZYij − λiλj)ψ i ψ j
− λiψj λ i ψ j − c.c. .
2 Gµν(Z λiσ µν ψ i + ¯ Z ¯ ψi¯σ µν ¯ λ i )
By virtue of eqs. (A.35) and (A.25) the four-fermion terms that remain when M ij is
inserted can be written as the product of an antisymmetric tensor with its dual,
1
4Z(ZMij − 4i λiψj)M ij + (ZYij − λiλj)ψ i ψ j − λiψj λ i ψ j − c.c. =
= − 1
2 (λiλj ψ i ψ j + λiψj λ j ψ i ) − c.c.
= − i
4 εµνρσ (λiσµνψ i − ¯ ψi¯σµν ¯ λ i ) (λjσρσψ j − ¯ ψj ¯σρσ ¯ λ j )
= i
2 (Σµν − LFµν) ( ˜ Σ µν − L ˜ F µν ) ,
which together with the relation
− 1
2 Gµν(Z λiσ µν ψ i + ¯ Z ¯ ψi¯σ µν ¯ λ i ) = − 1
2 (IGµν − R ˜ Gµν) (Σ µν − LF µν )