N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

26 Chapter 2. The Vector-Tensor Multiplet
Here we discover again the factor E accompanying the central charge generator, for the
covariant d’Alembertian acting on L may be written as
D µ DµL = L + U∂ µ Aµ + 2A µ ∂µU + A µ AµδzU .
The last term then combines with |Z| 2 δzU on the left-hand side of eq. (2.35) into EδzU.
It remains to consider the part symmetric in αβ. Using eq. (A.21) we readily obtain
with the abbreviations
At last we require
δz IGµν − R ˜ Gµν − ˜
Σµν = −2 D[µWν] + i
6εµνρσ D i σ ρ D¯ j σ
Nij , (2.36)
I ≡ Im Z , R ≡ Re Z (2.37)
Σµν ≡ LFµν + i(λiσµνψ i − ¯ ψi¯σµν ¯ λ i ) . (2.38)
[ δz , D i α ] ¯ ψ j !
˙α = 0 .
We again decompose the commutator into SU(2) irreducible parts. Symmetrized in ij
the equation is fulfilled identically when the conditions (C.1–3) hold. Antisymmetrized
the real part provides the second Bianchi identity,
I Dν ˜ G µν + R DνG µν = − 1
2 U∂µ |Z| 2 − 1
2 δz(Zψ i σ µ¯ λi + ¯ Zλ i σ a ψi) ¯
− i
12 Z Diσ µ DjM ¯ ij − i
12 ¯ Z Diσ µ Dj
¯ ¯ M ij
− 1
12
ZDiDj + ¯ Z ¯ Di ¯ a ij
Dj N ,
while the imaginary part gives rise to the central charge transformation of W µ ,
2 µ i
δz |Z| W + 2L(Z∂µ Z¯ − Z∂ ¯ µ i Z) + 2 (Zψiσ µ¯ λi − ¯ Zλ i σ µ ψi) ¯ =
= I DνG µν − R Dν ˜ G µν + 1
12 Z Diσ µ DjM ¯ ij − 1
12 ¯ Z Diσ µ Dj
¯ ¯ M ij
− i
ZDiDj −
12
¯ Z ¯ Di ¯ a ij
Dj N .
(BI.2)
(2.39)
With this the evaluation of the supersymmetry algebra on ψ i is completed. We could
already determine all the supersymmetry and central charge transformations of the
covariant components of the vector-tensor multiplet. Evidently we cannot obtain any
information on the gauge fields Vµ and Bµν as long as the deformations M ij and N ij
α ˙α
have not been specified and the Bianchi identities solved. It is now a tedious exercise
to check that the algebra holds also on W µ , Gµν and U and that we obtain no further
consistency conditions.