N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

24 Chapter 2. The Vector-Tensor Multiplet
and finally the part DαiWα ˙α, which together with eq. (2.30) gives the complete supersymmetry
transformation of W µ ,
D i αWββ ˙ = −Dββ˙ ψ i
α − εαβ 2iZ¯ δz ¯ ψ i + ¯ λ i U
i
˙β + 2εαβ D γ ij
j N
γ ˙ i −
β 3 D ij
j(αN β) ˙ . (2.32)
β
Before going any further, let us examine eqs. (2.31). We observe a structure similar
to that of the central charge transformations of the spinors in the hypermultiplet, eqs.
(1.53), namely the equations for δzψ i and δz ¯ ψ i are coupled by virtue of the covariant
derivative. Let us try to solve for δzψ i . For convenience we introduce the abbreviation
and calculate
η i α ≡ λ i αU − 1
3 Dαj ¯ M ij − i
3 ¯ D ˙α j N ij
α ˙α
|Z| 2 δzψ i α = iAα ˙α ¯ Zδz ¯ ψ ˙αi + i ¯ Z(∂α ˙α ¯ ψ ˙αi − η i α)
= Aα ˙α(D ˙αβ ψ i β − ¯η ˙αi ) + i ¯ Z(∂α ˙α ¯ ψ ˙αi − η i α)
= Aα ˙αA ˙αβ δzψ i β + Aα ˙α(∂ ˙αβ ψ i β − ¯η ˙αi ) + i ¯ Z(∂α ˙α ¯ ψ ˙αi − η i α) .
Again the prefactor E, defined in eq. (1.55), emerges. We have now eliminated δz ¯ ψ i ,
provided that η i or its complex conjugate does not contain such a term. We assume
this to be the case 3 . Then the action of the central charge generator on ψ i reads
δzψ i α = 1
iZ(∂α ¯
˙α
E
¯ ψ ˙αi − η i α) + Aα ˙α(∂ ˙αβ ψ i β − ¯η ˙αi ) . (2.33)
This expression does not appear to be covariant with respect to local central charge
transformations, as the gauge potential occurs explicitly. However, from eq. (2.31) it
should be clear that δzψ i is indeed a tensor, and one may verify that all the differentiated
gauge parameters cancel when calculating the central charge transformation of δzψ i
proceeding from eq. (2.33). In what follows it is advantageous to use the manifestly
covariant expression (2.31) rather than the complicated equation (2.33).
We resume the evaluation of the supersymmetry algebra with the anticommutator
{D i α , D j
β } ¯ ψ k ˙α
!
= εαβε ij ¯ Z δz ¯ ψ k ˙α .
From eq. (2.27) we see that this involves D j
β Wα ˙α, which is given in eq. (2.32). We find
that the equation is fulfilled identically provided the consistency condition (C.2) holds,
so we obtain no new information.
Next we investigate the consequences of the requirement that the central charge generator
δz commute with the supersymmetry generator D i α,
[ δz , D i α ] ψ j
β
!
= 0 .
In this equation we encounter two generators Di α, ¯ Di ˙α acting on the deformations M ij
. From now on we assume the supersymmetry transformations of these fields
and N ij
α ˙α
3 All consistent constraints that we present in the following chapters have this property.