N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

8 Chapter 1. Gauging the Central Charge
It seems the dimensions have gone awry. If the generators δI are taken to be dimensionless,
the corresponding vector superfields must have mass dimension unity. As is
clear from its representation as a space derivative, however, δz has the dimension of an
inverse length, which results in a shift of the dimension of Z. To compensate for this,
the central charge coupling constant gz that will be introduced with the Lagrangian
(see next section) carries mass dimension −1.
To distinguish the components of the central charge vector multiplet from those of
ordinary gauge multiplets, we denote them by
Z , Aµ , λ i α | Y ij ,
and the abelian field strength of Aµ we write as
Fµν = ∂µAν − ∂νAµ . (1.30)
The tensor components are invariant under an infinitesimal central charge transformation
∆ z (C), while Aµ transforms into the gradient of the parameter C(x),
∆ z (C) Z , λ i α , Fµν , Y ij = 0 , ∆ z (C) Aµ = −∂µC . (1.31)
Note that the above discussion implies that the central charge multiplet is invariant
under gauge transformations ∆ g , while the ordinary gauge multiplets are invariant
under central charge transformations ∆ z .
The supersymmetry transformations can be copied from above. They are linear due to
the abelian nature of the central charge,
D i αZ = λ i α , D i α ¯ Z = 0
D i αλ j
β = εαβ Y ij + ε ij Fµν σ µν
αβ , Di α ¯ λ j
˙α = iεij ∂α ˙α ¯ Z
D i αY jk = iε i(j ∂α ˙α ¯k) ˙α
λ
D i αAµ = i
2 (σµ ¯ λ i )α , D i αFµν = i ∂[µ(σν] ¯ λ i )α .
(1.32)
Note that the nonvanishing vev of Z does not break supersymmetry spontaneously.
Since the vector-tensor multiplet transforms trivially under gauge transformations ∆ g
(see next chapter), the algebra we shall be dealing with mostly in this thesis includes
only the central charge generator δz, and for reference we list the commutation relations
as they are to hold on tensor components,
{D i α , ¯ D ˙αj} = −iδ i j σ µ
α ˙α Dµ
{D i α , D j
β } = εαβ ε ij Zδz
¯
{ ¯ D ˙αi , ¯ Dβj ˙ } = ε ˙α β˙ εijZδz
The commutators involving δz vanish.
[ Dµ , Dν ] = Fµν δz
[ D i α , Dµ ] = i
2 (σµ ¯ λ i )αδz
[ ¯ D ˙αi , Dµ ] = i
2 (λiσµ) ˙αδz .
(1.33)