N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

6 Chapter 1. Gauging the Central Charge
The commutator of two covariant derivatives involves the field strength F I µν,
which is given by
and which satisfies the Bianchi identity
[ Dµ , Dν ] = F I µν δI , (1.17)
F I µν = ∂ µA I ν − ∂ νA I µ − A J µA K ν fJK I , (1.18)
ε µνρσ D νF I ρσ = 0 . (1.19)
The field strength is a tensor that also transforms in the adjoint representation of the
Lie algebra G,
∆ g (C) F I µν = −C J fJK I F K µν . (1.20)
Hence, in the abelian case the field strength is gauge invariant.
In [5] Grimm et al. have shown how to embed 1-form gauge fields into N = 2 supersymmetry
multiplets. Let us first discuss the case without an explicit central charge,
δz = 0. In analogy to eq. (1.16) one extends the flat supercovariant derivatives D i α to
super- and gaugecovariant derivatives D i α and imposes constraints on the field strengths
such that only a minimal number of components survives. The Bianchi identities then
fix the algebra to read
{D i α , D j
β } = εαβε ij ¯ φ I δI
{ ¯ D ˙αi , ¯ D ˙ βj } = ε ˙α ˙ β ε ij φ I δI
{D i α , ¯ D ˙αj} = −iδ i jσ µ
α ˙α Dµ
[ D i α , Dµ ] = i
2 (σµ ¯ D i ¯ φ I )α δI
[ ¯ D ˙αi , Dµ ] = i
2 (Diφ I σµ) ˙α δI .
(1.21)
The generators δI act trivially on the supercovariant derivatives, i.e. they commute.
We do not give the explicit realization of the D i α as in the following we need only
their commutation relations. The calligraphic D i α shall always generate supersymmetry
transformations of component fields, and we remark that the relations (1.11) and (1.12)
apply also to the present case when the D i α are replaced by D i α.
The so-called vector superfields φ I transform as tensors in the adjoint representation
under gauge transformations. The Bianchi identities imply that they are subject to
constraints
¯D ˙αiφ I = 0 = D i α ¯ φ I , D (i D j) φ I = ¯ D (i ¯ D j) ¯ φ I , (1.22)
which lead to the field content of a complex scalar, a doublet of Weyl spinors, a real
SU(2) triplet of auxiliary scalars and a real antisymmetric tensor 1 ,
φ I | , χ iI
α = D i αφ I | , D ijI = 1
2 D(i D j) φ I |
F I µν = 1
4 (Di σµνDiφ I − ¯ Di¯σµν ¯ D i ¯ φ I )| ,
(1.23)
the latter providing the field strength for the gauge potentials A I µ. Here and henceforth
we shall employ the convention of labeling a superfield and its lowest component by the
1 A bar denotes projection to the lowest component of a superfield by setting θ = ¯ θ = 0.