N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

2.1. Introducing the Multiplet 19
This allows to identify these components with the field strengths 1 (2.1) of the gauge
potentials Vµ and Bµν. The reason for using different labels in the definition of the
component fields will become clear when we investigate deformations of the superfield
constraints (2.3) in the next chapters. There the relation between Gµν and W µ and
the field strengths Vµν and H µ will be more complicated as the differential constraints
(2.5) on the former also get modified. As we shall see, the transformations and actions
can be formulated most easily in terms of components defined as in eq. (2.4) when
interactions are introduced. In this section, however, in which only the free case is
presented, there is no distinction between Gµν, W µ and Vµν, H µ .
The supersymmetry transformations of the components (2.4) read
D i αL = −iψ i α , D i αU = −i(σ µ ∂µ ¯ ψ i )α
D i αVµν = −2 ∂[µ(σν] ¯ ψ i )α , D i αH µ = 2(σ µν ∂νψ i )α (2.6)
D i αψ j 1
β = 2εij (εαβU + iVµν σ µν αβ) , D i α ¯ ψ j 1
˙α = 2εijσ µ
α ˙α (∂µL − iHµ) ,
while those of the potentials are given by
D i αVµ = −(σµ ¯ ψ i )α , D i αBµν = −2i (σµνψ i )α . (2.7)
The commutation relations of these involve a global central charge. The action of the
generator δz reads
δzL = U , δzU = L , δzψ i = σ µ ∂µ ¯ ψ i
δzVµν = −2 ∂[µHν] , δzH µ = ∂νV µν
on the tensors, and on the gauge fields one has
(2.8)
δzVµ = −Hµ , δzBµν = − 1
2 εµνρσV ρσ . (2.9)
We refrain from giving a detailed derivation of these results, as this will be done later
on in the more general case of an algebra with a gauged central charge.
On the potentials Vµ and Bµν the algebra (1.9) holds modulo gauge transformations
(2.2). The commutator of two rigid supersymmetry transformations is given on all the
fields 2 by
[ ∆(ξ) , ∆(ζ) ] = ɛ µ ∂µ + ∆ z (C) + ∆ V (Θ) + ∆ B (Ω) , (2.10)
with ɛ µ as in eq. (1.28), C = i(ξiζ i + ¯ ξ i ¯ ζi), and field dependent parameters
Θ = ɛ µ Vµ − L(ξiζ i − ¯ ξ i ¯ ζi)
Ωµ = ɛµL − Bµνɛ ν − Vµ(ξiζ i − ¯ ξ i ¯ ζi)
(2.11)
1 µ Occasionally we call H the field strength of Bµν for short, hoping not to confuse the reader by
this abuse of denotation.
2 V B It is understood that ∆ and ∆ act nontrivially only on Vµ and Bµν, respectively.