N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

1.1. The N=2 Supersymmetry Algebra 7
same symbol. As we are going to deal with up to three multiplets simultaneously and
introduce a fair amount of abbreviations, a large number of symbols is needed, which
calls for an economical notation. It should be clear in each equation which is which;
when ambiguities might occur, we explicitly state whether the full superfield or merely
a component field is meant.
From eq. (1.11) and the algebra (1.21) one derives the supersymmetry transformations
of the tensor components of φ I . The action of D i α is found to be
D i αφ I = χ iI
α , D i α ¯ φ I = 0
D i α χjI
β = εαβ D ijI + ε ij F I µν σ µν 1
αβ + 2εαβε ij φ J φ¯ K
fJK I , D i α ¯χjI
˙α = iεij Dα ˙α ¯ φ I
D i αD jkI = iε i(j (Dα ˙α ¯χk) ˙αI + iχ k)J
α ¯ φ K fJK I ) (1.24)
D i αF I µν = i D[µ(σν] ¯χiI )α
while the action of ¯ D ˙αi is readily obtained by complex conjugation. Since the gauge
fields AI µ do not occur linearly and undifferentiated in a θ-expansion of the φI , their
supersymmetry transformations cannot be derived from eq. (1.11). We define the action
of Di α on AI µ by
D i αA I µ = i
2 (σµ ¯χiI )α , (1.25)
which is compatible with the transformation of F I µν. Then the supersymmetry algebra
reads on all component fields
{D i α , ¯ D ˙αj} = −iδ i j σ µ
α ˙α
∂µ + ∆ g (Aµ)
{D i α , D j
β } = εαβ ε ij ∆ g ( ¯ φ) { ¯ D ˙αi , ¯ D ˙ βj } = ε ˙α ˙ β εij∆ g (φ) .
(1.26)
On tensors the combination ∂µ + ∆ g (Aµ) is just the covariant derivative. We conclude
that the commutator of two supersymmetry transformations yields a translation and a
field dependent gauge transformation,
with parameters
[ ∆(ξ) , ∆(ζ) ] = ɛ µ ∂µ + ∆ g (C) , (1.27)
ɛ µ = i(ζiσ µ ¯ ξ i − ξiσ µ ¯ ζ i ) , C I = ɛ µ A I µ − ξiζ i ¯ φ I + ¯ ξ i ¯ ζi φ I . (1.28)
Now let us compare the anticommutator of two spinor derivatives in eqs. (1.9) (substituting
δz for ∂z = ∂¯z) and (1.21). Evidently, an operator δz generating a rigid central
charge transformation may formally be incorporated into the latter algebra by first extending
the gauge group by an extra U(1) factor, the generator of which one identifies
with δz, and then replacing the corresponding superfield with the constant background
value i. Accordingly, the central charge is promoted to a local transformation by reintroducing
the full vector superfield, denoted in the following by Z. This differs from
the other φ I in that it has a nonvanishing vacuum expectation value (vev),
〈Z〉 = i . (1.29)