N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

4 Chapter 1. Gauging the Central Charge
In the N = 2 case the presence of central charges reduces this symmetry from U(2) to
SU(2), under which ε ij is an invariant tensor.
The algebra (1.1) with N = 2 can be represented on a so-called central charge superspace
[4] with coordinates x µ , θ α i , ¯ θ ˙αi and a further bosonic complex variable z. On
superfields infinitesimal supersymmetry transformations are generated by differential
operators
Q i α = ∂
∂θ α i
− i
2 (σµ¯ θ i )α∂µ + i
with commutation relations
{Q i α , Q j
β } = iεαβε ij ∂z
2θi α∂z , Q¯ ˙αi = − ∂
∂ ¯ θ
{Q i α , ¯ Q ˙αj} = iδ i jσ µ
α ˙α ∂µ
˙αi + i
2 (θiσ µ )α∂µ + i
2 ¯ θ ˙αi∂¯z
{ ¯ Q ˙αi , ¯ Q ˙ βj } = −iε ˙α ˙ β ε ij ∂¯z .
The commutator of two rigid supersymmetry transformations
(1.4)
(1.5)
∆(ξ) = ξ α i Q i α + ¯ ξ i ˙α ¯ Q ˙α i , (ξ α i ) ∗ = ¯ ξ ˙αi , (1.6)
yields global translations in the bosonic directions,
[ ∆(ξ) , ∆(ζ) ] = i(ξiσ µ ¯ ζ i − ζiσ µ ¯ ξ i )∂µ + iξ i ζi∂z + i ¯ ξi ¯ ζ i ∂¯z . (1.7)
Supercovariant spinor derivatives
D i α = ∂
∂θ α i
+ i
2 (σµ¯ θ i )α∂µ − i
2θi α∂z , D¯ ˙αi = − ∂
∂ ¯ i −
θ ˙αi 2 (θiσ µ )α∂µ − i
2 ¯ θ ˙αi∂¯z
(1.8)
anticommute with Q i α and ¯ Q ˙αi and therefore map superfields into superfields. Their
algebra involves a minus sign relative to the algebra of the Q’s,
{D i α , D j
β } = −iεαβε ij ∂z
{D i α , ¯ D ˙αj} = −iδ i jσ µ
α ˙α ∂µ
{ ¯ D ˙αi , ¯ D ˙ βj } = iε ˙α ˙ β ε ij ∂¯z .
(1.9)
The coefficient functions in the θ-expansion of a superfield constitute a supersymmetry
multiplet. Their supersymmetry transformations are generated by differential operators
D i α and ¯ D ˙αi, whose action can be read off the relation
D i α Φ(x, θ, ¯ θ, z) = Q i α Φ(x, θ, ¯ θ, z) , (1.10)
where Di α acts only on the components and anticommutes with the θ-variables. The
algebra of Di α and ¯ D ˙αi is the same as for the supercovariant derivatives. The components
of a superfield Φ may be regarded as the lowest components of superfields obtained
from applying an appropriate polynomial P (D, ¯ D) of supercovariant derivatives to Φ.
Eq. (1.10) implies that the generators Di α, ¯ D ˙αi act on components P (D, ¯ D) Φ | θ= θ=0 ¯
according to
D i
α P (D, D) ¯ Φ θ= ¯ θ=0 = D i αP (D, ¯ D) Φ
θ= ¯ . (1.11)
θ=0