N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

10 Chapter 1. Gauging the Central Charge
What happens when we consider local central charge transformations instead of ordinary
gauge transformations? We can simply replace the generators δI with δz and the
multiplets φ I with Z in the above equations. Then the constraint on K µ reads
DµK µ = − 1
2 δz
ZS + Z¯ S ¯ + 2λiϱ i − 2¯ λ i ¯ϱi + 3Yijϕ ij ≡ − 1
2 δz ˆ L , (1.39)
for now the gauge multiplet transforms trivially under the generator δz. While it cannot
be solved unless δzϕ ij = 0, the constraint implies the existence of a gauge invariant
action. Let us consider the expression
L = ˆ L + 2AµK µ .
Applying a local central charge transformation, we can replace δz ˆ L with the covariant
derivative of K µ using the constraint and then combine this with the transform of the
second term into a total derivative,
∆ z (C) L = Cδz ˆ L − 2∂µC K µ + 2CAµδzK µ
= −2C DµK µ − 2∂µ(CK µ ) + 2C(∂µK µ + AµδzK µ )
= −2∂µ(CK µ ) .
Thus upon integration over spacetime d 4 x L is invariant under gauged central charge
transformations. Amazingly, it is even supersymmetric, for we find after a short calcu-
lation
D i
αL = −i ∂µ 2Zσ ¯ µ i ij
¯ϱ + 3ϕ σ
µ¯ λj − 4iAνσ µν ϱ i
. (1.40)
α
Altogether we have found a general prescription to construct invariant actions: If the
components of the multiplets under consideration can be combined into a superfield
Lij = Lji = (Lij) ∗ such that it satisfies the constraints
then the Lagrangian
D (i
αL jk) = 0 = ¯ D (i
˙α Ljk) , (1.41)
L = 1
ZDiDj +
12
¯ Z ¯ Di ¯ Dj + 4λiDj − 4¯ λi ¯ Dj + 6Yij + 2iAµDiσ µ
Dj
¯ ij
L | (1.42)
provides us with a supersymmetric action that is invariant under local central charge
transformations. Note that if the linear superfield L ij is also invariant under gauge
transformations ∆ g , this rule extends to ordinary gauge theories as well. When we do
not consider fields that are subject to local central charge transformations, we replace
the superfield Z by its background value 〈Z〉 = i, and the Lagrangian reduces to
L = i
DiDj −
12
¯ Di ¯ ij
Dj L | . (1.43)
Occasionally we shall call L ij the “pre-Lagrangian”. There is no guarantee, however,
that one can always find an L ij which gives rise to a nontrivial Lagrangian, i.e. one
which is not merely a total derivative.