N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

2.3. The Ansatz 31
Note that when ¯ D ˙α(i N jk)
α ˙α
ij
= 0, for instance in the special case Nα ˙α = 0, the combination
κM ij + ¯κ ¯ M ij , κ ∈ C
is real and hence a linear superfield by itself according to eqs. (C.1) and (C.3),
¯D ˙α(i N jk)
α ˙α
= 0 ⇒ D(i α(κM jk) + ¯κ ¯ M jk) ) = 0 . (2.60)
Thus it will turn up as a particular solution to the conditions (2.59).
2.3.2 Solutions for Z = i
The general Ansatz (2.40), (2.41) does not reduce to the free constraints (2.3) in the
limit Z = i but there remain terms quadratic in D i αL. This suggests that the constraints
(2.3) may not be the only possible description of the vector-tensor multiplet, and indeed,
as mentioned in the introduction, Claus et al. have shown in [16] that there exists a
nontrivial deformation 5 which gives rise to self-interactions. With the set of consistency
conditions given above we can reproduce this result:
The case Z = i corresponds to
a = b = A = B = C = D = E = 0 , ∂ = ¯ ∂ ≡ 0 .
The equations (2.48) are satisfied identically, whereas (2.46) and (2.50) each provide a
single condition on the remaining functions c(L), F (L) and G(L), namely
0 = (∂L + c)G − (F − c)G
0 = (∂L + c)(F − c) − G ¯ G .
These are invariant under field redefinitions L = L( ˆ L) and transformations
ĉ = c L ′ − L ′′ /L ′ , ˆ F = F L ′ − L ′′ /L ′ , ˆ G = GL ′
(2.61)
as in (2.54). Since c transforms inhomogeneously, we may choose a gauge in which
c = 0. Note that this does not fix the gauge completely, we are still free to shift and
rescale L by real constant parameters,
For c = 0, i.e. N ij
α ˙α
reduces to
L = κ ˆ L + ϱ , κ ∈ R ∗ , ϱ ∈ R ⇒ L ′′ /L ′ = 0 . (2.62)
= 0, the consistency condition (C.4) can easily be evaluated. It
(i j)
0 = δz D D L − D ¯ (i
D¯ j)
L
= δz (F − G) ¯ i j
D L D L + (G − F ¯) D¯ i
L D¯ j
L .
Hence G = ¯ F , and the equations (2.61) both imply
(2.63)
∂LF = F ¯ F . (2.64)
5 Nontrivial in the sense that it may not be removed by a field redefinition.