N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

32 Chapter 2. The Vector-Tensor Multiplet
From this we infer first of all that the imaginary part of F is L-independent, thus
F = f(L) + iκ , κ ∈ R ⇒ ∂Lf = f 2 + κ 2 . (2.65)
When κ = 0, we have two solutions. On the one hand F1 = 0, which corresponds to
the free constraints (2.3) as then all coefficients vanish. The second solution is
F2 = − 1
L + µ
, µ ∈ R , (2.66)
where µ may be removed by a field redefinition (2.62). In the case κ = 0 the general
solution reads
F3 = κ tan(κL + ϱ) + i , ϱ ∈ R . (2.67)
We may choose κ = 1 and ϱ = 0. Since we have fixed the gauge, the three solutions
evidently yield distinct constraints that are not connected by a field redefinition. This
may also be seen from the transformation law of the coefficient G: If G = 0 for one
representative of a class of constraints, then this holds in the whole class. Moreover,
there is no transition from the second to the third solution since G2 is real while G3 is
not.
The constraints that correspond to F3 were first discovered by Ivanov and Sokatchev
in [21]. However, these are inconsistent, for the Bianchi identities (BI.1) and (BI.2)
admit no local solution. We shall not demonstrate this fact, but remark that it shows
that the conditions (C.1–4) are by no means sufficient.
The solution F2 implies constraints
D (i
α ¯ D j)
˙α L = 0 , D(iD j) L = − 1 i j
D L D L + D¯ i
L D¯ j
L , (2.68)
L
which are indeed consistent and describe what is known as the nonlinear vector-tensor
multiplet. We shall first generalize these constraints to admit a gauged central charge
before investigating them in any more detail. This will be done in chapter 4. At this
point we just emphasize that the constraints may be rendered regular for ˆ L = 0 by a
field redefintion
L = exp(−κ ˆ L) , κ ∈ R ∗ , (2.69)
which gives (omitting the hat)
D (i
α ¯ D j)
˙α
L = κ D(i
α L ¯ D j)
˙α L , D(iD j) L = 2κ D i L D j L + κ ¯ D i L ¯ D j L . (2.70)
In this form they were first derived in [20] and are evidently a deformation of the free
theory. While here the coefficients are constant, we shall nevertheless generalize the
constraints (2.68), for these have the useful property of vanishing N ij
α ˙α .
2.3.3 Generalization to Z(x)
In the general case of an x-dependent field Z the consistency conditions (C.1–3), which
we have translated into a set of differential equations, do not determine completely the