N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

34 Chapter 2. The Vector-Tensor Multiplet
Since v does not depend on ¯ Z and u is real, we conclude that there is a function w(Z)
such that
u(Z, ¯ Z) = w(Z) + ¯w( ¯ Z) , v(Z) = ∂w(Z) . (2.76)
Now let us perform a field redefinition (2.72) in eq. 25). With the transformation of C
as in eq. (2.54), we calculate
ˆL + ZĈ + ¯ Z ˆ¯ 1
C =
L ′
L − f − f¯ Z
+
L ′
Z¯
C − ∂f +
L ′
= 1
L ′
w + ¯w − f − f¯ − Z∂f − Z¯ ∂¯ f¯ .
The same redefinition applied to eq. 26) gives
¯C − ¯ ∂ ¯ f
Z ˆ D + ¯ Z ˆ Ē = 1
L ′
ZD + ZĒ ¯ + ∂f (ZA + ¯ Z ¯ B) + ∂f ∂f (ZF + ¯ Z ¯ G) − Z∂ 2 f
= 1
L ′ ∂w − f − Z∂f ,
where eqs. 23) and 24) have been used. So provided there is a function g(Z) such that
∂g = w, a field redefinition with f = g/Z yields (omitting the hats)
25) 0 = L + ZC + ¯ Z ¯ C 26) 0 = ZD + ¯ ZĒ . (2.77)
Note that there remains a residual gauge freedom
L = κ ˆ L + ϱ
Z
+ ¯ϱ
¯Z , κ ∈ R∗ , ϱ ∈ C , (2.78)
for g may be shifted by a complex constant ϱ.
Working in a gauge where u = v = 0, the 26 consistency conditions can be reduced to
a set of 11 independent equations,
1) 0 = 2∂F − ∂LA
2) 0 = ∂LC − CF − 1
2 A
3) 0 = ∂C − 1
2 AC + ¯ Z
Z Ē
4) 0 = ∂L ¯ F − F ¯ F
5) 0 = 2∂ ¯
F − A + 2
¯F
Z
6) 0 = ∂LE − EF (2.79)
7) 0 = ∂E − 1
2AE
1
9) 0 = ∂Ā − A + 2
2
2
Ā +
Z ¯Z
10) 0 = ∂A − 1
2A2 − 2ĒF − ¯ F ¯ Z
Z
19) 0 = ∂LA − A + 2
¯F
Z