N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

20 Chapter 2. The Vector-Tensor Multiplet
in the gauge transformations of Vµ and Bµν. Furthermore, ∆ z commutes with a supersymmetry
transformation only modulo gauge transformations,
where the parameters now read
[ ∆ z (C) , ∆(ξ) ] = ∆ V (Θ) + ∆ B (Ω) , (2.12)
Θ = C(ξiψ i + ¯ ξ i ¯ ψi) , Ωµ = i C(ξ i σµ ¯ ψi + ψ i σµ ¯ ξi) . (2.13)
To construct an invariant action for the vector-tensor multiplet it suffices to combine
its components into a linear superfield, as shown in the previous chapter. From the
constraints (2.3) on L it follows that the field
L ij = κ D i L D j L + ¯κ ¯ D i L ¯ D j L (2.14)
with κ ∈ C constant has the desired properties, i.e. it is real, symmetric and satisfies
D (i
α L jk) = 0. When calculating the Lagrangian using eq. (1.43) we find that the real
part of κ gives rise to a total derivative, while the imaginary part provides the kinetic
terms for the multiplet components. For κ = i one obtains
LfreeVT = 1
2 ∂µ L ∂µL − 1
2 Hµ Hµ − 1
4 V µν Vµν − iψ i ↔
µ
σ ∂µ ¯ ψi + 1
2 U 2 . (2.15)
We observe that the central charge transformations (2.8) of the tensor fields vanish by
virtue of the equations of motion. The gauge fields, however, transform nontrivially
even on-shell. The conserved current that corresponds to this global symmetry of the
action is given by
J µ z = V µν Hν . (2.16)
Upon gauging the central charge transformations we therefore anticipate a coupling of
this current to the gauge field Aµ to first order in the deformation of the free theory.
At last we would like to clarify the above statement about the conversion of the vectortensor
multiplet into an abelian vector multiplet. The equation of motion for Bµν may
be solved in terms of a real scalar field a(x), which is then constrained by virtue of the
Bianchi identity of the dual field strength H µ ,
∂[µHν] ≈ 0 ⇒ Hµ ≈ ∂µa
∂µH µ = 0 ⇒ a ≈ 0 .
(2.17)
Hence, the antisymmetric tensor Bµν describes one spin- and massless degree of freedom.
Alternatively, one may consider H µ to be a fundamental field and incorporate
the Bianchi identity by means of a Lagrange multiplier,
− 1
2 Hµ Hµ − a ∂µH µ = 1
2 ∂µ a ∂µa − 1
2 (Hµ − ∂µa) 2 − ∂µ(aH µ ) .
The supersymmetry transformation of H µ ,
D i αH µ = 2(σ µν ∂νψ i )α = (σ µ ¯σ ν ∂νψ i − ∂ µ ψ i )α ≈ −∂ µ ψ i α ,