N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

Conclusions and Outlook
In the present thesis we have given a derivation of the superfield constraints which
describe the two versions of the vector-tensor multiplet in presence of a gauged central
charge. Key to this was the formulation of consistency conditions every deformation
of the free model has to meet. We stress that these may be, and have been to a
certain extent in [2], employed to determine superfield constraints that yield even more
general models than the ones presented here, like for instance the linear vector-tensor
multiplet with global scale and chiral invariance first obtained in [17] by means of the
superconformal multiplet calculus. This involves a coupling to another abelian vector
multiplet with a nonvanishing background value (the nonlinear one with gauged central
charge possesses these invariances without further modifications).
Even in the case of a single vector multiplet, however, the consistency conditions turned
out to be insufficient when starting from a completely general Ansatz for the constraints.
While we were able to find solutions to the differential equations on the coefficients
that provide the sought generalizations of the two different vector-tensor multiplets,
we cannot exclude further solutions which may not be obtained from the known ones
merely by a field redefintion. However, what has been shown is that each solution
must reduce in the limit Z = i to either of two possible versions with global central
charge. Since we have found two corresponding classes of deformations, we venture the
assertion that no third one exists.
Unfortunately, as yet we do not know how to determine in a manifestly supersymmetric
way whether a given set of constraints is really compatible with the supersymmetry
algebra. While the consistency conditions (C.1–4) provide a preliminary selection of
superfield constraints, it is still necessary in each case to solve the Bianchi identities at
the component level in order to verify their validity.
Of course, the ultimate goal is to describe the vector-tensor multiplet in terms of an
unconstrained superfield, as it is possible for the hypermultiplet in harmonic superspace
at the expense of a finite number of off-shell components [1].
What we consider the most exciting feature of the vector-tensor multiplet (rather than
its relevance to certain string theory compactifications, which is beyond the scope of
this thesis), is the similarity of its local central charge transformations to the kind
of gauge transformations that occur in the new class of theories by Henneaux and
Knaepen. It is natural to ask for supersymmetric versions of these models. While this
problem could be solved completely for N = 1, in the case of two supersymmetries the
only known example we have presented here suffers from the explicit nonpolynomial
dependence on the central charge gauge field. As is clear in view of the complexity of
our component calculations, a first-order superfield formulation is indispensable. It is
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