Hi Jack,

the problem is, as you imply, that there is no easy way of writing the

N=2 SYM Lagrangian in terms of N=2 superfields. Writing it in terms of

N=1 superfields, however, disguises some symmetries. Namely, only half

of the N=2 SUSYs and only a U(1)_J subgroup of the (classical) SU(2)_R

X U(1)_R symmetry can be made manifest in a given N=1 formulation.

But you won't have to guess the requisite term in the N=1

superpotential. One way of deriving this is as follows. A suitable 180

degree SU(2)_R rotation, call it R_0, interchanges the N=2 fermionic

superspace coordinates. If you act with this rotation on a Lagrangian

that makes manifest the first half of the SUSYs, you must find an

equivalent Lagrangian that is obviously invariant under the second

half. This requirement is enough to fix the superpotential.

To do this in practice you need to know explicitly how R_0 acts on the

component fields. This can be worked out by decomposing the N=2 vector

multiplet into an N=1 vector and an N=1 chiral multiplet. That is,

write down the N=2 V-plet SUSY transformations and identify what parts

of it look like the N=1 V-plet transformations and N=1 chiral multiplet

transformations with respect to certain component fields. Of course,

the role of a given component field may be different under the

respective two sets of SUSY transformations. Now, R_0 interchanges the

two sets of SUSY tranformations and therefore must interchange the

fields accordingly. This observation tells you how R_0 acts on the

component fields.

The idea of requiring invariance under R_0 goes all the way back to P.

Fayet's pioneering work [Nucl.Phys.B113:135,1976], who incidentally

referred to extended supersymmetry as hypersymmetry. It's interesting

how some names stick and others don't...

Best,

Tobias