d=4 N=2 SYM
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Jack Tremarco
2005-07-01 09:29:01 UTC
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I'm reading Seiberg-Witten's "Monopole condensation and
confinement..."-paper and I have a basic question.

When you write down the N=2 SUSY Lagrangian in terms of N=1
superfields, there is a requisite N=1 superpotential term of the form

\sqrt{2} \tilde{Q} \Phi Q ,

which is required for N=2 SUSY.

I understand this fact and I can explicitly verify that it's true. But
suppose we didn't know it, how can we derive it? Do we have to guess
the presence of such a term and find its prefactor by explicit
computation? I hope there is a simpler and more straightforward way...

2005-07-01 16:17:08 UTC
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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...