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Two question on N=4 Super-Yang-Mills:

1) As far as I understand, the attempt to extend the usual N=1

superspace to a superspace that corresponds to extended (N=2, 4)

supersymmetry meets the difficulty that additional bosonic coordinates

appear, corresponding to central charges. This means we have an

infinite set of auxilary fields in the theory. This complication may in

certain cases be overcomed in the N=2 case but not in the N=4 case. In

fact, we do have the so-called "analytic superspace" which allows

bypassing the problem for N=4 but it only works on-shell.

A) What does it mean to have an "on-shell" formulation of the theory?

That we can describe observables, but we cannot write down a

path-integral? What is the "on-shell" formulation of N=4 SYM in

analytic superspace?

B) Nevertheless, can we write a path-integral for N=4 SYM in extended

superspace (with the infinite set of auxilary fields included)?

2) The quantum potential for the scalar fields in N=4 SYM has

non-renormalization properties due to supersymmetry. In particular,

there is a non-trivial moduli space of vacua (with dimension 6 times

the rank of the gauge group) corresponding to the vanishing of the

quantum potential. In a generic sector the gauge group is broken down

to a maximal Abelian subgroup (the so-called Coloumb phase). All the

fields of the theory are in the adjoint representation. In

hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb

phase theory is in fact _free_ (since the adjoint representation is

trivial for an Abelian group). However, what about the interactions

associated with massive gauge bosons?

Thx in advance for any help!

Best regards,

Squark

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# N=4 SYM: Extended Superspace, Coloumb Phase

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