1. What are the distinctions between Non-extended SUSY and Extended SUSY?

(Is that just a non-extended SUSY has $\mathcal{N}=1$ while the extended SUSY has $\mathcal{N}>1$ ? Then there is no conceptual upgrade?)


  • If we start from a non-extended SUSY, how to obtain an extended SUSY?

  • If we start from an extended SUSY , how to obtain a non-extended SUSY?

(via SUSY breaking? what else?)

  • 2
    $\begingroup$ For 1, yes it is just a matter of $\mathcal{N}$ I believe. Regarding 2, it is probably not what you are interested in, but note that you can construct (maximally) extended $\mathcal{N}=4$ SUSY in $4$d by dimensional reduction from $\mathcal{N}=1$ SUSY in $10$d. $\endgroup$ – Jxx Oct 26 '20 at 21:15
  1. One can get extended SUSY by combining two supersymmetric multiplets into one with the appropriate choice of coupling constant. For example - in order to get $\mathcal{N} = 2$ SYM (Super Yang-Mills) one combines the gauge multiplet with gluon $A_\mu$ and gluino $\lambda_\alpha$ with the chiral multiplet with a fermion $\psi_\alpha$ and a scalar $\phi$, all fields in the adjoint representation of the gauge group. \begin{gather*} A_\mu \\ \lambda \qquad \psi \\ \phi \end{gather*}
  2. Yes, one can "downgrade" the SUSY by symmetry breaking - for example in the famous artice by E.Witten and N.Seiberg "Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory" https://arxiv.org/abs/hep-th/9407087 - SUSY is broken from $\mathcal{N} = 2$ to $\mathcal{N} = 1$ by the superpotential term : $$ W = m \text{Tr} \ \Phi^2 $$ And this breaking is crucial for the monopole condensation and the description of confinement. I am not aware, unfortunately, of the other mechanisms of getting a symmetry with less SUSY in other way than the symmetry breaking by introducing soft or hard SUSY breaking terms or spontaneous SUSY breaking.

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