How to make an action supersymmetric?

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Discussion Overview

The discussion revolves around the process of making a given action supersymmetric, particularly focusing on actions involving scalar fields. Participants explore the necessary conditions and methods for incorporating fermionic terms alongside bosonic ones to achieve supersymmetry.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about simply adding fermionic terms to a bosonic action to achieve supersymmetry.
  • Another participant asserts that there are restrictions on bosonic interactions, indicating that not all bosonic actions can be supersymmetrized by merely adding fermions.
  • A participant suggests that for spin-0 scalar fields, one would typically add a spin-1/2 spinor field to create a Wess-Zumino supersymmetric field, noting the requirement for equal degrees of freedom between bosonic and fermionic fields.
  • It is mentioned that the supersymmetrization of certain scalar theories, like a non-linear sigma model, is contingent on specific geometric conditions, such as the manifold being a Kaehler manifold.
  • Another participant describes a traditional method for achieving supersymmetry by applying supersymmetry variations to the fields and adjusting the action accordingly, highlighting the iterative and complex nature of the process.
  • Areas of Agreement / Disagreement

    Participants express differing views on the feasibility of supersymmetrizing any given bosonic action, with some asserting that restrictions exist while others propose methods for achieving supersymmetry.

    Contextual Notes

    Participants note that the process of supersymmetrization can be intricate and may depend on specific properties of the fields and the underlying geometry of the models involved.

yola
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Hello,

i am given an action (only scalars) and i want to make it supersymmetric. Any hints?
Do i simply add fermionic terms similar to those of the bosonic?

Thanks
 
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No, in general there are restrictions on the bosonic interactions and you cannot supersymmetrize any given bosonic action by adding fermions. It is in general a non-trivial exercise.
 
For spin-0 scalar fields, one would add a spin-1/2 spinor field to each one to make a Wess-Zumino supersymmetric field (superfield). Since the on-shell numbers of bosonic and fermionic degrees of freedom must be equal, the scalar field must be a complex one.

Introductions to supersymmetry usually get into Wess-Zumino fields rather quickly, so it should not be much trouble to discover how to add superpartner fields.
 
Ineed not only the supermultiplet structure must work, but as said, there are restrictions on the bosonic interactions; not all scalar theories can be supersymmetrized. For example, a bosonic non-linear sigma model on some manifold M can only be supersymmetrized if M is a Kaehler manifold.
 
The way this was traditionally done is by writing down the supersymmetry variations of all of your fields. Then see what happens to the action when you apply these SUSY variations. It will spit out some extra terms. So then you adjust your action and SUSY variations by adding more terms. Repeat this process until you don't have to add any more terms.

The whole thing is quite tedious, but that's how 11-dimensional SUGRA was found.
 
Thanks
 

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