On p.249 of Freedman and Van Proeyen's Supergravity, the following is stated:

"Given the action of a supergravity theory, it is generally useful to search for solutions of the classical equations of motion. It is most useful to obtain solutions that can be interpreted as backgrounds or vacua. Fluctuations above the background are then treated quantum mechanically. The backgrounds that are considered have vanishing values of fermions, and are thus determined by a value of the metric, the vector fields (or higher forms) and scalar fields."

My question is: why do the fermionic fields have to vanish for solutions to a supergravity theory that preserve some (or all) supersymmetry?

  • $\begingroup$ The way your question is written is at least two, if not three, completely logically separated questions. You are firstly asking why you may not have non vanishing background "classical" fermions. You are then wondering how supersymmetry is realized in vacuum modes with non vanishing boson backgrounds but not fermion ones. $\endgroup$ – Cosmas Zachos Mar 31 at 20:33
  • $\begingroup$ Near Duplicate. $\endgroup$ – Cosmas Zachos Mar 31 at 20:38
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    $\begingroup$ @CosmasZachos ??? I wonder how do you read that. The question is clear and is quite specific $\endgroup$ – OON Mar 31 at 21:38
  • $\begingroup$ @OON Nontrivial background fermion solutions are excluded by Lorentz invariance, with or without supersymmetry. The issue of whether supersymmetry is realized in the Wigner-Weyl or Nambu-Goldstone mode is a separate one. The duplicate question answer parses it out more succinctly. $\endgroup$ – Cosmas Zachos Apr 1 at 0:41
  • $\begingroup$ @CosmasZachos in the "duplicate answer", it just states that "We set all the fermions to zero--they are Grassmann valued and must vanish in a classical configuration." Nothing more. I am essentially asking why this is the case. [Moreover, the "duplicate question" refers to the vanishing of the variation of the gravitino, not the vanishing of the gravitino. If I have misunderstood something, I am sorry.] $\endgroup$ – LORENTZo_lamas Apr 1 at 0:49

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