quantum field theory - Supersymmetric background and fermion variations

I'm trying to understand some basic questions about supersymmetric theories in curved backgrounds and supergravity. If I understand it correctly, there's a condition for a background to preserve SUSY, namely the gravitino SUSY variations should vanish. This seems to come from demanding that the vacuum $|0\rangle$ is anihilated by the supercharge $Q$, hence:

$$\langle \delta_Q \Psi_\mu \rangle \equiv \langle \{Q, \Psi_\mu\} \rangle = 0$$ where $\Psi_\mu$ denotes the gravitino.

My questions:

1. 
   
   

   Is this correct?

   
   


2. 
   

   If in addition to the gravitino I have further fermionic background fields, should their SUSY variations vanish as well? For example, if in my SUSY theory I promote some couplings $\lambda$ to background superfields $\Lambda(x)$, should I require that the SUSY variations of the fermionic component, say $\chi(x)$, of $\Lambda(x)$ vanish, i.e.

   $$\delta_Q \chi = 0?$$
   
   


3. 
   

   In a Lorentz invariant background, the gravitino should have a zero vacuum expectation value, since it's not a Lorentz scalar. Does it mean that the gravitino is zero in the background? I'm actually also confused about the metric -- the vev of the metric should be zero, but we usually take it to be nonzero in the vacuum. Maybe my understanding here is completely wrong.

   
   

Answer

Let me first describe the basic idea without even mentioning supergravity. Consider some classical field theory of two fields $\phi$ and $\sigma$, with an action $S[\phi,\sigma]$. Suppose this theory has a continuous symmetry (with parameter $\epsilon$):

$$S[\phi,\sigma]=S[\phi+\delta_\epsilon\phi,\sigma+\delta_\epsilon\sigma].$$ Say I find some particular field configuration $\sigma=\sigma_0$ which is invariant under the symmetry action: $\delta_\epsilon \sigma_0 = 0$. Then the field theory of $\phi$ alone with the action $\tilde S[\phi] := S[\phi,\sigma=\sigma_0]$ obtained by freezing $\sigma$ to the background configuration $\sigma_0$ will be invariant: $$\tilde S[\phi] = \tilde S[\phi+\delta_\epsilon \phi].$$ Now let's get to supersymmetry. I have some supersymmetric field theory in flat space $\mathbb{R}^d$, and I want to put the theory on a curved manifold $M$ (like $S^d$, $S^{d-1}\times S^1$, or whatever). In other words, I want to deform my flat-space theory by coupling it to some background metric $g_{\mu\nu}$ for the geometry $M$. First of all, note that this procedure is ambiguous; I can always add terms (e.g. proportional to the Ricci scalar of $g_{\mu\nu}$) to the action of the curved-space theory which vanish in the flat space limit. The idea is to find $\textit{some}$ action in curved space which reduces to our original theory in the flat space limit and preserves some amount of supersymmetry in curved space.

Our original theory has a bunch of fields transforming in supersymmetry multiplets. We want to couple this theory to the background metric $g_{\mu\nu}$. But doing so naively will just break all the supersymmetry. That's not surprising; $g_{\mu\nu}$ couples to the stress-tensor $T_{\mu\nu}$. But in a supersymmetric theory the stress-tensor belongs to a whole supersymmetry multiplet of currents including the supersymmetry current, any R-symmetry currents, etc. So if we want to preserve supersymmetry in the curved background, we likely need to turn on background sources for the other currents, such as a background R gauge field. In other words, we will couple the theory to a whole background supergravity multiplet.

Comparing with our initial example, $\phi$ played the role of all the dynamical fields in the flat-space theory, and $\sigma$ corresponds to the supergravity multiplet. The idea then is to start with a theory of supergravity (played by $S[\phi,\sigma]$) in which $\textit{all}$ the fields are dynamical and transform under supersymmetry, freeze the supergravity fields into background configurations (including the desired metric) which are invariant under some subset of the supersymmetries (this is the $\sigma\to\sigma_0$ step), and thereby obtain a theory ($\tilde S[\phi]$) on the curved manifold $M$ which is invariant under some global supersymmetries acting only on the dynamical fields "$\phi$."

The dynamical supergravity multiplet will consist of bosonic fields like the metric $g_{\mu\nu}$ and R gauge field $A_\mu$, as well as fermionic fields like the gravitino $\psi_\mu$. Our job is to choose these fields such that their supersymmetry variations vanish. Of course, we want to freeze the dynamical metric to be the desired metric of $M$. We set all the fermions to zero--they are Grassmann valued and must vanish in a classical configuration. The supersymmetry variations of the bosonic fields are proportional to the fermions and therefore vanish. So the only constraints come from the fermion variations. In particular, the gravitino variation will take the schematic form

$$\delta_\epsilon \psi_\mu = (\nabla_\mu-A_\mu)\epsilon.$$ (There can be additional terms on the RHS coming from other fields, but let's focus on the R gauge field for clarity). We choose an appropriate background value for $A_\mu$ such that the equation $\nabla_\mu \epsilon = A_\mu \epsilon$ has some solutions. The $\epsilon$'s which solve this equation are called Killing spinors, and they determine the preserved supersymmetries of the curved-space theory (if any).

This procedure for obtaining supersymmetric theories on curved manifolds was first explored systematically here. Appendix B of this paper contains a nice review which might be helpful. I also learned a lot from this talk by Thomas Dumitrescu.