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I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism.

As a motivation to learn more about frame-fields would someone sketch me why this is necessary? If possible give nice references.

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The Lorentz group $O(3,1)$ has spinor representations (actually $SL(2,\mathbb C)$, that is the universal cover of $O(3,1)$), as well known. The problem is that now, in general relativity, we want to deal with generic transformations. So we are working with G$L(4)$.

Roughly speaking, the associated Lie Algebra $\mathfrak{gl}(4)$ doesn't admit spinor representations. See for instance: No Spinor The way out is to go to a local inertial frame, in which you recover the flat space and you can define the spinors.

For supergravity I can suggest to you D. Freedman and A. V. Proeyen (Supergravity, 2012)

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