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It is said that In theories such as Supergravity where there are fermions coupled to gravity, one must use an auxiliary quantity, the frame field (vielbein).

  1. In supergravity, can a boson be coupled to gravity? If not, then why?

  2. Why is it that because fermions couple to gravity, then vielbein must be used?

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(1) Of course bosons can be coupled to gravity. The graviton is a spin 2 field.

(2) We want to use vierbien to couple fermions to gravity because the Dirac equation is formulated in Minkowski space and we want the behavior of fermions in a general spacetime to be locally like their behavior in Minkowski space. Therefore we use the local frame fields/tetrads/vielbien $e_I^{\mu}(x)$ to write the Dirac equation as

$$ (i \gamma^I e_I^{\mu} D_{\mu} - m)\Psi = 0$$

Of course if we use the constant coordinate $e_I^{\mu}$ this reduces to the usual Dirac equation.

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  • $\begingroup$ The difficulty with fermions was that they have a more complicated spinor structure so we need tetrads to write an action/equation of motion for the fermion field. With bosons on the other hand there is no such difficulty. That previous statement was just saying that the gravitational field is itself bosonic, though that's probably not what you were asking. A quick search of the literature will tell you that there are various techniques for embedding the Standard Model gauge bosons in SUGRA. $\endgroup$ – Jordan May 22 '15 at 7:44

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