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I'm trying to understand why there are theoretical models predicting that spin and gravity should couple.

Specifically, what I'm asking is if such Standard Model extension comes from gauging the Poincaré group as per the Cartan variant of GR, where additionally torsion is sourced by spin instead of just energy and momentum.

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Gauge field of local supersymmetry (=supergravity) - gravitino - leads to non-trivial torsion. Gravitino is a spin-$\frac{3}{2}$ field satisfying Rarita-Schwinger equation.

Update.

No, there is no violation of equivalence principle, since bosonic components of torsion still vanish. Whereas fermionic components give gravitino-matter coupling. See this answer for gravitino contribution https://physics.stackexchange.com/a/134689/85745

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  • $\begingroup$ Can it be seen somehow as a 'yes' answer, if we assume a first order approach to supergravity, reformulated as an Einstein-Cartan theory of gravity coupled to spin-$\frac{3}{2}$ massless Rarita-Schwinger fields? $\endgroup$
    – Xlsx2020
    Jun 1 '17 at 13:21
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    $\begingroup$ supergravity automatically assumes presence of fermions (gravitino in pure supergravity), that's the point of supersymmetry. And because of fermions, first-order formulation must be used. $\endgroup$
    – Kosm
    Jun 1 '17 at 13:35
  • $\begingroup$ well, you can technically add fermions without using vielbeins, but you will have to use spin base group, spin metric, and that kind of stuff. $\endgroup$
    – Kosm
    Jun 1 '17 at 13:44
  • $\begingroup$ Let me double check: does the fact that in supergravity (with gravitino) the non-vanishing torsion (see this answer ) implies a violation of the Equivalence Principle? See another answer saying that Einstein-Cartan theory does not obey the Equivalence Principle as "spin" components could result in a different measure in a gravitational field. Notice that recent experiments follow this test approach indeed $\endgroup$
    – Xlsx2020
    Jun 5 '17 at 0:05
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    $\begingroup$ No, there is no violation of equivalence principle, since bosonic components of torsion still vanish. Whereas fermionic components give gravitino-matter coupling. See this answer for gravitino contribution physics.stackexchange.com/a/134689/85745 $\endgroup$
    – Kosm
    Jun 5 '17 at 2:09

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