Einstein-Cartan theory ($(e^a,{\omega }^{ab})$)

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Solve the algebraic connection equation for $\hat{\omega }(e,\chi )$, but perform the remaining variations as though the connection were independent. The chain-rule contribution $(\delta S/\delta \omega )\delta \hat{\omega }$ vanishes on that solution. The resulting equations are those of the connection-eliminated second-order action. This is especially useful for local supersymmetry calculations, where the gravitino-induced contorsion is quadratic in fermions and the omitted induced connection variation would otherwise be lengthy.

Use $K_{abc}=e_a{}^{\mu }(\omega -\omega (e){)}_{\mu bc}$ and $T_{abc}=K_{bac}-K_{cab}$. For the symmetrized Dirac action $-e\bar{\chi }{\gamma }^{\mu }{\overleftrightarrow{\nabla }}_{\mu }\chi /2-em\bar{\chi }\chi$, varying the spin connection gives $-e\bar{\chi }{\gamma }^{\mu ab}\chi \delta {\omega }_{\mu ab}/4$. Only totally antisymmetric contorsion is sourced. Its action is $-e{K}^2/(2{\kappa }^2)-e{K}_{abc}\bar{\chi }{\gamma }^{abc}\chi /4$, hence the stated solution. Substitution generates $e{\kappa }^2(\bar{\chi }{\gamma }^{abc}\chi )(\bar{\chi }{\gamma }_{abc}\chi )/32$. The torsion is an auxiliary field, not an independent propagating degree of freedom.