  Unimodular gravity is an interesting approach to address the cosmological
constant problem, since the vacuum energy density of quantum fields does not
gravitate in this framework, and the cosmological constant appears as an
integration constant. These features arise as a consequence of considering a
constrained volume element 4-form that breaks the diffeomorphisms invariance
down to volume preserving diffeomorphisms. In this work, the first-order
formulation of unimodular gravity is presented by considering the spin density
of matter fields as a source of spacetime torsion. Even though the most general
matter Lagrangian allowed by the symmetries is considered, dynamical
restrictions arise on their functional dependence. The field equations are
obtained and the conservation laws associated with the symmetries are derived.
It is found that, analogous to torsion-free unimodular gravity, the field
equation for the vierbein is traceless; nevertheless, torsion is algebraically
related to the spin density as in standard Einstein-Cartan theory. The
particular example of massless Dirac spinors is studied, and comparisons with
standard Einstein-Cartan theory are shown.
