The role of torsion and a scalar field in gravitation, especially, in the presence of a Dirac field in the background of a particular class of the Riemann-Cartan geometry is considered here. Recently, a Lagrangian density with Lagrange multipliers has been proposed, which has been obtained by picking some particular terms from the SO (4, 1) Pontryagin density, where the scalar field causes the de Sitter connection to have the proper dimension of a gauge field. In this formalism, conserved axial vector matter current can be constructed, irrespective of any gauge choice, in any space-time manifold having arbitrary background geometry. This current is not a Noether current.

It is a remarkable result of differential geometry that certain global features of a spacetime manifold are determined by some local invariant densities. These topological invariants have an important property in common. They are total divergences and in any local theory, these invariants, when treated as Lagrangian densities, contribute nothing to the Euler-Lagrange equations.

Hence in a local theory only a few parts, not the whole part, of these invariants can be kept in a Lagrangian density. A gravitational Lagrangian was proposed in this direction sometime ago (Mahato, 2002), where a Lorentz invariant part of the de Sitter Pontryagin density, i.e., having broken de Sitter symmetry, was treated as the Einstein-Hilbert Lagrangian. In this way the role of torsion in the underlying spacetime manifold has become multiplicative rather than additive one and the Lagrangian looks like torsion curvature. In other words—the additive torsion is decoupled from the theory but not the multiplicative one. This indicates that torsion is uniformly nonzero everywhere. In the geometrical sense, this implies that micro local spacetime is such that, at every point there is a direction vector (vortex line) attached to it. This effectively corresponds to the noncommutative geometry having the manifold M₄ x Z₂, where the discrete space Z₂ is just not the two point space (Connes, 1994) but appears as an attached direction vector. This has direct relevance in the quantization of a fermion where the discrete space appears as the internal space of a particle (Ghosh and Bandyopadhyay, 2000).

Considering torsion and torsion-less connections as independent fields (Mahato, 2004), it has been found that K of Einstein-Hilbert Lagrangian appears as an integration constant in such a way that it has been found to be linked with the topological Nieh-Yan density of U₄ space. If we consider axial vector torsion, together with a scalar field connected to a local scale factor (Mahato, 2005 and 2007a), then the Euler-Lagrange equations not only give the constancy of the gravitational constant, but they also the link, in laboratory scale, the mass of the scalar field with the Nieh-Yan density and, in cosmic scale of FRW-cosmology, they predict only three kinds of the phenomenological energy density representing mass, radiation and cosmological constant. Recently, it has been shown that (Mahato, 2007b) using field equations of all fields except the frame field, the starting Lagrangian reduces to a generic f (R) gravity Lagrangian which, for FRW metric, gives standard FRW cosmology.

Physics Journal, Nieh-Yan Density, Torsion, Axial Current, Anomaly, Lagrangian Density, Sitter Symmetry, Einstein-Hilbert Lagrangian, Euler-Lagrange Equations, Pontryagin Density, Riemann-Cartan Geometry, Dimensionless Coupling , Geroch's Theorem, Dirac Field.