  We find a large internal symmetry within 4-dimensional Poincare gauge theory.
In the Riemann-Cartan geometry of Poincare gauge theory the field equation and
geodesics are invariant under projective transformation, just as in affine
geometry. However, in the Riemann-Cartan case the torsion and nonmetricity
tensors change. By generalizing the Riemann-Cartan geometry to allow both
torsion and nonmetricity while maintaining local Lorentz symmetry the
difference of the antisymmetric part of the nonmetricity Q and the torsion T is
a projectively invariant linear combination $S = T - Q$ with the same symmetry
as torsion. The structure equations may be written entirely in terms of S and
the corresponding Riemann-Cartan curvature. The new description of the geometry
has manifest projective and Lorentz symmetries, and vanishing nonmetricity.
Torsion, S and Q lie in the vector space of vector-valued 2-forms. Within the
extended geometry we define rotations with axis in the direction of S. These
rotate both torsion and nonmetricity while leaving S invariant. In n dimensions
and (p, q) signature this gives a large internal symmetry. The four dimensional
case acquires SO(11,9) or Spin(11,9) internal symmetry, sufficient for the
Standard Model. The most general action up to linearity in second derivatives
of the solder form now includes combinations quadratic in torsion and
nonmetricity, torsion-nonmetricity couplings, and the Einstein-Hilbert action.
Imposing projective invariance reduces this to dependence on S and curvature
alone. The new internal symmetry decouples from gravity in agreement with the
Coleman-Mandula theorem.
