Densitized unimodular formulation with torsion

Determine whether the local Weyl sector of the densitized unimodular formulation factors out as a pure gauge mode in Riemann-Cartan geometry when torsion variables and the functional measure are treated consistently, and characterize the resulting equivalence with the constrained unimodular formulation.

Background

The paper identifies the densitized formulation of unimodular gravity as a distinct extension. In this formulation, an auxiliary metric carries a local Weyl redundancy, while the physical metric remains unimodular.

For a Riemann-Cartan theory, the unresolved issue is how this Weyl redundancy interacts with torsion and with the functional measure. If torsion is Weyl invariant, the authors expect equivalence after accounting for the Weyl gauge volume and Jacobians; if the torsion trace transforms under Weyl rescalings, additional mixing and ghost determinants may appear. The paper does not settle which scenario is realized.

References

In a Riemann-Cartan setting, the relevant question is whether this Weyl sector factors out as a pure gauge mode once the torsion variables and the functional measure are treated consistently. If torsion is chosen to be Weyl invariant, one expects the densitized and constrained unimodular formulations to be equivalent after the Weyl gauge volume and Jacobians are included. If, instead, the torsion trace participates in the Weyl transformation, additional mixing and ghost determinants may arise. A separate analysis is required to settle this point.

— Towards Perturbative Unimodular Poincaré Gauge Theories with Propagating Torsion  (2608.22669 - Vieira et al., 24 Aug 2026) in Section 4, Discussions and Conclusions, paragraph beginning “Another natural extension is the densitized formulation of unimodular gravity”

The special coupling relations that remove ghost or tachyonic excitations at the classical level are not generically expected to define renormalization-group-invariant subspaces unless they are protected by additional symmetries or dynamical mechanisms. A calculation resolving the complete dimension-four operator basis would therefore determine whether such classically healthy parameter subspaces are preserved by quantum corrections in either the Diff-invariant or unimodular formulation.

— Towards Perturbative Unimodular Poincaré Gauge Theories with Propagating Torsion  (2608.22669 - Vieira et al., 24 Aug 2026) in Section 4, Discussions and Conclusions, paragraph beginning “A further direction is to enlarge the class of (extended) PGT actions”