Anomaly footprints in SM+Gravity

This is a follow-up of arXiv:2412.07470 [hep-th]. A simplified version of the SM plus gravity, put forward there, is presented here and some of its aspects delved into. The basic structure consists of two sectors, left and right, with chirally mirror fermions and scalars, as well as $SU(3)$ and $U(1)$ gauge fields, while the $SU(2)$ gauge fields as well as the metric are in common to both sectors. This structure is dictated by the request to cancel all dangerous anomalies. The left sector consists of the fermion, gauge and scalar fields of the SM, now minimally coupled to gravity. The right sector is a mirror image of the left, with distinct fields, except the metric and the $SU(2)$ gauge potentials. The first new aspect is the proposed and motivated interpretation of the right sector as the dark matter one. The second new subject covered here is Weyl symmetry and its possible application to cosmology and its theoretical fallout on unitarity and renormalization of the model. A background solution of the Weyl invariant theory is derived, which may apply to the very early stages of the universe. This solution also suggests interesting applications to the cosmological constant problem. On the quantum field theory side the subject of Weyl symmetry and Weyl anomalies is reviewed and, among other things, an application of the WZ terms is illustrated to the problem of one-loop quantization of the model which may avoid negative norm states.