Locally Weyl invariant massless bosonic and fermionic spin-1/2 action in the (Y-4, g) space-time

We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of c ontortion and non-metricity tensor. In the presence of the non-metricity te nsor the investigation is extended to (W-n,g) space-time while when the tor sion is encountered we are restricted to the (U-4, g) space-time. Our resul ts hold in general for the (Y-4, g) space-time and we also calculate extra contributions to the conformal gravity.