Becchi-Rouet-Stora-Tyutin structure for the mixed Weyl-diffeomorphism residual symmetry

We use the denomination introduced in Ref. 17.

Global aspects require careful consideration which might lead one to look for a better adapted geometrical framework.

As stated in Ref. 43, the Lie algebra of diffeomorphisms is anti-isomorphic to the Lie algebra of vector fields. This explains why the factor 1/2 occurs without a minus sign. Thus upon substituting ξ by −ξ one recovers variations obtained in Ref. 6.

This means that G′ is to be suitable for this definition, in particular, it shares the same representations as H (at least the adjoint representation and ρ).

To some extent, H′ is identified to be a subgroup of H along which the gauge invariance can be restored. One may also check that the complement $H∖H′$ is stable under H.

Likewise, performing the same substitution starting from second presentation (2.15) of BRST^ξ would result in an algebra for the composite fields formally identical to (2.15) but with pure gauge ghost (u⁻¹vu + u⁻¹σu) − u⁻¹L_ξu.

The reader is referred to Ref. 17, (Chap.12).

Indeed this does not depend on the expression of the shifted ghost.

Due to the successive dressings, the “hat” symbol is dropped out to the benefit of a lower index.