Becchi-Rouet-Stora-Tyutin structure for the mixed Weyl-diffeomorphism residual symmetry (Q2798700)

In this paper the authors study the relation and compatibility between the shifting and dressing methods for constructing extensions of the BRST algebra of pure gauge transformations (i.e., symmetries in pure Yang-Mills theory) to a similar algebra of gauge and diffeomorphism transformations (i.e. symmetries in Yang-Mills on a dynamical gravitational background).NEWLINENEWLINERevealing the similarities and differences between classical Yang-Mills theory and classical general relativity theory have been the target of intense research through decades. At first sight both theories share similar concepts: the most important of these is the common principle that these theories are subject to local gauge transformations. However more thorough investigations often discover subtle differences in the manifestations of these concepts: for example the ``entering level'' of local gauge symmetries in these theories is different.NEWLINENEWLINEAn insightful treatment of local gauge symmetry in pure Yang-Mills theory is the celebrated BRST approach which is based upon introducing a graded differential algebra generated by the original Yang--Mills field, its field strength, the matter fields (possibly present and coupled to the pure Yang--Mills theory) and an artificial object, the so-called Faddeev-Popov ghost field. Apart from this, the gauge group of general relativity is the diffeomorphism group of the underlying manifold hence its gauge algebra is generated by vector fields. The natural question therefore arises how to extend the original pure Yang-Mills BRST algebra in order to contain vector fields among its generators, too. Two methods are known: the so-called shifting method and the so-called dressing method (summarized in Sections II and III in the paper). Both are based upon including vector fields by suitable modifications (shifting and dressing, respectively) of the original Faddeev-Popov ghost field. However the relations of the resulting extended BRST algebras are not obviously clear. In this paper the authors combine together these methods by introducing \textit{shifted dressed ghosts} and then derive necessary conditions for the two methods to be compatible i.e., to commute with each other when applied (see Section III of the paper).NEWLINENEWLINEFinally (see Section IV of the paper) the authors illustrate their construction on two examples: the first is classical general relativity (in which they combine together fiberwise Lorentz symmetry with diffeomorphisms) and the second is the case of manifolds carrying so-called second order conformal structures (in which they combine together fiberwise Weyl conformal symmetries with diffeomorphisms).