Physics Letters B 307 ( 1993 ) 77-82 North-Holland PHYSICS LETTERS B On the BRST charge for topological BF theories M.I. Caicedo and A. Restuccia Departarnento de Fisica, UmversidadStm6n Bolivar, Apartadopostal 89000, Caracas 1080-A, Venezuela Received 9 February 1993 Editor: M. Dine Topological quantum field theories are discussed using the generalized canonical approach to constrained systems. We give the general structure of the canomcal effect:re action of hnearly dependent first class constrained systems When this construction is apphed to metric independent classical actions, one is able to ensure - under some exphcltly given boundary conditions - metric independence of the associated partltmn function. The quantum analysis of the non-abehan BF theories in four dimensions is performed on these hues, their off-shellmlpotent BRST charge :s constructed and a new direct proof of the topological nature of the associated quantum theories :s presented. We do also comment on some properties of the BF actmns in higher dimensions. BF topological actions, which may be considered as a generalization of the three dimensional Chern- Simons theory to higher dimensions, were intro- duced in [ 1,2] where they were analysed from the canonical and covariant point of view. A complete analysis of the abelian BF theories was presented in [ 3 ], where it was shown that the partition function is a topological invarlant related [4] to the Ray- Singer torsion [ 5 ]. Moreover, it was also proven that other observables, namely the Wilson "surfaces" de- termine linking and intersection numbers of mani- folds in any dimension. The problem of constructing higher-dimensional link invariants in the framework of non-abelian BF theory was addressed in [ 6 ]. The metric independence of quantum non-abelian BF theory was first proposed in [ 3 ] and proven later on in [ 7-9 ] in a more direct way. A solution of the master equation of the Batalin-Vllkovxski approach for the BF action was previously presented in [ 10]. In all these approaches the general program of Ba- talin-Vilkoviski [ 11 ] to linearly dependent con- strained systems was used. Here we analyse the problem of the metric inde- pendence of the partition function of TQFTs using the generalized canonical approach to linearly de- pendent constrained systems first developed by Ba- talin and Fradkin [ 12 ]. By using this approach, we construct the off-shell nilpotent BRST charge and the Off shell BRST invariant effective action for the four dimensional non abelian BF theory. Such results have not been obtained in any of the references given above. The formalism we are going to use [ 13,14 ] has the same minimal sector of the Batalin-Fradkin ap- proach [ 12 ] but a different ghosts-antighosts and Lagrange multipliers structure in the non-minimal sector. The fields in the minimal sector are the only ones that we regard as canonical, this allows the usage of the covariant gauge fixing conditions in a straight- forward way. The distinction is important for infi- nite reducible systems where infinite sets of ghosts, antighosts and Lagrange multipliers have to be orga- nized into generating functions in order to allow a correct treatment of the effective action. In [ 13 ] we define the admissibility conditions for the gauge fix- ing functions at any level of reducibility and show the off-shell reduction of the functional integral to the in- dependent physical degrees of the freedom (which was not presented in [ 12 ]. For the sake of completeness we first present the BRST invariant construction of the effective action of linearly constrained systems by using the general- ized canonical approach. We then discuss general conditions under which the partition function is met- ric independent. After the discussion of these results, we apply the formalism to the non-abelian BF theory Elsev:er Science Pubhshers B.V. 77