Comments on the Equivalence between Chern - Simons Theory and Topological Massive Yang - Mills Theory in 3 D

The classical formal equivalence upon a redefinition of the gauge connection between Chern-Simons theory and topological massive Yang-Mills theory in three-dimensional Euclidean spacetime is analyzed at the quantum level within the BRST formulation of the Equivalence Theorem. The parameter controlling the change in the gauge connection is the inverse λ of the topological mass. The BRST differential associated with the gauge connection redefinition is derived and the corresponding Slavnov-Taylor (ST) identities are proven to be anomaly-free. Hence they can be restored order by order in the loop expansion by a recursive choice of non-invariant counterterms. The Green functions of local operators constructed only from the (λ-dependent) transformed gauge connection, as well as those of BRST invariant operators, are shown to be independent of the parameter λ, as a consequence of the validity of the ST identities. The relevance of the antighost-ghost fields, needed to take into account at the quantum level the Jacobian of the change in the gauge connection, is analyzed. Their rôle in the identification of the physical states of the model within conventional perturbative gauge theory is discussed. It is shown that they prove to be essential in keeping the correspondence between the degrees of freedom of the theory at λ = 0 (Chern-Simons theory) and at λ 6= 0. Key-words: BRST quantization, BRST symmetry, Chern-Simons theory E-mail: [email protected]