The antibracket formalism for gauge theories, at both the classical and quantum level, is reviewed. Gauge transformations and the associated gauge structure are analyzed in detail. The basic concepts involved in the antibracket formalism are elucidated. Gauge-fixing, quantum effects, and anomalies within the field-antifield formalism are developed. The concepts, issues and constructions are illustrated using eight gauge-theory models.

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An algebraic program of computation and characterization of higher loop BRST anomalies is presented. We propose a procedure for disentangling a genuine local higher loop anomaly from the quantum dressings of lower loop anomalies. For such higher loop anomalies we derive a local consistency condition, which is the generalisation of the Wess-Zumino condition for the one-loop anomaly. The development is presented in the framework of the field-antifield formalism, making use of a nonlocal regularization method. The theoretical construction is exemplified by explicitly computing the two-loop anomaly of chiral W 3 gravity. We also give, for the first time, an explicit check of the local two-loop consistency condition that is associated with this anomaly. ♭

Quantization of anomalous gauge theories with closed, irreducible gauge algebra within the extended Field-Antifield formalism is further pursued. Using a Pauli-Villars (PV) regularization of the generating functional at one loop level, an alternative form for the anomaly is found which involves only the regulator. The analysis of this expression allows to conclude that recently found ghost number one cocycles with nontrivial antifield dependence can not appear in PV regularization. Afterwards, the extended Field-Antifield formalism is further completed by incorporating quantum effects of the extra variables, i.e., by explicitly taking into account the regularization of the extra sector. In this context, invariant PV regulators are constructed from non-invariant ones, leading to an alternative interpretation of the Wess-Zumino action as the local counterterm relating invariant and non-invariant regularizations. Finally, application of the above ideas to the bosonic string reproduces the well-known Liouville action and the shift (26 − D) → (25 − D) at one loop.

Anomalies and BRST invariance are governed, in the context of Lagrangian Batalin-Vilkovisky quantization, by the master equation, whose classical limit is (S, S) = 0. Using Zimmerman's normal products and the BPHZ renormalisation method, we obtain a corresponding local quantum operator equation, which is valid to all orders in perturbation theory. The formulation implies a calculational method for anomalies that is useful also outside this context and that remains completely within regularised perturbation theory. It makes no difference in principle whether the anomaly appears at one loop or at higher loops. The method is illustrated by computing the one-and two-loop anomalies in chiral W 3 gravity.