The main issue of this Master Thesis is to explore the Adler-Bardeen theorem in the presence of external fields, which states that gauge anomalies vanish to all orders if they vanish at one-loop, while the anomalies that are quadratic or cubic in the external fields are one-loop exact quantities. The first virtue of the Adler-Bardeen theorem is that it allows us to prove that a theory is gauge-anomaly free to all orders, using only the condition that anomalies vanish at one-loop. If this theorem did not hold, one would have to prove the anomaly cancellation order by order in perturbation theory. The Standard Model is the most important example of theory that is free of gauge anomalies to all orders. The second virtue of the theorem is that it provides the simplest examples of quantities that can be calculated exactly to all orders.
We prove the theorem in the framework of perturbative Quantum Field Theory making use of the renormalization properties of gauge theories, employing the languange of the Batalin-Vilkovisky formalism.