Unitary, Anomalous Master Ward Identity and its Connections to the Wess–Zumino Condition, BV Formalism and \(L_\infty \)-algebras

The C*-algebraic construction of QFT by Buchholz and one of us relies on the causal structure of space-time and a classical Lagrangian. In one of our previous papers, we have introduced additional structure into this construction, namely an action of symmetries, which is related to fixing renormalization conditions. This action characterizes anomalies and satisfies a cocycle condition which is summarized in the unitary anomalous Master Ward identity. Here (using perturbation theory) we show how this cocycle condition is related to the Wess–Zumino consistency relation and the consistency relation for the anomaly in the BV formalism, where the latter follows from the generalized Jacobi identity for the associated \(L_\infty \)-algebra. In addition, we give a proof that perturbative agreement (i.e., independence of a perturbative QFT on the splitting of the Lagrangian into free and interacting parts) can be achieved by finite renormalizations.