On the realization of impossible anomalies

TL;DR

This paper investigates exotic anomalies permitted by the Wess-Zumino consistency condition in conformal field theories, demonstrating that some previously thought impossible anomalies can indeed exist through non-primary operators or semi-local terms.

Contribution

It reveals that certain exotic anomalies are realizable in conformal field theories via mechanisms involving non-primary operators or semi-local terms, challenging previous no-go assumptions.

Findings

Some anomalies are not primary operators, allowing their existence.

Semi-local terms can realize anomalies like the Green-Schwarz mechanism.

Impossible anomalies can be consistent with conformal invariance through these mechanisms.

Abstract

The Wess-Zumino consistency condition allows more exotic forms of anomalies than those we usually encounter. For example in two-dimensional conformal field theories in the curved background with space-time dependent coupling constant $\lambda^i(x)$, a $U(1)$ current could possess anomalous divergence of the form $D^\mu J_\mu = \tilde{c} R + \chi_{ij} \partial^\mu \lambda^i \partial_\mu\lambda_j + \tilde{\chi}_{ij} \epsilon^{\mu\nu} \partial_\mu \lambda^i \partial_\nu \lambda^j + \cdots $. Another example is the CP odd Pontryagin density in four-dimensional Weyl anomaly. We could, however, argue that they are impossible in conformal field theories because they cannot correspond to any (unregularized) conformally invariant correlation functions. We find that this no-go argument may be a red herring. We show that some of these impossible anomalies avoid the no-go argument because they are not primary operators, and the others circumvent it because they are realized as semi-local terms as is the case with the conformally invariant Green-Schwartz mechanism and in the higher dimensional analogue of Liouville or linear dilaton theory.