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1

Of course an anomalous global symmetry destroys the associated Ward identity, but...we don't care so much about that. The Ward identity of global symmetries is not needed for consistency of the theory. However, a broken local Ward identity completely destroys the associated gauge theory, in particular since the decoupling of the unphysical degrees of freedom ...


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In fact there are two different meaning for the term "global anomaly", which is a pity: global anomaly as opposed to gauge anomaly; global gauge anomaly as opposed to local gauge anomaly. An anomaly can arise from global and gauge symmetries. So here global refers to the fact that the symmetry group is not gauged: these symmetries have a physical ...


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You now want to go back to position space. I'll do that here very schematically which gives the answer without keeping track of the overall factor. Essentially, under the Fourier transform, $q_{i\mu} \to \partial_\mu$ and $\epsilon_{i\mu} \to A_\mu$. Then $$ g^2 \text{Tr}(T^aT^b) \varepsilon_{\mu\nu\rho\sigma} q_1^\mu q_2^\nu \epsilon_1^\rho ...


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The axial (also called "chiral") current is the Noether current for the axial transformation $$ \psi(x) \mapsto \mathrm{e}^{\mathrm{i}\alpha \gamma^5}\psi(x),\alpha\in\mathbb{R}$$ of a massless fermionic theory with the usual Dirac action $$ S_D[\psi] = \int \bar\psi(x)\mathrm{i}D_\mu\gamma^\mu\psi(x)\mathrm{d}^d x$$ so it is classically non-conserved only ...



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