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Suppose the simple theory with chiral fermions possessing non-trivial gauge anomalies cancellation: $$ S = \int d^4 x \big(\bar{\psi}i\gamma_{\mu}D^{\mu}_{\psi}\psi + \bar{\kappa}i\gamma_{\mu}D^{\mu}_{\kappa}\kappa\big), $$ where $$ D^{\mu}_{\psi} = \partial^{\mu} - iA^{\mu}_{L}P_{L} -iA^{\mu}_{R}P_{R}, \quad D^{\mu}_{\kappa} = \partial^{\mu}+iA^{\mu}_{L}P_{L} +iA^{\mu}_{R}P_{R} $$ Although separately $\psi, \kappa$ sectors are anomalous, together their gauge anomalies are cancelled: $$ \partial_{\mu}J^{\mu}_{L/R,\psi, \kappa} = \pm \frac{1}{96\pi^2}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}^{L/R}F_{\alpha\beta}^{L/R}, \quad \partial_{\mu}(J^{\mu}_{L/R,\psi}-J^{\mu}_{L/R,\kappa}) = 0 $$ Lets generate the mass for $\kappa$ fermion (by using spontaneous symmetry breaking with higgs singlet $fe^{i\varphi}$ with infinite mass for $f$) and integrate it out in the limit $m_{\kappa}\to \infty$. Corresponding effective field theory has to be free from anomalies, so there must be (possibly non-local) term $\Gamma[A_{L}, A_{R},\varphi ]$ which reproduces the anomalous structure of $\kappa$ sector; I can call it the Wess-Zumino term. It is possible to write its explicit form, and it turns out that this form is local (a polynomial in $A, \varphi$ and their derivatives)!

However, as I know, the anomaly is the local expression given by the variation of the non-local action. So where the non-locality is hidden?

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  • $\begingroup$ Your question looks too broad and open-ended to have a cogent answer here. Your basic conceit about the interchangeability of chiral fermions with WZW effective terms is standard practice in the industry. We demonstrate chiral delocalization and the bulk-boundary interplay in our Hill & Zachos AnnPhys 323 (2008) 3065. $\endgroup$ Commented Dec 15, 2016 at 14:41
  • $\begingroup$ ...and also Hill & Zachos PhysRev D71 (2005) 046002. $\endgroup$ Commented Dec 15, 2016 at 14:42
  • $\begingroup$ Above Ann Phys Link that works. $\endgroup$ Commented Dec 16, 2016 at 18:50
  • $\begingroup$ @CosmasZachos : so, You mean that above model with anomalies cancellation by generating the local term can be thought as the 5-d model, in which the anomaly cancellation is, of course, non-local? $\endgroup$
    – Name YYY
    Commented Dec 22, 2016 at 9:53
  • $\begingroup$ You might see it that way... An interplay of the boundary and the bulk.... Anything topological always has a whiff of higher dimensions, by dint of the generalized Stokes' theorem... Strict separation of locality and non locality may be subjective, unless precisely defined... $\endgroup$ Commented Dec 22, 2016 at 14:39

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