Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted

We discuss the Hopf-Wess-Zumino term in the effective action of the 6d (2, 0) theory of the type AN−1 in a generic Coulomb branch. For such terms, the supergravity calculation could be trusted. We calculate the WZ term on supergravity side and show that it could compensate the anomaly deficit, as is required by the anomaly matching condition. In contrast with the SYM theory, in which each WZ term involves one root ei − ej, here, the typical WZ term involves two roots ei − ej and ek − ej. Such kind of triple interaction may come from the integrating out of the massive states carrying three indices. A natural candidate is the recently proposed 1/4 BPS objects in the Coulomb phase of the 6d (2, 0) theories. The WZ term could be derived from the field theory by the integration out of massive degrees of freedom. Without the 6d (2, 0) theory at hand, we take the supersymmetric equations for the 3-algebra valued (2, 0) tensor multiplet as the prototype to see how far we can go. The H3 ∧ A3 part of the WZ term is obtained, while the A3 ∧ F4 part, which is the term accounting for the anomaly matching, cannot be produced by the standard fermion loop integration.