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

TL;DR

This paper revisits the Hopf-Wess-Zumino term in the effective action of 6d (2,0) theories, calculating its form via supergravity and exploring its role in anomaly matching and interactions involving three roots.

Contribution

It provides a supergravity calculation of the WZ term in 6d (2,0) theories and discusses its implications for anomaly matching and the structure of interactions.

Findings

Supergravity calculation confirms the WZ term can compensate anomaly deficits.

The WZ term involves interactions with two roots, e_{i}-e_{j} and e_{k}-e_{j}.

The A_{3} F_{4} part of the WZ term cannot be derived from standard fermion loops.

Abstract

We discuss the Hopf-Wess-Zumino term in the effective action of the 6d (2, 0) theory of the type A_{N-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 e_{i}-e_{j}, here, the typical WZ term involves two roots e_{i}-e_{j} and e_{k}-e_{j}. 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 H_{3}\wedge A_{3} part of the WZ term is obtained, while the A_{3}\wedge F_{4} part, which is the term accounting for the anomaly matching, cannot be produced by the standard fermion loop integration.