General anomaly matching by Goldstone bosons

Kazuya Yonekura

doi:10.1007/jhep03(2021)057

Abstract

We describe how Goldstone bosons of spontaneous symmetry breaking G → H can reproduce anomalies of UV theories under the symmetry group G at the nonpertur- bative level. This is done by giving a general definition of Wess-Zumino-Witten terms in terms of the invertible field theories in d + 1 dimensions which describe the anomalies of d-dimensional UV theories. The hidden local symmetry hat{H} , which is used to describe Goldstone bosons in coset construction G/H , plays an important role. Our definition also naturally leads to generalized θ-angles of the hidden local gauge group hat{H} . We illustrate this point by SO(Nc) (or Spin(Nc)) QCD-like theories in four dimensions.

Highlights

In [15, 16], it is shown that the WZW terms of four dimensional QCD-like theories with odd color numbers require spin structure of spacetime
We describe how Goldstone bosons of spontaneous symmetry breaking G → H can reproduce anomalies of UV theories under the symmetry group G at the nonperturbative level
The Atiyah-(Patodi)-Singer index theorem plays an important role for the description of ‘t Hooft anomalies of UV fermions, and it is possible to show the well-definedness of the WZW terms in QCD-like theories by using the index theorem [17] which requires spin structure

Summary

Anomalies and invertible field theories

We would like to review the fact that anomalies of d-dimensional theories are described by (d + 1)-dimensional bulk theories, which are sometimes called symmetry protected topological phases or invertible field theories. An invertible field theory I in d+1 dimensions is defined by the property that its Hilbert space on any closed d-manifold X is one dimensional. A d-dimensional anomalous theory T whose anomaly is given by the invertible field theory I has the property that its partition function ZT (X) on a closed d-manifold X is not a complex number, but takes values in H(X)∗, i.e. the dual of the Hilbert space H(X) of the invertible field theory. If we want to make the partition function ZT (X) of the anomalous theory T to be a well-defined complex number in C, we must add the bulk Y with the invertible field theory I on it. We can reach to the above understanding; see [22] and section 5.4 of [13]

General WZW terms and the anomaly matching

Topological theta angles