The aetiology of sigma model anomalies
Certain nonlinear sigma models with fermions are ill-defined due to an anomaly which exhibits characteristics of both the nonabelian gauge theory anomaly and the SU(2) anomaly. The simplest way to diagnose the anomaly involves consideration of the global topology of the theory. In this paper, the authors review the mathematical methods needed for this analysis and apply them to several supersymmetric sigma models. Some of these are found to be anomalous. This paper includes eight sections and four appendices. In Section 2, the authors characterize the anomaly as an obstruction to a continuous definition of the functional on the map space \({\mathcal C}\). The condition obtained is that an integer \(\nu\) should vanish. A physical interpretation to this obstruction is given in Section 3. It is shown that for \(\nu\) \(\neq 0\) it is impossible to find well-behaved local counterterms which render different perturbative expansions of the same Green function physically equivalent. In Section 4, the authors compute \(\nu\), arriving at the final anomaly criterion which essentially says that (\({\mathcal F})^ 3\), \({\mathcal F}\) being the curvature or ''field strength'', should be an exact 6-form. Here and in the appendices, brief and useful descriptions of the needed mathematical tools can be found. Sections 5-7 are devoted to the applications of the anomaly criterion to some models.
scientific article; zbMATH DE number 3956061
scientific article