The evaluation map in field theory, sigma-models and strings. II
This paper presents a systematic analysis of local anomalies in sigma- models and strings and of global anomalies for both sigma-models and field theories. As in part I of the series [ibid. 112, 237-282 (1987; Zbl 0645.53071)], which dealt with local anomalies in field theories, the basic tool is a certain evaluation map, in this case \(ev: M\times Map(M,T)\to T,\) where M is a compact connected oriented n-dimensional Riemannian manifold (``space-time''), T the target space (connected Riemannian) of the sigma-model and Map(M,T) the space of \(C^{\infty}\)- maps from M to T. The authors point out a certain parallelism of \(\sigma\)-models with field theories, where T is replaced by the universal classifying space of the relevant principal fibre bundle. There is an important difference, however, as a generalized Wess-Zumino term may cancel an anomaly. In the case of strings both field-theory anomalies (gravitational and conformal, the latter being interpreted as a special case of holomorphic anomalies) and sigma-model anomalies are considered and the conditions on the background manifold necessary for their cancellation are derived. Global anomalies are shown to be connected with the differential characters of Cheeger and Simons. Their absence in sigma-models imposes restrictions on the target space that are also detailed.
scientific article; zbMATH DE number 4069780
scientific article