Anomalies of Dirac Type Operators on Euclidean Space

We develop by example a type of index theory for non-Fredholm operators. A general framework using cyclic homology for this notion of index was introduced in a separate article (Carev and Kaad, Topological invariance of the homological index. arXiv:1402.0475 [math.KT], 2014) where it may be seen to generalise earlier ideas of Carey–Pincus and Gesztesy–Simon on this problem. Motivated by an example in two dimensions in Bollé et al. (J Math Phys 28:1512–1525, 1987) we introduce in this paper a class of examples of Dirac type operators on R²ⁿ that provide non-trivial examples of our homological approach. Our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles, with an important difference, namely, we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our Dirac type operators.