Showing papers by "Harald Grosse published in 2015"

Anomalies of Dirac Type Operators on Euclidean Space

Abstract: 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:14020475 [mathKT], 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 Bolle et al (J Math Phys 28:1512–1525, 1987) we introduce in this paper a class of examples of Dirac type operators on \({\mathbb{R}^{2n}}\) 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

On a spectral flow formula for the homological index

Abstract: Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the assumption that the operators {D(t)= D+A(t)} all have discrete spectrum then the spectral flow along the path { D(t)} can be shown to be equal to the index of d/dt+D(t) when the latter is an unbounded Fredholm operator on L^2(R, H). In \cite{GLMST11} an investigation of the index=spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In \cite{CGPST14a} the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in \cite{CGPST14a} much further. We show how to generalise a formula known from the setting of the L^2 index theorem to the non-Fredholm setting. Our main theorem gives a trace formula relating the homological index of \cite{CaKa:TIH} to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalisation of spectral flow to the non-Fredholm setting.