The Residue Determinant

The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators (ψdos) associated to the Guillemin-Wodzicki residue trace. The resulting residue determinant functional is multiplicative and unique up to taking powers. It is a local invariant and not defined by a regularization procedure. The residue determinant is consequently a quite different object to the zeta function (quasi-) determinant, which is non-local and non-multiplicative. Indeed, the residue determinant does not arise as the derivative of a trace on the complex power operators, and does not depend on a choice of spectral cut. The identification of a certain residue determinant with the index of an elliptic ψdo shows the residue determinant to be topologically significant. 1 1This work arose following conversations with Steve Rosenberg concerning higher Chern-Weil invariants, my thanks to him for his support and interest. I am also indebted to Kate Okikiolu for a helpful suggestion, the essential role of [Ok1, Ok2] in the current work is evident. I am grateful to Gerd Grubb and Sylvie Paycha for numerous interesting discussions and helpful comments. 1