The Residue Determinant

ABSTRACT 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, a local invariant, and not defined by a regularization procedure. The residue determinant is consequently a quite different object from the zeta function determinant, which is nonlocal and nonmultiplicative. 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.