An anomaly associated with 4-dimensional quantum gravity (Q1924022)

This article studies invariants of the Paneitz operator, a conformally invariant fourth order linear differential operator defined on any four-dimensional conformal manifold. This operator is a natural analogue of the Laplacian in two dimensions, its symbol being the square of the Laplacian. Though the operator is conformally invariant, its functional determinant depends on a choice of metric in the conformal class. (This determinant arises as an anomaly associated to a possible action for quantum gravity.) One can deform a particular metric in its conformal class and ask how the determinant changes. The author of the present article answers the question. In particular, he computes the first variation, a local conformal invariant. The method of proof is interesting, using a kind of `dimensional regularisation'. There are results of \textit{P. Gilkey} [Duke Math. J. 47, 511-528 (1980; Zbl 0448.58026)] giving integrated asymptotic expansions of the corresponding heat operator. By a method of the author and \textit{B. Ørsted} [Compos. Math. 60, 261-293 (1986; Zbl 0608.58039)], one can deduce the local asymptotics from their integrated counterparts but only for dimension \(m\neq 4\). However, one can show that the dependence on dimension must be rational in \(m\) and hence obtain the four-dimensional results by meromorphic continuation.