Dimensional asymptotics of effective actions on S^n, and proof of Bär-Schopka's conjecture
Abstract
Description
We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians Δ+βR on round S^n. For Laplacians the behavior depends on ``the coupling strength'' β, and one cannot in general expect a finite limit of ζ'(0), and for the ordinary Laplacian, β=0, we prove it to be +\infty, for odd dimensions. For the Dirac operator, Bär and Schopka conjectured a limit of unity for the determinant ([BS]), i.e. \lim_{n\to\infty}\det(D, S^n_{\mathrm{can}})=1.
We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having ``enough scalar curvature'' and no kernel. Thus for the important (conformally covariant) Yamabe operator, β=(n-2)/(4(n-1)), the determinant tends to unity.
For the ordinary Laplacian it is natural to rescale spheres to unit volume, since \lim_{k\to\infty}\det(Δ, S_\mathrm{rescaled}^{2k+1})=\frac{1}{2πe}.
14 pages
14 pages