Dimensional asymptotics of effective actions on S^n, and proof of B\"ar-Schopka's conjecture

TL;DR

This paper investigates the asymptotic behavior of effective actions for Dirac and Laplacian operators on high-dimensional spheres, proving B"ar-Schopka's conjecture that the Dirac determinant approaches one as dimension increases.

Contribution

It rigorously proves B"ar-Schopka's conjecture on the Dirac operator determinant and analyzes the asymptotics of Laplacian determinants on spheres, revealing their dependence on geometric parameters.

Findings

Dirac determinant tends to 1 as dimension increases.

Laplacian determinants diverge or tend to specific limits depending on parameters.

Established inequalities and asymptotic patterns for determinants.

Abstract

We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians \Delta +\beta R on round S^n. For Laplacians the behavior depends on ``the coupling strength'' \beta, and one cannot in general expect a finite limit of \zeta'(0), and for the ordinary Laplacian, \beta=0, we prove it to be +\infty, for odd dimensions. For the Dirac operator, B\"ar 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, \beta=(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(\Delta, S_\mathrm{rescaled}^{2k+1})=\frac{1}{2\pi e}.