# Dimensional asymptotics of effective actions on S^n, and proof of Bär-Schopka's conjecture - Mathematical Physics

Dimensional asymptotics of effective actions on S^n, and proof of Bär-Schopka's conjecture - Mathematical Physics - Download this document for free, or read online. Document in PDF available to download.

Abstract: We study the dimensional asymptotics of the effective actions, or functionaldeterminants, for the Dirac operator D and Laplacians \Delta +\beta R on roundS^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 theordinary Laplacian, \beta=0, we prove it to be +\infty, for odd dimensions. Forthe Dirac operator, B\-ar and Schopka conjectured a limit of unity for thedeterminant BS, i.e. \lim {n\to\infty}\detD, S^n {\mathrm{can}}=1.We prove their conjecture rigorously, giving asymptotics, as well as apattern of inequalities satisfied by the determinants. The limiting value ofunity is a virtue of having ``enough scalar curvature- and no kernel. Thus forthe important conformally covariant Yamabe operator, \beta=n-2-4n-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\pie}.

Author: ** Niels Martin Møller**

Source: https://arxiv.org/