Dimensional asymptotics of effective actions on S^n, and proof of Bär-Schopka's conjecture
| dc.creator | Møller, Niels Martin | |
| dc.date | 2007-09-01 | |
| dc.date.accessioned | 2026-07-07T12:46:24Z | |
| dc.date.available | 2026-07-07T12:46:24Z | |
| dc.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}. | |
| dc.description | 14 pages | |
| dc.identifier | https://arxiv.org/abs/0709.0067 | |
| dc.identifier | http://arxiv.org/abs/0709.0067 | |
| dc.identifier | Math. Ann. 343 (2009), no. 1., 35--51. | |
| dc.identifier | doi:10.1007/s00208-008-0264-x | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/221370 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 58J52 | |
| dc.title | Dimensional asymptotics of effective actions on S^n, and proof of Bär-Schopka's conjecture | |
| dc.type | text |