arXiv 0709.00672007-09-01math-phmath.MP

S^n上有效作用量的维数渐近性及Bär-Schopka猜想的证明

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

Niels Martin Møller

备注：: 14 pages
发表信息：: Math. Ann. 343 (2009), no. 1., 35--51.
DOI：: 10.1007/s00208-008-0264-x

摘要：

我们研究了Dirac算子D和Laplacian算子Δ+βR在轮S^n上的有效作用量或泛函行列式的维数渐近性。对于拉普拉斯算子，其行为取决于“耦合强度”β，一般不能期望有一个有限的极限β '（0），对于普通拉普拉斯算子，β=0，我们证明它是+ \infty，对于奇数维。对于狄拉克算子，Bär和Schopka证明了行列式（[BS]）的单位极限，即\lim_{n\to\infty} \det（D，S^n_{ \mathrm{can} }）=1。我们证明他们的猜想严格，给出渐近性，以及一个模式的不等式所满足的决定因素。单位的极限值是具有“足够的标量曲率”和没有内核的优点。因此，对于重要的（共形协变的）Yamabe算子β=（n-2）/（4（n-1）），行列式趋于1。对于普通的拉普拉斯算子来说，将球体重新标度为单位体积是很自然的，因为\lim_{k\to\infty} \det（Δ，S_ \mathrm{rescaled}^{2k+1}）= \frac{1}{2πe}。

英文摘要：

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}.

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