  
  
Stat 
Members: 3658 Articles: 2'599'751 Articles rated: 2609
02 November 2024 

   

Article overview
 

Dimensional asymptotics of effective actions on S^n, and proof of B"arSchopka's conjecture  Niels Martin Møller
;  Date: 
1 Sep 2007  Abstract:  We study the dimensional asymptotics of the effective actions, or functional
determinants, for the Dirac operator D and Laplacians Delta +eta R on round
S^n. For Laplacians the behavior depends on ’’the coupling strength’’ eta,
and one cannot in general expect a finite limit of zeta’(0), and for the
ordinary Laplacian, eta=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 oinfty}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, eta=(n2)/(4(n1)),
the determinant tends to unity.
For the ordinary Laplacian it is natural to rescale spheres to unit volume,
since lim_{k oinfty}det(Delta, S_mathrm{rescaled}^{2k+1})=frac{1}{2pi
e}.  Source:  arXiv, 0709.0067  Services:  Forum  Review  PDF  Favorites 


No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.

 


