Regularized determinant of the Laplacian on forms over spheres
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Regularized determinant of the Laplacian on forms over spheres F. S. Rafael1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract We establish formulae for calculating the regularized determinant of the Laplacian on forms over spheres. This study is a continuation of the ones previously developed for theses spaces, since the previous formulae could only be applied to the study of the Laplacian on forms of degree zero, i. e. functions, and our formulae may be applied to arbitrary degree. Moreover, we use a different approach to the problem, based on a work done by Weng and You and obtain new formulae on the case of degree zero. Keywords Regularized determinant · Laplacian · Forms · Spheres Mathematics Subject Classification 11M36
1 Introduction Regularized determinant is a concept that arises from the study of spectral zeta funcp tions. In the specific case of the Laplacian ΔM on p-forms over closed Riemannian manifolds these functions are defined by ∑ p p 𝜁(s, ΔM ) = dim(E(𝜆, ΔM ))𝜆−s , 𝜆>0
where is the eigenspace associated to the eigenvalue 𝜆 . Since the spectrum of the Laplacian is discrete in this case, the sum converges when Re(s) is sufficiently large, so that the spectral zeta function is well defined for Re(s) ≫ 0. p E(𝜆, ΔM )
Communicated by Mauro Spreafico. * F. S. Rafael [email protected] 1
Universidade Federal de Rondônia - Brazil, BR 364 Km 9,5, Porto Velho Postal Code 76801 ‑ 059, Brazil
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
It is possible to prove that the spectral zeta function admits meromorphic extension to the whole complex plane and that this extension is holomorphic at s = 0 . The regularized determinant of the Laplacian is then defined by: p
p
det(ΔM ) = exp(−𝜁 � (0, ΔM )). This definition is rather coherent with the definition of the determinant of positive definite symmetric operators on finite dimensional vector spaces, since for such an operator T we have: (( )� | ) ∑ | −s | . det(T) = exp dim(E(𝜆, T))𝜆 | | 𝜆>0 |s=0 Moreover, this concept has an important role in modern physics, as exemplified in chapter 7 of [1] and an important role in differential topology, since it may be used to define and study Analytic Torsion. The expression for the Analytic Torsion Ta (M) of a Riemannian manifold M is given by: ) ( n 1∑ k+1 k (−1) k(log det ΔM ) . Ta (M) = exp − 2 k=0 The main objective of this paper is to establish formulae for calculating the determinant of the Laplacian on forms over spheres, Theorems 4.1 and 4.2. This work is a generalization of Theorem 1 of [4], Theorem 1 of [6] and Theorem 2.1 of [2], which produced formulae for the determinant of the Laplacian on functions (forms of degree zero) over these spaces. Beyond obtaining new versions of these Theorems that may be applied to forms of higher degree we also adopt a different approach to the problem, which was inspired by [12] and leads us to formulae that are smaller than the ones presented by [2, 4] and [6] when dealing with functions. This approach also prov
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