The Dirac operator on space forms of positive curvature

Bär, Christian

doi:10.2969/jmsj/04810069

Cited by 73 publications

(105 citation statements)

References 11 publications

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“…One then writes the expression for Tr(f (D/Λ)) as a finite number of sums over lattices, by separating the spectrum of D into a union of arithmetic progressions λ n,i , parameterized by the integers n ∈ Z. One also needs to be able to rewrite the multiplicities computed with the method of [3] as polynomial functions m λn,i = P i (λ n,i ) evaluated at the corresponding eigenvalue. This step is computationally challenging, but it can be obtained with the help of computer calculations with Mathematica.…”

Section: 3mentioning

confidence: 99%

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Building Cosmological Models via Noncommutative Geometry

Marcolli

2011

Int. J. Geom. Methods Mod. Phys.

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Abstract. This is an overview of new and ongoing research developments aimed at constructing cosmological models based on noncommutative geometry, via the spectral action functional, thought of as a modified gravity action, which includes the coupling with matter when computed on an almost commutative geometry. This survey is mostly based on recent results obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various aspects of cosmological models of the very early universe, developed by the author and Pierpaoli, based on the asymptotic expansion of the spectral action functional and on renormalization group analysis of the associated particle physics model (an extension of the standard model with right handed neutrinos and Majorana mass terms previously developed in collaboration with Chamseddine and Connes). We also describe non-perturbative results, more recently obtained by Pierpaoli, Teh, and the author, which extend to the more modern universe, which show that, for different candidate cosmic topologies, the form of the slow-roll inflation potentials obtained from the non-perturbative calculation of the spectral action are strongly coupled to the underlying geometry. We discuss some ongoing directions of research and open questions in this new field of "noncommutative cosmology". The paper is based on the talk given by the author at the conference "Geometry and Quantum Field Theory" at the MPI, in honor of Alan Carey.

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Section: 3mentioning

confidence: 99%

“…One needs to know explicitly the spectrum of the Dirac operator, which can be computed in the case of the spherical space forms from the spectrum on the 3-sphere, using the generating functions technique of [3] to compute the correct multiplicities. It is also known for the flat tori.…”

Section: 3mentioning

confidence: 99%

Building Cosmological Models via Noncommutative Geometry

Marcolli

2011

Int. J. Geom. Methods Mod. Phys.

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“…On the other hand, the Poisson summation may be used only for operators with polynomial spectra (see Section 4.1), and only for those with eigenvalues given by first order polynomial. 3 This is quite restrictive, however many symmetric sphere-like manifolds and their deformations provide Dirac operators with spectra of the desired form [57]. Poisson summation also generalises easily to the multi-index case, which can be applied to obtain spectral action on tori.…”

Section: Examplementioning

confidence: 99%

“…, K } the eigenvalues λ k n and their respective multiplicities M k n are given by polynomials in n. Since the (inverse) Mellin transform is linear, one can apply the general theory to each sequence λ k n n∈N separately. One can find examples of such spectra in the framework of Dirac operators on some homogeneous spaces [3,57,68], like the Poincaré sphere or lens spaces. Indeed, [57, (6.3), (6.11)] give the decomposition of Dirac spectrum into four 1 (K = 4) polynomial sequences for quaternionic space SU (2)/Q8.…”

Section: Operators Of Polynomial Spectrummentioning

confidence: 99%

Asymptotic and Exact Expansions of Heat Traces

Eckstein

Zając

2015

Math Phys Anal Geom

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We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zetafunctions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.

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“…with multiplicity 2 [(n−1)/2] k+n−2 k (see, e.g., [2]). The eigenspinors for the smallest eigenvalues ±(n − 1)/2 are simply restrictions of parallel (i.e., constant) positive, respectively negative spinors from R n .…”

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confidence: 99%

On the L^pindex of spin Dirac operators on conical manifolds

Legrand

Moroianu

2006

Studia Math.

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Abstract. We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from L p (Σ + ) to L q (Σ − ) with p, q > 1. When 1 + n/p − n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n + 1)/2 − n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the L 2 index. For 1 + n/p − n/q ≤ 0 the index formula contains an extra term related to the Calderón projector.

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The Dirac operator on space forms of positive curvature

Cited by 73 publications

References 11 publications

Building Cosmological Models via Noncommutative Geometry

Building Cosmological Models via Noncommutative Geometry

Asymptotic and Exact Expansions of Heat Traces

On the L^pindex of spin Dirac operators on conical manifolds

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The Dirac operator on space forms of positive curvature

Cited by 73 publications

References 11 publications

Building Cosmological Models via Noncommutative Geometry

Building Cosmological Models via Noncommutative Geometry

Asymptotic and Exact Expansions of Heat Traces

On the Lpindex of spin Dirac operators on conical manifolds

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On the L^pindex of spin Dirac operators on conical manifolds