Spectral action for Bianchi type-IX cosmological models

Fan, Wentao; Fathizadeh, Farzad; Marcolli, Matilde

doi:10.1007/jhep10(2015)085

Cited by 9 publications

(36 citation statements)

References 53 publications

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“…The rationality of the spectral action for a general triaxial Bianchi type-IX metric with an SU (2)-symmetry, obtained in [19], suggested the existence of a rich arithmetic structure in the Seeley-de Witt coefficients associated with the square of the Dirac operator of these cosmological models. Here the rationality assertion means that each coefficient is the time integral of an expression presented by a several variable polynomial with rational coefficients evaluated on the expansion factors and their derivatives, up to a certain order with respect to time.…”

Section: Introductionmentioning

confidence: 99%

“…In [2], following the work carried out in [48,27], these equations are solved by using the τ -function of the Schlesinger system formulated in terms of theta functions [30], and an explicit parametrization of the Bianchi IX gravitational instantons is given in terms of theta functions with characteristics. Considered along with our rationality result about the spectral action [19], this parametrization of the gravitational instantons will be the main ingredient in our construction of the modular expression for the terms appearing in the expansion of the spectral action in the energy scale. We will also describe an explicit connection between the modular functions that arise in the spectral action and well-known classical modular forms.…”

Section: Introductionmentioning

confidence: 99%

“…Here, the metric ds 2 is the Bianchi IX model with general cosmic expansion factors w 1 , w 2 and w 3 , and SU (2)-invariant 1-forms σ 1 , σ 2 and σ 3 . We found it convenient to recall from [19] the expression for the Dirac operator D of the Bianchi IX model ds 2 and to use it for the presentation of the Dirac operatorD of the conformally equivalent metric ds 2 = F ds 2 . We then obtain a rationality statement for the general terms a 2n appearing in the small time asymptotic expansion of the heat kernel,…”

Section: Introductionmentioning

confidence: 99%

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Modular forms in the spectral action of Bianchi IX gravitational instantons

Fan

Fathizadeh

Marcolli

2019

J. High Energ. Phys.

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Following our result on rationality of the spectral action for Bianchi type-IX cosmological models, which suggests the existence of a rich arithmetic structure in the action, we study the arithmetic and modular properties of the action for an especially interesting family of metrics, namely SU (2)-invariant Bianchi IX gravitational instantons. A variant of the previous rationality result is first presented for a time-dependent conformal perturbation of the triaxial Bianchi IX metric which serves as a general form of Bianchi IX gravitational instantons. By exploiting a parametrization of the gravitational instantons in terms of theta functions with characteristics, we then show that each Seeleyde Witt coefficientã 2n appearing in the expansion of their spectral action gives rise to vector-valued and ordinary modular functions of weight 2. We investigate the modular properties of the first three termsã 0 ,ã 2 andã 4 by preforming direct calculations. This guides us to provide spectral arguments for general termsã 2n , which are based on isospectrality of the involved Dirac operators. Special attention is paid to the relation of the new modular functions with well-known modular forms by studying explicitly two different cases of metrics with pairs of rational parameters that, in a particular sense, belong to two different general families. In the first case, it is shown that by running a summation over a finite orbit of the rational parameters, up to multiplication by the cusp form ∆ of weight 12, each termã 2n lands in the 1-dimensional linear space of modular forms of weight 14. In the second case, it is proved that, after a summation over a finite orbit of the parameters and multiplication by G 4 4 , where G 4 is the Eisenstein series of weight 4, eachã 2n lands in the 1-dimensional linear space of cusp forms of weight 18.Mathematics Subject Classification (2010). 58B34, 11F37, 11M36.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modular forms in the spectral action of Bianchi IX gravitational instantons

Fan

Fathizadeh

Marcolli

2019

J. High Energ. Phys.

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“…where the summation runs over all α ∈ Z 4 + , j ∈ 0, 1, ..., n − 1, k ∈ {0, 1, 2} such that |α| + j + 2 − k = n, see [20], [16].…”

Section: Spectral Gravity and Robertson-walker Metricsmentioning

confidence: 99%

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Fathizadeh

Kafkoulis

Marcolli

2020

Ann. Henri Poincaré

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We obtain an explicit formula for the full expansion of the spectral action on Robertson-Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman-Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson-Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson-Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function. Contents

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“…The Seeley-deWitt coefficients as residues. For any n ∈ Z ≥1 , the Seeley-deWitt coefficients a 2n can be computed as a noncommutative residue (see [5])…”

Section: Seeley-dewitt Coefficients and Periodsmentioning

confidence: 99%

Motives and periods in Bianchi IX gravity models

2018

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We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat kernel expansion of the Dirac-Laplacian on SU (2) Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.

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Spectral action for Bianchi type-IX cosmological models

Cited by 9 publications

References 53 publications

Modular forms in the spectral action of Bianchi IX gravitational instantons

Modular forms in the spectral action of Bianchi IX gravitational instantons

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Motives and periods in Bianchi IX gravity models

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