Heat kernel coefficients on the sphere in any dimension

Kluth, Yannick

doi:10.1140/epjc/s10052-020-7784-2

Cited by 16 publications

(10 citation statements)

References 76 publications

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“…The Q n [ f ] are in general easy to determine either numerically or even analytically. We are left with determining the heat-kernel coefficients B k (∆), which was already done for many different Laplacians in the literature [296][297][298]. We do not go into details here and simply state the results, if we choose the sphere as background, where the results tremendously simplify.…”

Section: Ttmentioning

confidence: 99%

Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity

Reichert¹

2020

Proceedings of XV Modave Summer School in Mathematical Physics — PoS(Modave2019)

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These lecture notes give an introduction to the functional renormalisation group and asymptotically safe quantum gravity. We cover the basics of generating functionals in quantum field theories and derive the Wetterich equation. The anharmonic oscillator is presented as a simple example and compared to perturbation theory with resummation methods. A special focus is set on the implementation of symmetries in the functional renormalisation group. The second half of these notes is dedicated to quantum gravity, explaining the failure of perturbative gravity and how these issues are addressed in the non-perturbative approach. After detailing the implementation of the diffeomorphism symmetry, we present a simple computation in the Einstein-Hilbert truncation and give a short outlook on the state-of-the-art in the field. These notes are based on lectures given at the XV Modave Summer School in Mathematical Physics.

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Section: Ttmentioning

confidence: 99%

Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity

Reichert¹

2020

Proceedings of XV Modave Summer School in Mathematical Physics — PoS(Modave2019)

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“…In particular, in two dimensions with constant curvature, one again gets the result (4π)E 6 = 1 630 R 3 . More recently, Kluth and Litim [53] computed up to the R 3 , R 4 , and R 5 coefficients in E 6 , E 8 , and E 10 , respectively, in 2 to 6 dimensions for a sphere (actually for any maximally symmetric space). This is done using quite a different method, which leverages simplifications specific to maximal symmetry from the beginning.…”

Section: Jhep04(2021)178mentioning

confidence: 99%

New heat kernel method in Lifshitz theories

Melby-Thompson

Yan

2021

J. High Energ. Phys.

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We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Hořava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.

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“…Integer quantum numbers (n) enumerating these eigenfunction are of course different for tensor, vector and scalar modes, but we will not introduce for them different notations, for in what follows we will need for each (n) only the eigenvalue ∆ n and its degeneracy D n -the dimensionality of eigenvalue subspace. In generic dimension d, which we need for the sake of dimensional regularization, they read on the sphere of unit radius [52,53,54,55]…”

Section: Tensor and Vector Operators On Smentioning

confidence: 99%

Beta functions of (3+1)-dimensional projectable Horava gravity

Barvinsky,

Kurov,

Sibiryakov

2021

Preprint

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We derive the full set of beta functions for the marginal essential couplings of projectable Hořava gravity in (3 + 1)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. The calculation uses symbolic computer algebra and is performed in four different gauges yielding identical results for the essential beta functions. We additionally check the calculation by evaluating the effective action on a special background with spherical spatial slices using an alternative method of spectral summation. We conclude with a preliminary discussion of the properties of the beta functions and the resulting renormalization group flow, identifying several candidate asymptotically free fixed points.

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Heat kernel coefficients on the sphere in any dimension

Cited by 16 publications

References 76 publications

Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity

Lecture notes: Functional Renormalisation Group and Asymptotically Safe Quantum Gravity

New heat kernel method in Lifshitz theories

Beta functions of (3+1)-dimensional projectable Horava gravity

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