Clebsch-Gordan coefficients of the SL(2, C) group

We investigate structural aspects of JT gravity through its BF description. In particular, we provide evidence that JT gravity should be thought of as (a coset of) the noncompact subsemigroup SL + (2, R) BF theory. We highlight physical implications, including the famous Plancherel measure sinh 2π √ E. Exploiting this perspective, we investigate JT gravity on more generic manifolds with emphasis on the edge degrees of freedom on entangling surfaces and factorization. It is found that the one-sided JT gravity degrees of freedom are described not just by a Schwarzian on the asymptotic boundary, but also include frozen SL + (2, R) degrees of freedom on the horizon, identifiable as JT gravity black hole states. Configurations with two asymptotic boundaries are linked to 2d Liouville CFT on the torus surface.

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“…are the well-known 3j-symbols of SL(2, C) [108], identifiable as conformal three-point functions as recently discussed in [109].…”

Section: A22 Quantum Mechanics On Sl(2 C)mentioning

confidence: 99%

Fine structure of Jackiw-Teitelboim quantum gravity

Blommaert

Mertens

Verschelde

2019

J. High Energ. Phys.

189

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“…Analytical and numerical properties of these functions are work in progress [26,28,29,30]. On the other hand, the explicit evaluation of booster functions in spite of their rather simple form is still a very involved task: For n = 3 we employ an expression for (3) in terms of finite sums of Γ functions, for details see [32,26]; for n ≥ 4 a similar formula exists but features an integration over virtual labels 4 , and in the end we found it less time consuming to numerically integrate directly the boost integrals. A C numerical code for the virtual irreps formula has been recently developed in [31].…”

Section: The Eprl Model and Its Connection With Bf Theorymentioning

confidence: 99%

Infrared divergences in the EPRL-FK spin foam model

Donà

2018

Class. Quantum Grav.

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We provide an algorithm to estimate the divergence degree of the Lorentzian EPRL-FK spin foam amplitudes for arbitrary 2-complexes. We focus on the "self-energy" and "vertex renormalization" diagrams and find an upper bound estimate. We argue that our upper bound must be close to the actual value, and explain what numerical improvements are needed to verify this numerically. For the self-energy, this turns out to be significantly more divergent than the lower bound estimate present in the literature. We support the validity of our algorithm using 3-stranded versions of the amplitudes (corresponding to a toy 3d model) for which our estimates are confirmed numerically. We also apply our methods to the simplified EPRLs model, finding an utterly convergent behavior, and to BF theory, independently recovering the divergent estimates present in the literature. * pxd81@psu.edu

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“…Kerimov and Verdiev first got interested in the generalisation of the Clebsch-Gordan coefficients to the irreps of SL 2 (C) [KVM78]. The SL 2 (C)-Clebsch-Gordan coefficients are defined by the relation |p, k; j, m = dp 1 dp 2 k 1 j 1 m 1 k 2 j 2 m 2 C pkjm p 1 k 1 j 1 m 1 ,p 2 k 2 j 2 m 2 |p 1 , k 1 ; j 1 m 1 ⊗ |p 2 , k 2 ; j 2 , m 2 .…”

Section: Sl 2 (C) Wigner's Matrixmentioning

confidence: 99%

“…χ is a function of 9 variables which can be computed by the following expression (found initially in [KVM78] but corrected slightly in [Spe17]):…”

Section: Sl 2 (C) Wigner's Matrixmentioning

confidence: 99%

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A primer of group theory for Loop Quantum Gravity and spin-foams

Martin-Dussaud

2019

Gen Relativ Gravit

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A primer of group theory for Loop Quantum Gravity and Spin-foamsThe aim of a theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, our students and their students and so on, will be able to absorb the essential aspects in as painless a way as possible, and this is the only way in which you can go on cumulatively building up any kind of scientific activity without eventually coming to a dead end.M. F. Atiyah, [Ati74].

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Clebsch-Gordan coefficients of the SL(2, C) group

Cited by 18 publications

References 5 publications

Fine structure of Jackiw-Teitelboim quantum gravity

Fine structure of Jackiw-Teitelboim quantum gravity

Infrared divergences in the EPRL-FK spin foam model

A primer of group theory for Loop Quantum Gravity and spin-foams

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