Symplectic and semiclassical aspects of the Schläfli identity

Hedeman, Austin; Kur, Eugene; Littlejohn, Robert G.; Haggard, Hal M.

doi:10.1088/1751-8113/48/10/105203

Cited by 10 publications

(6 citation statements)

References 78 publications

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“…The δu a contributions cancel between the two branches. Hence this has exactly the form of the Schläfli variation [40,41]. The Schläfli identity then allows us to conclude that…”

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confidence: 85%

Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

et al. 2016

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“…The δu a contributions cancel between the two branches. Hence this has exactly the form of the Schläfli variation [40,41]. The Schläfli identity then allows us to conclude that…”

mentioning

confidence: 85%

Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

et al. 2016

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“…However, the main reason why this works surprisingly well is the fundamental interplay between the Regge equations of motion and the generalization of the Schlaefli identities to curved simplices, see e.g. [35,77] for classical applications of this idea, and [78] for a quantum geometrical one involving Chern-Simons theory.…”

Section: Flat Space Perturbative Quantum Regge Calculusmentioning

confidence: 99%

Quasi-local holographic dualities in non-perturbative 3d quantum gravity I – Convergence of multiple approaches and examples of Ponzano–Regge statistical duals

et al. 2019

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This is the first of a series of papers dedicated to the study of the partition function of three-dimensional quantum gravity on the twisted solid torus with the aim to deepen our understanding of holographic dualities from a non-perturbative quantum gravity perspective. Our aim is to compare the Ponzano-Regge model for non-perturbative three-dimensional quantum gravity with the previous perturbative calculations of this partition function. We begin by reviewing the results obtained in the past ten years via a wealth of different approaches, and then introduce the Ponzano-Regge model in a self-contained way. Thanks to the topological nature of three-dimensional quantum gravity we can solve exactly for the bulk degrees of freedom and identify dual boundary theories which depend on the choice of boundary states, that can also describe finite, non-asymptotic boundaries. This series of papers aims precisely at the investigation of the role played by the different quantum boundary conditions leading to different boundary theories. Here, we will describe the spin network boundary states for the Ponzano-Regge model on the twisted torus and derive the general expression for the corresponding partition functions. We identify a class of boundary states describing a tessellation with maximally fuzzy squares for which the partition function can be explicitly evaluated. In the limit case of a large, but finely discretized, boundary we find a dependence on the Dehn twist angle characteristic for the BMS 3 character. We furthermore show how certain choices of boundary states lead to known statistical models as dual field theories -but with a twist. * Electronic address: bdittrich@perimeterinstitute.ca † Electronic address: christophe.goeller@ens-lyon.fr ‡ Electronic address: etera.livine@ens-lyon.fr § Electronic address: ariello@perimeterinstitute.ca arXiv:1710.04202v1 [hep-th]

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“…The equations of motion impose that is, flatness of the simplicial complex. This is due to the Schläfli identity [86] (for a modern symplectic proof see [87])…”

Section: Area Regge Calculusmentioning

confidence: 99%

The degrees of freedom of area Regge calculus: dynamics, non-metricity, and broken diffeomorphisms

Asante

Dittrich

Haggard

2018

Class. Quantum Grav.

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Discretization of general relativity is a promising route towards quantum gravity. Discrete geometries have a finite number of degrees of freedom and can mimic aspects of quantum geometry. However, selection of the correct discrete freedoms and description of their dynamics has remained a challenging problem. We explore classical area Regge calculus, an alternative to standard Regge calculus where instead of lengths, the areas of a simplicial discretization are fundamental. There are a number of surprises: though the equations of motion impose flatness we show that diffeomorphism symmetry is broken for a large class of area Regge geometries. This is due to degrees of freedom not available in the length calculus. In particular, an area discretization only imposes that the areas of glued simplicial faces agrees; their shapes need not be the same. We enumerate and characterize these non-metric, or 'twisted', degrees of freedom and provide tools for understanding their dynamics. The non-metric degrees of freedom also lead to fewer invariances of the area Regge action-in comparison to the length action-under local changes of the triangulation (Pachner moves). This means that invariance properties can be used to classify the dynamics of spin foam models. Our results lay a promising foundation for understanding the dynamics of the non-metric degrees of freedom in loop quantum gravity and spin foams. * Electronic address: sasante@perimeterinstitute.ca † Electronic address: bdittrich@perimeterinstitute.ca ‡ Electronic address: haggard@bard.edu arXiv:1802.09551v1 [gr-qc]

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Symplectic and semiclassical aspects of the Schläfli identity

Cited by 10 publications

References 78 publications

Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks

Quasi-local holographic dualities in non-perturbative 3d quantum gravity I – Convergence of multiple approaches and examples of Ponzano–Regge statistical duals

The degrees of freedom of area Regge calculus: dynamics, non-metricity, and broken diffeomorphisms

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