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arXiv · 1701.02439

Discretization of 3d gravity in different polarizations

Maïté Dupuis·Laurent Freidel·Florian Girelli
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Abstract

We study the discretization of 3d gravity with $\Lambda=0$ following the loop quantum gravity framework. In the process, we realize that different choices of polarization are possible. This allows to introduce a new discretization based on the triad as opposed to the connection as in the standard loop quantum gravity framework. We also identify the classical non-trivial symmetries of discrete gravity, namely the Drinfeld double, given in terms of momentum maps. Another choice of polarization is given by the Chern-Simons formulation of gravity. Our framework also provides a new discretization scheme of Chern-Simons, which keeps track of the link between the continuum variables and the discrete ones. We show how the Poisson bracket we recover between the Chern-Simons holonomies allows to recover the Goldman bracket. There is also a transparent link between the discrete Chern-Simons formulation and the discretization of gravity based on the connection (loop gravity) or triad variables (dual loop gravity).

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Maïté Dupuis, Laurent Freidel, Florian Girelli. 2017-01-10. Discretization of 3d gravity in different polarizations. https://doi.org/10.1103/physrevd.96.086017

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