Anyonic statistics and large horizon diffeomorphisms for loop quantum gravity black holes

Abstract

We investigate the role played by large diffeomorphisms of quantum isolated horizons for the statistics of loop quantum gravity (LQG) black holes by means of their relation to the braid group. To this aim the symmetries of Chern-Simons theory are recapitulated with particular regard to the aforementioned type of diffeomorphisms. For the punctured spherical horizon, these are elements of the mapping class group of S 2 , which is almost isomorphic to a corresponding braid group on this particular manifold. The mutual exchange of quantum entities in two dimensions is achieved by the braid group, rendering the statistics anyonic. With this we argue that the quantum isolated horizon model of LQG based on SUð2Þ k -Chern-Simons theory exhibits non-Abelian anyonic statistics. In this way a connection to the theory behind the fractional quantum Hall effect and that of topological quantum computation is established, where non-Abelian anyons play a significant role.

Fock Representation of Gravitational Boundary Modes and the Discreteness of the Area Spectrum

“…For large gauge transformations, on the other hand, we expect that they have a non-trivial action in the quantum theory [24]. This observation could be used, in fact, to determine the statistics of the N particle wave function (the exchange of two punctures can always be made undone by performing a large diffeomorphism).…”

Effective Chern–Simons actions of particles coupled to 3D gravity

“…This allows us to establish links between these areas and gravity, see e.g. [43] (which also shows how the Chern-Simons theory can model a black hole horizon in the quantum theory) and references therein.…”

Loop Quantum Gravity meets Topological Phases of Matter at the Black Hole horizon

“…It could be investigated to what extent this quantitatively reduces the entropy. Apart from the qualitative arguments given in [4], that the black hole radiance spectrum should display traces of the non-trivial braiding statistics, it could also be interesting to exactly quantify how much the emission spectra proposed by LQG are then actually modified.…”

“…In this light, the Hilbert space of the horizon d.o.f. is reinterpreted as the fusion Hilbert space of non-abelian anyons [4]. The braiding amounts to non-local observable correlations among the horizon d.o.f.…”

“…could actually be different from the one assumed, motivated by the wellknown fact from solid state physics that quantum objects in 2d obey anyonic statistics. Indeed, we found that the LQG black hole model explicitly exhibits non-abelian anyonic/braiding statistics [4]. From statistical physics one knows that particle exchange is equivalent to an (adiabatic) parallel transport along paths that fall into the first homotopy group π 1 of the respective N-particle configuration space Q N .…”