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

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 $SU(2{)}_k$-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.

Figure 1

Parallel transport of the first puncture around the second one.

Figure 2

Two horizon punctures: unbraided vs upon the application of $\widehat{M}$ and $\widehat{B}$, respectively.

Figure 3

Two incident bulk edges piercing the horizon: unbraided vs upon the application of $\widehat{M}$ and $\widehat{B}$, respectively.

Figure 4

$R$-matrix.

Figure 5

$F$-matrix.

Figure 6

$B$-matrix.

Figure 7

Depiction of a braid.

Figure 8

Depiction of a pure braid.