The SU(2) Black Hole entropy revisited

Abstract

We study the state-counting problem that arises in the SU(2) black hole entropy calculation in loop quantum gravity. More precisely, we compute the leading term and the logarithmic correction of both the spherically symmetric and the distorted SU(2) black holes. Contrary to what has been done in previous works, we have to take into account "quantum corrections" in our framework in the sense that the level k of the Chern-Simons theory which describes the black hole is finite and not sent to infinity. Therefore, the new results presented here allow for the computation of the entropy in models where the quantum group corrections are important.

Figures (8)

• Figure 1: Pictorial representation of the 3-valent intertwiner $\iota(j_1,j_2;j_3)$ and its adjoint operator $\iota(j_3;j_1,j_2)$.
• Figure 2: Illustration of the normalization properties: the three relations are in fact equivalent to the condition that the $\theta$-graph is normalized to one.
• Figure 3: Pictorial representation of the R-matrix and its inverse $R^{-1}$. Both R-matrices are evaluated in $j_1\otimes j_2$.
• Figure 4: Representation of the Hopf-link. The evaluation of the associated quantum spin-network colored with the representations $j_1$ and $j_2$ gives the un-normalized Verlinde coefficient $\tilde{S}_{j_1j_2}$.
• Figure 5: Pictorial proof of the fusion relation. We start with the graph on the left. The two arrows are identities: the first one is obtained applying the decomposition of the identity "along the vertical dashed line"; the second one is obtained applying the decomposition of the identity "along the horizontal dashed line". Both lead to equivalent expressions and the equality between the evaluations of the graphs on the right is exactly the fusion relation. We made used of the identities represented in the picture (\ref{['norm']}).