  Several recent results have hinted that black hole thermodynamics in loop
quantum gravity simplifies if one chooses an imaginary Barbero-Immirzi
parameter $\gamma=i$. This suggests a connection with
$\mathrm{SL}(2,\mathbb{C})$ or $\mathrm{SL}(2,\mathbb{R})$ conformal field
theories at the "boundaries" formed by spin network edges intersecting the
horizon. I present a bit of background regarding the relevant conformal field
theories, along with some speculations about how they might be used to count
black hole states. I show, in particular, that a set of unproven but plausible
assumptions can lead to a boundary conformal field theory whose density of
states matches the Bekenstein-Hawking entropy.
