Conformal blocks on a 2-sphere with indistinguishable punctures and implications on black hole entropy

TL;DR

This paper explores the counting of conformal blocks on a 2-sphere with indistinguishable punctures in Chern-Simons theory, revealing implications for black hole entropy and showing that the logarithmic correction remains unchanged regardless of puncture distinguishability.

Contribution

It extends the counting of conformal blocks to the case of indistinguishable punctures and demonstrates the invariance of the black hole entropy correction coefficient.

Findings

The counting formula decouples into two parts in the large level limit.

Indistinguishability of punctures does not alter the logarithmic correction coefficient.

The entropy correction coefficient remains at -3/2 regardless of puncture statistics.

Abstract

The dimensionality of the Hilbert space of a Chern-Simons theory on a 3-fold, in the presence of Wilson lines carrying spin representations, had been counted by using its link with the Wess-Zumino theory, with level $k$, on the 2-sphere with points (to be called punctures) marked by the piercing of the corresponding Wilson lines and carrying the respective spin representations. It is shown, in the weak coupling (large $k$) limit, the formula decouples into two characteristically distinct parts; one mimics the dimensionality of the Hilbert space of a collection of non-interacting spin systems and the other is an effective overall correction contributed by all the punctures. The exact formula yield from this counting has been shown earlier to have resulted from the consideration of the punctures to be distinguishable. We investigate the same counting problem by considering the punctures to be indistinguishable. Although the full formula remains undiscovered, nonetheless, we are able to impose the relevant statistics for indistinguishable punctures in the approximate formula resulting from the weak coupling limit. As an implication of this counting, in the context of its relation to that of black hole entropy calculation in quantum geometric approach, we are able to show that the logarithmic area correction, with a coefficient of $-3/2$, that results in this method of entropy calculation, in independent of whether the punctures are distinguishable or not.