Interpretation of the auxiliary Hilbert space in the closed-universe limit

Ascertain the precise interpretation of the auxiliary Hilbert space labeled by three Virasoro representations when the two asymptotic AdS cylinders are taken to be infinitely long so that only a closed genus-2 universe remains, and explain how this auxiliary space is realized and probed in the absence of AdS boundaries.

Background

In the pinching limit where the asymptotic AdS boundaries are frozen to the vacuum, the geometry reduces to a closed universe without AdS regions. The associated auxiliary Hilbert space—arising from Virasoro TQFT conformal blocks—still appears to be present, yet its physical interpretation becomes murky.

The authors point out additional complications when introducing matter within a VTQFT perspective, which can change the auxiliary Hilbert space structure and suggest many superselection sectors, further motivating a clear interpretation.