Global PEE and bit-thread construction for compact BTZ

Construct a global PEE-thread and bit-thread flow for the compact BTZ black hole that consistently incorporates the annular topology, nontrivial periods around the horizon, winding-image minimization, gluing across identified faces, and the homology and winding data of all endpoint sectors.

Background

The paper treats the planar BTZ black brane, whose spatial direction is noncompact and whose geodesic-thread construction can be analyzed on a covering geometry without summing over angular images. The compact BTZ quotient is substantially more difficult because its spatial exterior is annular and geodesics can carry winding information around the horizon.

A prospective construction must reconcile several structures simultaneously: closed one-forms or equivariant potentials on the annulus or its universal cover, endpoint sectors that change when the minimizing winding image switches, gluing conditions on identified faces, and a consistent allocation of boundary, winding, and horizon fluxes. The authors suggest that a locking-multiflow formulation may provide suitable bookkeeping, but do not carry out the construction.

References

The compact BTZ black hole is qualitatively more subtle than the planar black brane studied here. Its spatial exterior is an annulus, so the flow may carry a nontrivial period around the horizon, while geodesic lengths and endpoint kernels require minimizing over winding images of the quotient. A vector field constructed in a cut-open fundamental domain must in addition satisfy the gluing condition on the identified faces. Consequently, a simple periodization of the planar boundary--horizon construction need not respect the homology and winding data of all sectors simultaneously. One possible route is to work with closed one-forms on the annulus, or quasi-periodic/equivariant potentials on the universal cover, and to divide endpoint space into image-switching chambers whose periods are fixed by the horizon flux. A locking-multiflow formulation may then provide the natural bookkeeping for simultaneous boundary, winding, and horizon sectors. We leave the detailed compact-BTZ construction to future work; related obstructions to a single global geodesic congruence were emphasized in .

— PEE threads and bit-threads in BTZ black brane and finite-cutoff AdS$_3$  (2608.25782 - Basu et al., 26 Aug 2026) in Section 6, “Limitations and outlook” (subsection “Limitations and outlook”)