Construct an explicit PEE-generated max-flow for disconnected BCFT phases

Construct an explicit PEE-generated bit-thread max-flow representative for the disconnected phases of AdS$_3$/BCFT$_2$, extending beyond the intersection-weighted PEE normalization already established for those phases.

Background

The paper analyzes disconnected BCFT phases in which the RT surface has multiple components ending on the end-of-the-world brane. In these configurations, a naive unweighted superposition of PEE-thread source currents does not reproduce the correct entropy because different thread classes intersect the relevant homologous surface with different multiplicities. The authors therefore establish an intersection-weighted, or chamber-projected, normalization prescription.

Although this weighted prescription correctly accounts for the entropy, the paper does not derive a closed-form bit-thread vector field generated by the PEE construction for the disconnected phase. Producing such a representative would complete the flow construction in the remaining BCFT homology chamber and clarify how the intersection weights are implemented at the level of a smooth, divergenceless, norm-bounded current.

References

In disconnected BCFT phases we establish the intersection-weighted PEE normalization and formulate the corresponding flow problem, but we do not claim a closed-form PEE-generated max-flow representative.

— PEE threads and bit threads in gravitational subregions of AdS  (2608.25783 - Basu et al., 26 Aug 2026) in Section 3.2, subsection “Disconnected phase”; Section 6, subsection “From PEE threads to extended bit-thread flows”