A Differential Form Description of Partial Entanglement Entropy: Testing a Killing Vector Construction in Covariant Phase Space

Abstract

The bit thread formulation provides a geometric description of holographic entanglement entropy, while partial entanglement entropy (PEE) and PEE threads resolve this structure with respect to individual boundary points. Motivated by the fact that PEE thread flows can be superposed to reconstruct conventional bit thread flows, we introduce a differential form description of PEE thread flow. For an interval in the vacuum AdS$_3$/CFT$_2$ setup, we show that the flux of the resulting form reproduces the known entanglement contour, providing a consistency check of the proposed description. We then investigate whether the PEE form can be related, possibly up to an exact form improvement, to a current constructed from the covariant phase space (CPS) formalism. As a first test, we consider exact Killing vectors in Rindler-AdS$_3$. The Rindler parameter $a$ parametrizes a one-parameter family of backgrounds. At each value of $a$, the Killing vector of the corresponding background is substituted into the Iyer-Wald surface charge form, and the full $a$-dependence of the resulting form is retained. We define a finite candidate current by integrating this surface charge form over $a$. For the reflection-symmetric PEE flow sourced at the central boundary point $r_0=0$, we find that no choice of the Killing parameters reproduces both components of the known PEE flow after transforming to Poincaré-AdS$_3$. This result shows that the direct Killing vector CPS construction considered here is not sufficient to reproduce the central PEE flow. Possible extensions include an exact form improvement, a different CPS current, or a more general choice of generator.