From off-shell strings to entropy flows

Abstract

Bit threads express holographic entropy as the maximal flux of a divergenceless vector field subject to a local capacity bound, but do not identify a physical carrier of that flux. In our previous work, we identified the projected string charge density with a bit-thread flow, reproduced the entropies of BTZ and ten-dimensional three-charge black holes, and showed that the maximal entropy flux is preserved under S- and T-dualities. However, the norm bound was imported from the bit-thread prescription, while the string-current configurations were prescribed along known bit-thread lines. In this paper, we give a stringy foundation for the local capacity and construct dynamical probe-string representatives of maximal flow. At genus zero and two-derivative order, the conical variation of the target-space action derived from the off-shell sphere fixes the string-frame entropy density. The gravitational replica construction and weighted flow-cut duality then yield the string-frame norm bound. We next find regular stationary F1 solutions in an NS-NS supported BTZ black hole whose spatial parts are logarithmic spirals. A uniform ensemble produces a conserved, constant-norm projected string charge density and realizes a maximal flow whose flux equals the BTZ entropy. We finally extend the construction to BTZ backgrounds with mixed NS-NS and RR flux. Varying the flux fraction changes the stationary F1 configurations at fixed metric and dilaton, while the corresponding conserved flows reproduce the same leading BTZ entropy. Thus, both the norm bound and its physical maximal-flow representative arise from string theory, with stationary strings providing physical channels for the entropic flux represented geometrically by bit threads.