  We study the geometric action of some modular conjugations in two dimensional
(2D) conformal field theories. We investigate the bipartition given by an
interval when the system is in the ground state, either on the line or on the
circle, and in the thermal Gibbs state on the line. We find that the
restriction of the corresponding inversion maps to a spatial slice is obtained
also in the gauge/gravity correspondence through the geodesic bit threads in a
constant time slice of the dual static asymptotically AdS background. For a CFT
in the thermal state on the line, the modular conjugation suggests the
occurrence of a second world which can be related through the geodesic bit
threads to the horizon of the BTZ black brane background. An inversion map is
constructed also for the massless Dirac fermion in the ground state and on the
line bipartite by the union of two disjoint intervals.
