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 conformal field theory 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.
