On minimal entanglement wedge cross section for holographic entanglement negativity

We demonstrate the equivalence of two different conjectures in the literaturefor the holographic entanglement negativity in AdS$_3$/CFT$_2$, modulo certainconstants. These proposals involve certain algebraic sums of bulk geodesicshomologous to specific combinations of subsystems, and the entanglement wedgecross section (EWCS) backreacted by a cosmic brane for the conical defectgeometry in the bulk gravitational path integral. It is observed that theformer conjectures reproduce the field theory replica technique results in thelarge central charge limit whereas the latter involves constants related to theMarkov gap. In this context we establish an alternative construction for theEWCS of a single interval in a CFT$_2$ at a finite temperature to resolve anissue for the latter proposal involving the thermal entropy elimination for theholographic entanglement negativity. Our construction for the EWCS correctlyreproduces the corresponding field theory results modulo the Markov gapconstant in the large central charge limit.