Topology of Born-Infeld-AdS Black Hole Phase Transitions: Bulk and CFT Sides

The thermodynamic criticality of the AdS black holes serves as an important structure during the thermal phase transition. This paper discusses about the critical points and their topology during thermal phase transitions of the Born-Infeld AdS black holes. We make such investigations using two different topological approaches, namely, using Duan's topological current $\phi$-mapping theory, and the off-shell free energy. Within Duan's formalism, we observe that for a given value of the Born-Infeld parameter $b$, there exists an associated electric charge parameter $Q$, which is highly sensitive to the topological phase transitions. This way we examine the connections of the first-order phase transition and the topological nature of the critical points. We find that the topological nature has a possible breakdown in certain parametric ranges. In effect, we determine the unconventional and the conventional phase critical points as the creation (topologically vortex) and annihilation (topologically anti-vortex) points (pairs). As the second approach, we call the off-shell free energy to determine the topological classes: of which one corresponds to the AdS-Schwarzschild black hole phases, while the other provides a possible topological phase transition. Here we also reveal a novel phase transition between two unstable phases, namely, the unstable small black hole and the intermediate black holes. For a certain parametric values of the Born-Infeld parameter and the pressure, we also study the different topological descriptions that inevitably correspond to the AdS-Reissner-Nordstr$\Ddot{o}$m black hole phases. As a consistency check of the critical points during the topological phase transitions, we study the vortex/anti-vortex annihilation thermodynamics from local as well as global thermodynamic viewpoint.