ModMax-AdS Black Holes with a Cloud of String immersed in Perfect Fluid Dark Matter and quintessence field: Thermodynamics and Photon Sphere

In this work, we present a detailed study of the thermodynamic and photon sphere of ModMax-AdS black holes surrounded by a string cloud, perfect fluid dark matter (PFDM), and a quintessence field (QF). Within the framework of extended phase-space thermodynamics, where the cosmological constant is interpreted as thermodynamic pressure, we derive the fundamental thermodynamic quantities, including the Hawking temperature, Gibbs free energy, internal energy, and specific heat at constant pressure. We also formulate the corresponding first law by incorporating the effects of the string cloud, PFDM, and quintessence field through their associated thermodynamic conjugates. We further investigate critical behavior by determining the critical temperature, pressure, and volume, and analyzing their dependence on the model parameters. The resulting equation of state exhibits van der Waalslike phase transition behavior, together with a rich thermodynamic stability structure. In addition, we explore the thermodynamic topology of the system using topological charge methods, providing insight into the global structure of the phase space. Finally, we analyze null geodesics and determine the photon-sphere radius, along with its associated topological properties. We show how the presence of the string cloud, PFDM, and quintessence field modifies the photon sphere and influences its topological charge, thereby encoding information about the underlying spacetime geometry