Topology of restricted phase space thermodynamics in Kerr-Sen-Ads black holes

In this study, we investigate the thermodynamic topology of the Kerr-Sen-Ads black hole in restricted phase space. In the restricted phase space, a new parameter, central charge $C$, and its conjugate parameter $\mu$ are introduced, omitting the well-known $PdV$ term in the first law of black hole thermodynamics. We study the local and global topology of the black hole by considering the black hole solution as topological defects in the free energy landscape. We compute the winding number and the total topological number at the thermodynamic defects. For our analysis, we have considered five ensembles of Kerr-Sen-Ads black holes in restricted phase space: fixed $(Q, J, C)$, fixed $(\phi, J, C)$, fixed $(Q,\Omega, C)$, fixed $(Q, J, \mu)$, and fixed $(\phi,\Omega, C)$, where $Q$ is the electric charge, $J$ is the angular momentum, $C$ is the central charge, $\phi$ is the electric potential conjugate to charge, $\Omega$ is the angular frequency conjugate to $J$, and finally, $\mu$ is the chemical potential. In the fixed $(Q, J, C)$, fixed $(\phi, J, C)$, and fixed $(Q, J, \mu)$ ensembles, we find a topological charge of $+1$. In the fixed $(Q,\Omega, C)$ and fixed $(\phi, \Omega, C)$ ensembles, depending on the values of the thermodynamic parameters, we find topological charges of $-1$, $0$, and $+1$. Interestingly, in ensembles where we find the topological charge to be $0$, we observe both Hawking-Page and Davies type phase transitions. We show that both types of these phase transitions can be studied using a common vector field, and the topological charges associated with Davies type and Hawking-Page phase transitions are $-1$ and $+1$, respectively.