Topological dyonic Taub-Bolt/NUT-AdS: Thermodynamics and first law

Motivated by the absence of Misner string in the Euclidean Taub-Bolt/NUT solutions with flat horizons, we present a new treatment for studying the thermodynamics of these spactimes. This treatment is based on introducing a new charge, $N=\sigma \, n$ (where $n$ is the nut charge and $\sigma$ is some constant) and its conjugate thermodynamic potential $\Phi_N$. Upon identifying one of the spatial coordinates, the boundary of these solutions contains two annulus-like surfaces in addition to the constant-r surface. For these solutions, we show that these annuli surfaces receive electric, magnetic and mass/energy fluxes, therefore, they have nontrivial contributions to these conserved charges. Calculating these conserved charges we find, $Q_e = Q^{\infty}_e-2N\Phi_m$, $Q_m =Q^{\infty}_m+2N\Phi_e$ and $\mathfrak{M} =M-2N\Phi_N$, where $Q^{\infty}_e$, $Q^{\infty}_m$, $M$ are electric charge, magnetic charge and mass in the $n=0$ case, while $\Phi_e$ and $\Phi_m$ are the electric and magnetic potentials. The calculated thermodynamic quantities obey the first law of thermodynamics while the entropy is the area of the horizon. Furthermore, all these quantities obey Smarr's relation. We show the consistency of these results through calculating the Hamiltonian and its variation which reproduces the first law.