Aspects of $T\bar{T}+J\bar{T }$ deformed Schwarzian: From gravity partition function to late-time spectral form factor

Abstract: In this paper, we investigate different thermodynamic properties of $T\bar{T}+J\bar{T}$ deformed Schwarzian theory and its different gravitational perspectives. First, we compute the partition function of $U(1)$ coupled 2D-gravity with fixed chemical potential, obtained from the dimensional reduction of the four-dimensional Einstein-Maxwell theory. Then, we compute the partition function of the gravity theory which is the dual to the deformed Schwarzian living on its boundary and study the genus expansion of the one and two-point correlation function of the partition function of the theory. Subsequently, we use the one-point function to compute the "Annealed" and "Quenched" free energy in low-temperature limits and make a qualitative comparison with the undeformed theory. Then, using the two-point function, we compute the Spectral Form Factor of the deformed theory in early and late time. We find a dip and ramp structure in early and late time, respectively. We also get a plateau structure in the $\tau$-scaling limit. Last but not least, we comment on the late-time topology change to give a physical interpretation of the ramp of the Spectral Form Factor for our theory.