We study the geometry of $Tbar{T}$-deformed BTZ black hole and find it can
be regarded as a quotient of hyperbolic space. We then consider the massive
scalar field propagating in the $Tbar{T}$-deformed BTZ black hole background.
The one-loop partition function of scalar field is calculated using the heat
kernel method and the Wilson spool proposal. These two methods give consistent
result which implies the Wilson spool proposal still holds under $Tbar{T}$
deformation. Moreover, we also calculate the one-loop partition function of
graviton in $Tbar{T}$-deformed BTZ black hole. We find the deformed one-loop
partition functions are modified in a simple way, which corresponds to a
replacement of the modular parameter. The result precisely matches the large
$c$ expansion of $Tbar{T}$-deformed CFT partition function. These results
provide a further check about the correspondence between $Tbar{T}$-deformed
CFT$_2$ and AdS$_3$ with mixed boundary condition.
Examining the Conclusions and Outlining a Future Roadmap
The study of the geometry of the $Tbar{T}$-deformed BTZ black hole has yielded interesting results that provide insight into the connection between $Tbar{T}$-deformed CFT (Conformal Field Theory) and AdS (Anti-de Sitter) spacetime with mixed boundary condition. In addition, the calculations of the one-loop partition functions for the massive scalar field and graviton in the deformed black hole background have shown consistent results and modifications that align with the $Tbar{T}$ deformation.
Potential Challenges
- Further investigation is required to understand the precise implications and consequences of the replacement of the modular parameter and its impact on the deformed one-loop partition functions.
- Exploring the physical interpretation of the modified partition functions and their connection to other aspects of quantum gravity and quantum field theory.
- Understanding the broader implications of the consistent results obtained from both the heat kernel method and the Wilson spool proposal.
Potential Opportunities
- Utilizing the insights gained from this study to further explore the correspondence between $Tbar{T}$-deformed CFT$_2$ and AdS$_3$ with mixed boundary condition.
- Investigating the potential applications of these findings in other areas of theoretical physics, such as holography and black hole physics.
- Considering the implications for quantum information theory and quantum gravity, particularly in relation to entanglement entropy and the holographic principle.
In summary, the examination of the $Tbar{T}$-deformed BTZ black hole geometry and the calculation of the one-loop partition functions for scalar fields and gravitons have provided valuable insights into the correspondences between $Tbar{T}$-deformed CFT and AdS spacetime with mixed boundary condition. However, further exploration and investigation are needed to fully understand the implications of these results and to explore their broader applications in theoretical physics.