Boundary Conditions for Extremal Black Holes from 2d Gravity

1. Connection between two and higher dimensional asymptotic symmetries.

2. Novel asymptotic symmetries for Kerr and Kerr-de Sitter spacetimes.

1. Physical meaning of results not entirely clear.

2. Presentation is not very clear.

The paper explores novel boundary conditions for black hole type spacetimes in three and four spacetime dimensions. The approach takes inspiration from a variety of analyses in two-dimensions leading to interesting asymptotic symmetry algebras. Of particular interest is the application to certain limits of Kerr-de Sitter black holes, which have not been analysed in the literature in much detail.

1. Provide a synthesis and summary of results, compared and contrasted to previous cases.

2. Make note of cases when central charges have non-standard reality properties, or are vanishing.

3. Provide some outlook on any uniqueness of lack thereof of the new boundary conditions.

New boundary conditions and asymptotic symmetry algebra are found for black holes in three and four dimensions including extremal Kerr black holes.

lack of physical interpretations

Boundary conditions are part of the definition of a quantum theory of gravity, and play an important role in the bottom-up approach of holography duality. The paper finds new boundary conditions for the near-horizon geometries of extremal black holes in three dimensions and four dimensions. The work is useful for understanding the holographic dual of extremal black holes including Kerr black holes. At the technical level, the paper is a higher dimensional uplift of boundary conditions in two dimensions. Expressions of the boundary conditions and asymptotic symmetry algebra have been written down explicitly and are useful for further study. The weak point of the paper is that there are not enough interpretations of the assumptions and results. I would like to recommend it for publication, provided that the authors consider the requested changes below.

1-In order to perform the mode expansion, the authors assume that $\tau$ in eq(34) is periodic. The authors should explain why such an identification is reasonable in a Lorentzian geometry, and what the implication in the putative dual theory. Relatedly, the central charges in eq.(25) depend on the Length $L$ which is arbitrarily chosen. The authors should explain why such central charges are physical.
2-The authors should provide more explanations on boundary conditions in different coordinates. Are these boundary conditions equivalent in the sense that they define the same quantum theory? If these boundary conditions are not equivalent, is there a preferred one? What is the interpretation of the relation eq.(39) that relates charges in different coordinates? How do we understand these from the perspective of the holographic dual?
3- What is the interpretation of imaginary level $\kappa$ such as in eq.(61)? Is the resulting dual theory non-unitary? If so, how do we make sense of it ?