On 3d Bulk Geometry of Virasoro Coadjoint Orbits: Orbit invariant charges and Virasoro hair on locally AdS3 geometries

TL;DR

<3-5 sentence high-level summary> This paper analyzes the most general locally AdS3 solutions (Bañados geometries) under Brown-Henneaux boundary conditions, encoding them by periodic left/right data $L_+(x^+)$ and $L_-(x^-)$ and organizing their physical content into orbit-invariant geometric data and Virasoro hair. It develops two families of conserved charges—Symplectic Exact Symmetries (SES) and Symplectic Non-exact Symmetries (SNS)—which furnish left/right Virasoro algebras and Killing-vector charges, respectively, and shows a one-to-one map between Virasoro coadjoint orbits and Bañados geometries. The work derives horizon structure, entropy, and first-law relations in terms of Floquet indices ${}$ and horizon data, and classifies geometries according to constant vs non-constant orbit representatives, including multi-BTZ-like configurations. This framework positions Virasoro coadjoint orbits as a natural phase-space classification for AdS3 gravity, suggesting a path toward quantization and a refined link to the dual 2d CFT through orbit invariants and hair.

Abstract

Expanding upon [arXiv:1404.4472, 1511.06079], we provide further detailed analysis of Bañados geometries, the most general solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions. We analyze in some detail the causal, horizon and boundary structure, and geodesic motion on these geometries, as well as the two class of symplectic charges one can associate with these geometries: charges associated with the exact symmetries and the Virasoro charges. We elaborate further the one-to-one relation between the coadjoint orbits of two copies of Virasoro group and Bañados geometries. We discuss that the information about the Bañados goemetries fall into two categories: "orbit invariant" information and "Virasoro hairs". The former are geometric quantities while the latter are specified by the non-local surface integrals. We elaborate on multi-BTZ geometries which have some number of disconnected pieces at the horizon bifurcation curve. We study multi-BTZ black hole thermodynamics and discuss that the thermodynamic quantities are orbit invariants. We also comment on the implications of our analysis for a 2d CFT dual which could possibly be dual to AdS3 Einstein gravity.

Figures (4)

• Figure 1: Penrose diagram for generic Bañados geometry. To draw this causal diagram we have used the analysis of the simpler case of BTZ (cf.) appendix A.1) and the analysis made in this section. In the Penrose diagram, as usual, we have suppressed a spacelike compact direction (here the one along $\Phi_+$ or $\Phi_-$ coordinates). Our discussions in this section reveal that Penrose diagrams for generic $n_\pm$ are essentially the same as those of usual BTZ geometries discussed in BTZ. However, we cut the regions which have CTCs and we should make appropriate identifications, and hence we remain with a geometry whose Penrose diagram is $(n_++1)(n_-+1)$ multiple repetition of that of a single BTZ geometry (Figure \ref{['rho-vs-r-BTZ']}).
• Figure 2: Left: Coordinate transformation \ref{['coortran']} plotted for the BTZ case of $L_+> L_->0$. Vertical axis denotes $\rho^2$ and horizontal axis $r^2$. The (red) dotted line, where $\rho^2<0$, is the location of CTC. Therefore, the CTC-free region in the Bañados coordinate system is $-\ell^2 L_+ <r^2< -\ell^2 L_-$ and $r^2>0$. The region on vertical axis in gray color, $\rho_-^2<\rho^2<\rho_+^2$, is not covered in the Bañados coordinate system, while the other $\rho^2>0$ regions are covered twice. The four regions $-\ell^2 L_+ <r^2\leq -r_0^2,\ -r^2_0\leq r^2<-\ell^2 L_-,\ 0<r^2\leq r_0^2$ and $r^2\geq r_0^2$ (with $r_0^2=\ell^2\sqrt{L_+L_-}$) which are also color-coded in the Left figure, correspond to the four regions, four diamonds, on the Penrose diagram (Right). Right: Penrose diagram for the BTZ case of $L_+> L_->0$BTZLoran. The region II (which lies between the inner and outer horizons) is not covered in the Bañados coordinate system. The regions I and I' respectively corresponds to $0<r^2\leq r_0^2$ and $r^2\geq r_0^2$ regions and the regions III and III' to $-\ell^2 L_+ <r^2\leq -r_0^2,\ -r^2_0\leq r^2<-\ell^2 L_-$. The shaded regions are where we have CTC's and correspond to the (red) dotted regions in the Left figure. We have used the same color-coding in the Left and Right figures to indicate the range of $r^2$ coordinate. This figure shows how the Bañados and BTZ coordinate systems are complementary to each other.
• Figure 3: Coordinate transformation \ref{['coortran']} plotted for the conic space with $L_+<L_- <0$. CTC region, where $\rho^2<0$, is denoted by (red) dotted curve. Extending coordinate to negative $r^2$ gives CTC. Positive values of $r^2$ with $r^2>-\ell^2L_+$ gives one cover denoted by yellow colour in above figure and $r^2<-\ell^2 L_-$ gives another cover which is denoted by by pink colour. Region $-\ell^2 L_-<r^2<-\ell^2L_+$ gives CTC. Therefore, in the CTC-free range there is no horizon. This is compatible with the fact that conic spaces correspond to particles on AdS$_3$ and not black holes.
• Figure 4: Coordinate transformation \ref{['coortran']} plotted for $L_+>0$ and $L_-<0$. The regions $r^2>-\ell^2 L_-$ and $-\ell^2 L_+ <r^2<0$ are the CTC-free regions. The geometry does not have horizon in this region. The two CTC-free pieces both correspond to the same coordinate range $\rho^2>0$ in the BTZ-coordinate system.