Kinematic space has been defined as the space of
codimension-

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2$. As far as we know, these have not been studied in depth. It would be interesting to study these orbits and their quantization further in future work. Finally, note that this is not the first time coadjoint orbits have appeared in holography; in the context of 3d gravity, for example, see~\cite{Maloney-ml-2007ud, Oblak-ml-2017ect, Barnich-ml-2017jgw, Cotler-ml-2018zff}, or in AdS$_2$/CFT$_1$ with connections to the SYK model see~\cite{Alekseev-ml-1988ce, Mandal-ml-2017thl, Stanford-ml-2017thb}. Coadjoint orbits of the Virasoro group have recently been used to define computational complexity in two-dimensional CFT~\cite{Caputa-ml-2018kdj}. Coadjoint orbits have also been used to study the asymptotic symmetry groups of asymptotically flat spacetime~\cite{Duval-ml-2014lpa,Barnich-ml-2014kra,Barnich-ml-2015uva,Oblak-ml-2015sea}. Our work differs from these, however, because we consider coadjoint orbits of the conformal group rather than the full asymptotic symmetry group. The resulting physical interpretation is also very different: we consider a phase space of geodesics on a fixed background, whereas coadjoint orbits of the full asymptotic symmetry group should be interpreted as a phase space of different \emph{metrics}. \paragraph{Outline:} the paper is structured as follows. We begin in section~\ref{sec:orbits} with a summary of the orbit method, including an illustration of these methods in the simple example of $\SO(3)$. In section~\ref{sec:symplectic}, we apply the orbit method to the case of a bulk AdS$_3$ spacetime where the kinematic space of geodesics on a slice is simply dS$_2$. We show that the Crofton form that computes bulk lengths matches the Kirillov-Kostant symplectic form on the coadjoint orbit, and discuss the K\"{a}hler structure. Section~\ref{sec:higherdimensions} gives a partial generalization of these results to the $d$-dimensional case. We conclude by discussing some directions for future work, including implications for generalizing kinematic space beyond highly symmetric cases. In appendix~\ref{sec:timelike} we discuss the case of the space of timelike geodesics, which is equal to a different coadjoint orbit of the conformal group so that much of these techniques can be applied there as well. ]]>

= -\frac{1}{2} \mbox{tr}(L_i L_j) = \delta_{ij}~.\ee We identify $\mathfrak{so}(3)\cong \mathfrak{so}(3)^*$ using the Killing form. The dual Lie algebra is $\lieg^*\cong \mathbb{R}^3$, and we can further identify $\mathfrak{so}(3)\cong \mathbb{R}^3$, in which case the Lie bracket is just the cross product. Each point represents an angular momentum vector, $\vec{J}=(J_1,J_2,J_3)$, for the rigid body. The total angular momentum, $J^2$, is conserved in time. So rigid body dynamics lies on two-spheres of the foliation $\mathbb{R}^3\cong S^2 \times \mathbb{R}^+$ of constant total angular momentum. Likewise, the coadjoint action, $g\cdot J = gJ$, is just the ordinary action of the rotation group on $\mathbb{R}^3$ and the orbits are just the two-spheres of the foliation. It is important to note that the definition of coadjoint orbits does not depend on our choice of Hamiltonian: any Hamiltonian flow on $\lieg^*$ must lie on a coadjoint orbit. From~\eqref{eq:kirillov} we can compute the Kirillov-Kostant symplectic form. It is just the area element on the two-sphere divided by its radius, $|\mu|$, \be \omega_K = \frac{dA}{|\mu|} = |\mu|\sin \theta d\theta \wedge d\phi\,,\ee where $\theta$ and $\phi$ are the usual polar angles on the sphere. Indeed, the area element acts on tangent vectors as \beq dA(\tilde{\xi},\tilde{\xi'}) = \hat{\mu} \cdot (\tilde{\xi} \times \tilde{\xi}')\,, \eeq where $\hat{\mu}$ is the unit outward normal at the point $\mu$ on the sphere. Identify $\tilde{\xi} = \xi \times \mu$ and $\tilde{\xi}' = \xi' \times \mu$. Then \beq\label{eq:proof} dA(\tilde{\xi},\tilde{\xi'}) = \hat{\mu} \cdot [(\xi\times\mu) \times (\xi'\times\mu)] = |\mu| \mu \cdot (\xi\times \xi') = |\mu| \omega_K(\tilde{\xi},\tilde{\xi'})~. \eeq Later, when we turn to kinematic space, we will study coadjoint orbits of $\SO(2,1)$. Again we will find $\omega_K=dA/|\mu|$, where $dA$ is the area form on the orbit. ]]>

0$. Let $q=e^\zeta$. The result is \beq \chi_\ell(q^{\partial_\eta}) = -\frac{q^{i\ell+1/2} + q^{-i\ell+1/2}}{1-q} = \frac{q^{i\ell}+q^{-i\ell}}{q^{1/2}-q^{-1/2}}~. \eeq These are precisely the characters of the $\SO(2,1)$ principal series~\cite{Vergne}. The characters blow up as $q\rightarrow 1$ because the representations are infinite dimensional. This is reflected in the infinite area of the hyperboloids. To match notation with Vergne~\cite{Vergne}, identify $\ell=s/2$ (the factor of 2 appears because our definition of the symplectic form differs from Vergne's by a factor of 2). Furthermore, recall the Lie group isomorphism $\SL(2,\mathbb{R})/\mathbb{Z}_2 \cong \SO(2,1)$ and identify the Lie algebra, $\mathfrak{sl}(2,\mathbb{R})$, with $\mink_3$ via the map \beq \tilde{X}=\begin{pmatrix} X_2 & X_1+X_0 \\ X_1-X_0 & -X_2 \\ \end{pmatrix} \leftrightarrow (X_0,X_1,X_2)~. \eeq Now consider the adjoint action, $\tilde{X}\rightarrow g_\lambda\tilde{X}g_\lambda^{-1}$, of \beq\label{eq:sl2boost} g_\lambda=\begin{pmatrix} e^\lambda & 0 \\ 0 & e^{-\lambda} \\ \end{pmatrix} \in \SL(2,\mathbb{R}) \eeq on $\mathfrak{sl}(2,\mathbb{R})$. This sends \begin{align} X_0 &\rightarrow X_0 \cosh 2\lambda + X_1 \sinh 2\lambda\,, \notag\\ X_1 &\rightarrow X_1 \cosh 2\lambda + X_0 \sinh 2\lambda\,, \notag\\ X_2 &\rightarrow X_2~. \end{align} In other words, $g_\lambda$ acts as a Lorentz boost with rapidity $\zeta=-2\lambda$. Trading parameters relates our notation and Vergne's. ]]>

2$, are not symplectic.} We will show that the kinematic space for AdS$_{d+1}$ is a coadjoint orbit of $\SO(d,2)$. It is promising that it takes the form of~\eqref{cosetspace}, since the coadjoint orbits of a Lie group, $G$, are homogenous spaces of the form $G/K$, where $K$ is a subgroup of $G$. When $G$ is compact and connected,\footnote{Proposition 5.3 of~\cite{kirillov2004lectures}.} coadjoint orbits correspond precisely to those subgroups $K\subset G$ containing the maximal torus of $G$. Since $\SO(d,2)$ is not compact, it is not so obvious that $\Gamma_{2d}$ is a coadjoint orbit. To check this, we need to find a coadjoint vector whose stabilizer is $\SO(d-1,1)\times \SO(1,1)$. First, note that $\mathfrak{so}(d,2)$ has a nondegenerate invariant bilinear form, $\tr(XY)$ (where $X,Y\in\mathfrak{so}(d,2)$), so linear functions on $\mathfrak{g}$ may be identified with elements of $\mathfrak{g}$ and adjoint and coadjoint vectors can be identified. So our task is to find an adjoint vector whose stabilizer is $\SO(d-1,1)\times \SO(1,1)$. We will work infinitesimally and exhibit an adjoint vector whose stabilizer is $\mathfrak{so}(d-1,1)\times \mathfrak{so}(1,1)$. To begin, let \be\label{eq:gform} g\equiv{\rm diag}(\underbrace{-1,\dots,-1}_{p},\underbrace{+1,\dots,+1}_{q})~. \ee The group $O(p,q)$ is the set of $n\times n$ matrices $A$ satisfying \be A^T g = g A^{-1}\,, \ee where $n=p+q$. So the Lie algebra $\mathfrak{o}(p,q)$ is the set of $n\times n$ matrices $X$ satisfying \be X^T g = -gX~. \ee $X_{ij}$ is antisymmetric for $0