Probes in AdS 3 Quantum Gravity
393
solutions and its properties in the Chern–Simons formulation of higher spin gravity.
This section is a collection of results in [11, 15, 20–23].
Any definition of black holes should include at least two inputs:
1. A quantitative definition of physical observables; in particular, a definition of
conserved charges (such as mass and angular momentum) and its counterparts
potentials (such as temperature and angular velocity).
2. A notion of regularity and smoothness. The aim here would be to find a notion
of horizon. But even more broadly, we need to clearly argue if a solution, at least
in Euclidean signature, lacks singularities.
Let us elaborate first on how to obtain conserved charges. The properties and
values of these observables are intimately tied to the boundary conditions we use.
For instance, in AdS spacetimes we are mostly accustomed to Dirichlet boundary
conditions and to implement a notion of asymptotically AdS spaces (AAdS). But
let me emphasize: there is more than one choice! This occurs even in AdS 3 gravity,
where some non-trivial examples are shown in [24, 25] and more recently a broad
analysis was presented in [26] which are important deviations from the standard
Brown–Henneaux boundary conditions [27].
In higher spin gravity we of course have similar choices, but in addition there
are further complexities as we turn on sources. More concretely, consider the
Chern–Simons connections in (7) and (8), and that we impose AAdS boundary
conditions. From the CFT perspective, it is natural to capture the currents in a z
and the sources in a ¯
z , and vice versa for ¯
a [11]. From the gravitational perspective,
the canonical prescription is to encode in (a φ , ¯
a φ ) the currents [28–31]. These two
choices, a z versus a φ , amount for different partition functions as shown in [23]:
the a z prescription, denoted holomorphic black hole, corresponds to a Lagrangian
deformation of the theory; the a φ prescription, denoted canonical black hole,
corresponds to a Hamiltonian deformation. It is important to make a distinction
between these two, since the Legendre transformation that connects these two
prescriptions is non-trivial.
In the remainder of these lectures we will mostly use the canonical description.
Moreover, we are interested in stationary black hole solutions, hence (a, ¯
a) are
constant flat connections that contain both charges and sources. This in particular
implies that the φ-component will be always written as
a φ = L 1 −
N
s=2
Q (s) W
(s)
−s+1 , ¯
a φ = L −1 −
N
s=2
¯
Q (s) W
(s)
s−1 ,
(9)
where (Q (s) , ¯
Q (s) ) are constants (not functions) and they represent the conserved
charges associated with the zero modes of each higher spin current (J (s) , ¯
J (s) ).
The a t component will contain the information about the potentials, which we will
393
solutions and its properties in the Chern–Simons formulation of higher spin gravity.
This section is a collection of results in [11, 15, 20–23].
Any definition of black holes should include at least two inputs:
1. A quantitative definition of physical observables; in particular, a definition of
conserved charges (such as mass and angular momentum) and its counterparts
potentials (such as temperature and angular velocity).
2. A notion of regularity and smoothness. The aim here would be to find a notion
of horizon. But even more broadly, we need to clearly argue if a solution, at least
in Euclidean signature, lacks singularities.
Let us elaborate first on how to obtain conserved charges. The properties and
values of these observables are intimately tied to the boundary conditions we use.
For instance, in AdS spacetimes we are mostly accustomed to Dirichlet boundary
conditions and to implement a notion of asymptotically AdS spaces (AAdS). But
let me emphasize: there is more than one choice! This occurs even in AdS 3 gravity,
where some non-trivial examples are shown in [24, 25] and more recently a broad
analysis was presented in [26] which are important deviations from the standard
Brown–Henneaux boundary conditions [27].
In higher spin gravity we of course have similar choices, but in addition there
are further complexities as we turn on sources. More concretely, consider the
Chern–Simons connections in (7) and (8), and that we impose AAdS boundary
conditions. From the CFT perspective, it is natural to capture the currents in a z
and the sources in a ¯
z , and vice versa for ¯
a [11]. From the gravitational perspective,
the canonical prescription is to encode in (a φ , ¯
a φ ) the currents [28–31]. These two
choices, a z versus a φ , amount for different partition functions as shown in [23]:
the a z prescription, denoted holomorphic black hole, corresponds to a Lagrangian
deformation of the theory; the a φ prescription, denoted canonical black hole,
corresponds to a Hamiltonian deformation. It is important to make a distinction
between these two, since the Legendre transformation that connects these two
prescriptions is non-trivial.
In the remainder of these lectures we will mostly use the canonical description.
Moreover, we are interested in stationary black hole solutions, hence (a, ¯
a) are
constant flat connections that contain both charges and sources. This in particular
implies that the φ-component will be always written as
a φ = L 1 −
N
s=2
Q (s) W
(s)
−s+1 , ¯
a φ = L −1 −
N
s=2
¯
Q (s) W
(s)
s−1 ,
(9)
where (Q (s) , ¯
Q (s) ) are constants (not functions) and they represent the conserved
charges associated with the zero modes of each higher spin current (J (s) , ¯
J (s) ).
The a t component will contain the information about the potentials, which we will