398
A. Castro
S = 2πk
λ 1 − λ 3
+ other sector
= 2πk
2λ 1 + λ 2
+ other sector ,
(25)
with λ 1 and λ 2 obtained by inverting (24) and choosing the branch of the solution
that connects smoothly to the BTZ black hole as one turns off the Q (3) charge.
4 Extremal Black Holes
In general relativity there is a wide variety of black holes which are not necessarily
Euclidean. For example, there are Lorentzian black holes that do not have a real
Euclidean continuation (such as five-dimensional black rings), there are eternal
black holes (which are the maximal extension of the Euclidean geometry in
Lorentzian signature), and black holes that arise from gravitational collapse. And
there are as well extremal black holes. Extremal black holes have undoubtedly
played a crucial role in string theory: due to their enhanced symmetries and their
capacity to preserve supersymmetry, they have become a landpost for microstate
counting and precursors to many aspects of holography. As such, it is very natural
to wonder what is the definition of extremality in higher spin gravity. This is the
question we will address in this section. The discussion here is a summary of the
results presented in [41]. See also [42] for a discussion on related properties.
4.1 A Practical Definition of Extremality
In conventional gravitational theories, the notion of extremality is tied to the
confluence of two horizons. This feature generically implies that the Hawking
temperature of the black hole is zero. We could declare that extremality in higher
spin theories is simply defined as a solution at zero temperature. However, our aim
is to propose a definition that is along the lines of confluence (degeneration) of the
parameters of the solution and that relies only on the topological formulation of the
theory, yielding in particular the zero temperature condition as a consequence.
In this spirit, in [41] we proposed that a 3d extremal higher spin black hole is a
solution of Chern–Simons theory corresponding to flat boundary connections a and
¯
a satisfying the following conditions:
1. They obey AAdS boundary conditions, 5
2. Their components are constant, and therefore correspond to stationary solutions,
5 In the literature these boundary conditions are commonly known as Drinfeld–Sokolov boundary
conditions.
A. Castro
S = 2πk
λ 1 − λ 3
+ other sector
= 2πk
2λ 1 + λ 2
+ other sector ,
(25)
with λ 1 and λ 2 obtained by inverting (24) and choosing the branch of the solution
that connects smoothly to the BTZ black hole as one turns off the Q (3) charge.
4 Extremal Black Holes
In general relativity there is a wide variety of black holes which are not necessarily
Euclidean. For example, there are Lorentzian black holes that do not have a real
Euclidean continuation (such as five-dimensional black rings), there are eternal
black holes (which are the maximal extension of the Euclidean geometry in
Lorentzian signature), and black holes that arise from gravitational collapse. And
there are as well extremal black holes. Extremal black holes have undoubtedly
played a crucial role in string theory: due to their enhanced symmetries and their
capacity to preserve supersymmetry, they have become a landpost for microstate
counting and precursors to many aspects of holography. As such, it is very natural
to wonder what is the definition of extremality in higher spin gravity. This is the
question we will address in this section. The discussion here is a summary of the
results presented in [41]. See also [42] for a discussion on related properties.
4.1 A Practical Definition of Extremality
In conventional gravitational theories, the notion of extremality is tied to the
confluence of two horizons. This feature generically implies that the Hawking
temperature of the black hole is zero. We could declare that extremality in higher
spin theories is simply defined as a solution at zero temperature. However, our aim
is to propose a definition that is along the lines of confluence (degeneration) of the
parameters of the solution and that relies only on the topological formulation of the
theory, yielding in particular the zero temperature condition as a consequence.
In this spirit, in [41] we proposed that a 3d extremal higher spin black hole is a
solution of Chern–Simons theory corresponding to flat boundary connections a and
¯
a satisfying the following conditions:
1. They obey AAdS boundary conditions, 5
2. Their components are constant, and therefore correspond to stationary solutions,
5 In the literature these boundary conditions are commonly known as Drinfeld–Sokolov boundary
conditions.