Probes in AdS 3 Quantum Gravity
395
r
t E
Fig. 1 Topology of the Euclidean higher spin black hole for a static solution, where the compact
direction is Euclidean time t = it E . The red curve depicts the cycle along which the smoothness
condition (14) is imposed, and it is independent of the radial position. In Euclidean signature, the
geometry ends at a finite value of r: in a metric-like formulation of gravity this end point would be
the horizon
and similarly in the barred sector; here L 0 denotes the Cartan element of sl(2), 3 and
C E is the thermal cycle z ∼ z + 2πiτ which is contractible in the bulk.
The smoothness condition (14) is a robust and successful definition of Euclidean
black holes. It reproduces in an elegant manner many properties that we expect from
a thermal state in the dual CFT 2 . This definition has also unveiled novel properties
of systems in the grand canonical ensemble of W N , such as microscopic features
of the entropy [30, 35, 36], ensemble properties [21, 23], and novel phase diagrams
[37], and it inspires new observables related to entanglement entropy [33, 38, 39].
It is perhaps worth emphasizing that there exist several ways to compute the
entropy of higher spin black holes, all giving the same result. In the original proposal
of [11], the entropy was inferred by demanding integrability of the thermodynamic
laws. For a Hamiltonian derivation of the entropy, see, e.g. [22, 28, 40]. The entropy
can also be understood as the on-shell value of the appropriate action functional in
a microcanonical ensemble, where the charges at infinity are held fixed [21]. The
punchline is that the entropy of a higher spin black hole reads
S = −2πikTr
(a z + a ¯
z ) (τ a z + ¯
τ a ¯
z ) − ( ¯
a z + ¯
a ¯
z ) (τ ¯
a z + ¯
τ ¯
a ¯
z )
.
(15)
More interestingly, one can exploit the holonomy conditions to cast the entropy
directly as function of the charges only. Using the smoothness conditions (14) one
3 Depending on the gauge group, the choice of center in the rhs of (14) is not unique [32].
The choice used here has the feature that it is smoothly connected to the BTZ solution. The
interpretations of other choices are discussed in [33, 34].
395
r
t E
Fig. 1 Topology of the Euclidean higher spin black hole for a static solution, where the compact
direction is Euclidean time t = it E . The red curve depicts the cycle along which the smoothness
condition (14) is imposed, and it is independent of the radial position. In Euclidean signature, the
geometry ends at a finite value of r: in a metric-like formulation of gravity this end point would be
the horizon
and similarly in the barred sector; here L 0 denotes the Cartan element of sl(2), 3 and
C E is the thermal cycle z ∼ z + 2πiτ which is contractible in the bulk.
The smoothness condition (14) is a robust and successful definition of Euclidean
black holes. It reproduces in an elegant manner many properties that we expect from
a thermal state in the dual CFT 2 . This definition has also unveiled novel properties
of systems in the grand canonical ensemble of W N , such as microscopic features
of the entropy [30, 35, 36], ensemble properties [21, 23], and novel phase diagrams
[37], and it inspires new observables related to entanglement entropy [33, 38, 39].
It is perhaps worth emphasizing that there exist several ways to compute the
entropy of higher spin black holes, all giving the same result. In the original proposal
of [11], the entropy was inferred by demanding integrability of the thermodynamic
laws. For a Hamiltonian derivation of the entropy, see, e.g. [22, 28, 40]. The entropy
can also be understood as the on-shell value of the appropriate action functional in
a microcanonical ensemble, where the charges at infinity are held fixed [21]. The
punchline is that the entropy of a higher spin black hole reads
S = −2πikTr
(a z + a ¯
z ) (τ a z + ¯
τ a ¯
z ) − ( ¯
a z + ¯
a ¯
z ) (τ ¯
a z + ¯
τ ¯
a ¯
z )
.
(15)
More interestingly, one can exploit the holonomy conditions to cast the entropy
directly as function of the charges only. Using the smoothness conditions (14) one
3 Depending on the gauge group, the choice of center in the rhs of (14) is not unique [32].
The choice used here has the feature that it is smoothly connected to the BTZ solution. The
interpretations of other choices are discussed in [33, 34].