402
A. Castro
It is clear that (36) and (37) do not agree. However, the W 3 unitarity
bound (37) encloses the bulk extremality bound (36), indicating that all sl(3)
black holes are dual to states allowed by unitarity in the dual CFT.
5. Conformal invariance: In two-derivative theories of gravity in D = 4, 5 all
extremal black holes contain an AdS 2 factor in its near horizon geometry [44, 45].
The enhancement of time translations to conformal transformations is non-trivial
and unexpected a priori; moreover, it is key to build microscopic models of
extremal black holes. Here we have not investigated this feature explicitly, but
we do expect that the connection at the extremal point is invariant a larger set of
gauge transformations relative to the non-extremal connection. Some evidence
was reported in [18].
6. Entropy bounds: The extremal limit of the spin-3 higher spin black hole was first
discussed in [11]. Their bound was found as the maximal value of Q (3) for a
given Q (2) such that the entropy is real, and it agrees with (32). Using the reality
of entropy as a bound which enhanced symmetries of the solution was also used
in [42]. It is not clear if this approach is always compatible with ours, and it will
be interesting to explore potential discrepancies.
7. Supersymmetry and Extremality: As we mentioned above, extremality can be
understood as the saturation of certain inequalities involving conserved charges,
and it is natural to contrast these inequalities with BPS bounds that appear
in supersymmetric setups. It is well known that in two-derivative theories of
supergravity these two types of conditions are intimately related: supersymmetry
always implies zero temperature and therefore extremality in the context of BPS
black holes. In supersymmetric theories of higher spin gravity this seems to not
be true!! In [41] we showed that there exist non-extremal solutions in the class of
diagonalizable connections that possess 4 independent Killing spinors. This is,
within the sl(3|2) theory, we managed to construct a smooth higher spin black
hole that is both at finite temperature and BPS. Understanding why higher spin
theories allow for this peculiar behavior is an open question that needs urgent
attention.
5 Wilson Lines
As we have mentioned throughout, higher spin gravity does not admit a conventional
geometric understanding. However, they do admit interesting higher spin invariant
probes. In this section we will consider the Wilson line operator constructed in [39,
46]. This object will allow us to address two important questions:
1. How to capture casual properties in higher spin gravity?
2. How to probe local bulk physics using the Chern–Simons formulation of gravity?
In the following we will only provide the basic definitions, features, and main results
obtained in this area. For a detailed discussion we refer to [16, 18, 39, 46].
A. Castro
It is clear that (36) and (37) do not agree. However, the W 3 unitarity
bound (37) encloses the bulk extremality bound (36), indicating that all sl(3)
black holes are dual to states allowed by unitarity in the dual CFT.
5. Conformal invariance: In two-derivative theories of gravity in D = 4, 5 all
extremal black holes contain an AdS 2 factor in its near horizon geometry [44, 45].
The enhancement of time translations to conformal transformations is non-trivial
and unexpected a priori; moreover, it is key to build microscopic models of
extremal black holes. Here we have not investigated this feature explicitly, but
we do expect that the connection at the extremal point is invariant a larger set of
gauge transformations relative to the non-extremal connection. Some evidence
was reported in [18].
6. Entropy bounds: The extremal limit of the spin-3 higher spin black hole was first
discussed in [11]. Their bound was found as the maximal value of Q (3) for a
given Q (2) such that the entropy is real, and it agrees with (32). Using the reality
of entropy as a bound which enhanced symmetries of the solution was also used
in [42]. It is not clear if this approach is always compatible with ours, and it will
be interesting to explore potential discrepancies.
7. Supersymmetry and Extremality: As we mentioned above, extremality can be
understood as the saturation of certain inequalities involving conserved charges,
and it is natural to contrast these inequalities with BPS bounds that appear
in supersymmetric setups. It is well known that in two-derivative theories of
supergravity these two types of conditions are intimately related: supersymmetry
always implies zero temperature and therefore extremality in the context of BPS
black holes. In supersymmetric theories of higher spin gravity this seems to not
be true!! In [41] we showed that there exist non-extremal solutions in the class of
diagonalizable connections that possess 4 independent Killing spinors. This is,
within the sl(3|2) theory, we managed to construct a smooth higher spin black
hole that is both at finite temperature and BPS. Understanding why higher spin
theories allow for this peculiar behavior is an open question that needs urgent
attention.
5 Wilson Lines
As we have mentioned throughout, higher spin gravity does not admit a conventional
geometric understanding. However, they do admit interesting higher spin invariant
probes. In this section we will consider the Wilson line operator constructed in [39,
46]. This object will allow us to address two important questions:
1. How to capture casual properties in higher spin gravity?
2. How to probe local bulk physics using the Chern–Simons formulation of gravity?
In the following we will only provide the basic definitions, features, and main results
obtained in this area. For a detailed discussion we refer to [16, 18, 39, 46].