Probes in AdS 3 Quantum Gravity
399
3. They carry charges and chemical potentials, which are manifestly real in the
Lorentzian section,
4. The angular component of at least one of a and ¯
a, say a φ , is non-diagonalizable.
Naturally, the key point of the definition is the non-diagonalizability of the a φ
component. The rationale behind this requirement is as follows. Suppose both the
a φ and ¯
a φ components were diagonalizable. Since the boundary connections are
assumed to be constant, by the equations of motion the (Euclidean) time components
of the connection commute with the angular components, and can be diagonalized
simultaneously with them. It is then possible to solve (14) and find a non-zero and
well-defined temperature and chemical potentials as function of the charges. On
the other hand, if at least one of a φ and ¯
a φ is non-diagonalizable, then a contract
will be non-diagonalizable as well. If we insist upon (14), then both features are
compatible if we take a zero temperature limit, because the smoothness condition
becomes degenerate as well. This is consistent with the usual notion that the solid
torus topology of the finite-temperature black hole should change at extremality.
The role of boundary conditions is crucial for our definition. For a general
connection the degeneration of eigenvalues does not imply non-diagonalizability.
However, the special form of the flat connections dictated by the AAdS boundary
conditions will guarantee that if two eigenvalues of a φ are degenerate, then the
connection is non-diagonalizable. From this perspective, we could interpret that
equating eigenvalues of a φ is in a sense analogous to the confluence of horizons
for extremal black holes in general relativity.
4.2 Example: Extremal sl(3) Black Holes
In [41] several supersymmetric and non-supersymmetric cases were studied. For
brevity, here we will only look at the extremal cousin of the Euclidean black hole
we considered in 3.1.
Let us write again (17) but will focus on the unbarred sector for concreteness;
recall that it is sufficient to impose our definition of extremality on one sector to
obtain the desired features. Using canonical boundary conditions, the connections
are given by
a φ = L 1 − Q (2) L −1 −
Q (3)
4
W −2 ,
(26)
ia t E + a φ = 2a − = 2μ 3
W 2 + 2Q (3) L −1 + Q
2
(2) W −2 − 2Q (2) W 0
.
(27)
It is also instructive to re-write the solutions to (14) for the general rotating case, i.e.
the generalization of (20). This gives
399
3. They carry charges and chemical potentials, which are manifestly real in the
Lorentzian section,
4. The angular component of at least one of a and ¯
a, say a φ , is non-diagonalizable.
Naturally, the key point of the definition is the non-diagonalizability of the a φ
component. The rationale behind this requirement is as follows. Suppose both the
a φ and ¯
a φ components were diagonalizable. Since the boundary connections are
assumed to be constant, by the equations of motion the (Euclidean) time components
of the connection commute with the angular components, and can be diagonalized
simultaneously with them. It is then possible to solve (14) and find a non-zero and
well-defined temperature and chemical potentials as function of the charges. On
the other hand, if at least one of a φ and ¯
a φ is non-diagonalizable, then a contract
will be non-diagonalizable as well. If we insist upon (14), then both features are
compatible if we take a zero temperature limit, because the smoothness condition
becomes degenerate as well. This is consistent with the usual notion that the solid
torus topology of the finite-temperature black hole should change at extremality.
The role of boundary conditions is crucial for our definition. For a general
connection the degeneration of eigenvalues does not imply non-diagonalizability.
However, the special form of the flat connections dictated by the AAdS boundary
conditions will guarantee that if two eigenvalues of a φ are degenerate, then the
connection is non-diagonalizable. From this perspective, we could interpret that
equating eigenvalues of a φ is in a sense analogous to the confluence of horizons
for extremal black holes in general relativity.
4.2 Example: Extremal sl(3) Black Holes
In [41] several supersymmetric and non-supersymmetric cases were studied. For
brevity, here we will only look at the extremal cousin of the Euclidean black hole
we considered in 3.1.
Let us write again (17) but will focus on the unbarred sector for concreteness;
recall that it is sufficient to impose our definition of extremality on one sector to
obtain the desired features. Using canonical boundary conditions, the connections
are given by
a φ = L 1 − Q (2) L −1 −
Q (3)
4
W −2 ,
(26)
ia t E + a φ = 2a − = 2μ 3
W 2 + 2Q (3) L −1 + Q
2
(2) W −2 − 2Q (2) W 0
.
(27)
It is also instructive to re-write the solutions to (14) for the general rotating case, i.e.
the generalization of (20). This gives