6.5 Symmetry breaking to U (1)L × U (1)R

The vector fields that generate the SL (2,ℝ ) × SL (2,ℝ ) symmetries are not globally defined. They are not periodic under the angular identification
ϕ′ ∼ ϕ′ + 2π. (226 )
Therefore, the SL (2,ℝ) symmetries cannot be used to generate new global solutions from old ones. In other words, solutions to the wave equation in the near region do not form SL (2,ℝ ) × SL (2,ℝ) representations. In the (ω+, ω− ) plane defined in (210View Equation), the identification (226View Equation) is generated by the SL (2,ℝ )L × SL (2,ℝ)R group element
− i4π2TRH0−i4π2TLH¯0 e , (227 )
as can be deduced from (217View Equation). This can be interpreted as the statement that the SL (2, ℝ)L × SL (2,ℝ )R symmetry is spontaneously broken to the U (1) × U (1) L R symmetry generated by ( ¯H ,H ) 0 0.

The situation is similar to the BTZ black hole in 2+1 gravity that has a SL (2,ℝ )L × SL (2,ℝ)R symmetry, which is spontaneously broken by the identification of the angular coordinate. This breaking of symmetry can be interpreted in that case as placing the dual CFT to the BTZ black hole in a density matrix with left and right-moving temperatures dictated by the SL (2,ℝ )L × SL (2,ℝ)R group element generating the 2π identification of the geometry [210Jump To The Next Citation Point].

In the case of non-extremal black-hole geometries, one can similarly interpret the symmetry breaking using a CFT as follows [68]. First, we need to assume that before the identification, the near region dynamics is described by a dual two-dimensional CFT, which possesses a ground state that is invariant under the full SL (2,ℝ )L × SL (2,ℝ )R symmetry. This is a strong assumption, since there are several (apparent) obstacles to the existence of a ground state, as we already discussed in the case of extremal black holes; see Section 4.4. Nevertheless, assuming the existence of this vacuum state, the two conformal coordinates + − (ω ,ω ) can be interpreted as the two null coordinates on the plane where the CFT vacuum state can be defined. At fixed r, the relation between conformal coordinates (ω+, ω− ) and Boyer–Lindquist (ϕ,t) coordinates is, up to an r-dependent scaling,

± ±t± ω = e , (228 )
t+ = 2πTR (ϕ′ − ΩRt ), (229 ) − ′ t = − 2πTL (ϕ − ΩLt). (230 )
This is precisely the relation between Minkowski (ω ±) and Rindler (t±) coordinates. The periodic identification (226View Equation) then requires that the Rindler domain be restricted to a fundamental domain under the identification
+ + 2 − − 2 t ∼ t + 4π TR, t ∼ t − 4 π TL, (231 )
generated by the group element (227View Equation).

The quantum state describing this accelerating strip of Minkowski spacetime is obtained from the SL (2,ℝ )L × SL (2,ℝ)R invariant Minkowski vacuum by tracing over the quantum state in the region outside the strip. The result is a thermal density matrix at temperatures (TL, TR). Hence, under the assumption of the existence of a CFT with a vacuum state, non-extremal black holes can be described as a finite temperature (TL,TR ) mixed state in a dual CFT.

It is familiar from the three-dimensional BTZ black hole that the identifications required to obtain extremal black holes are different than the ones required to obtain non-extremal black holes [27, 210]. Here as well, the vector fields (214View Equation) – (215View Equation) are not defined in the extremal limit because the change of coordinates (210View Equation) breaks down. Nevertheless, the extremal limit of the temperatures TL and TR match with the temperatures defined at extremality in Section 5.4. More precisely, the temperatures T L and T R defined in (222View Equation), (223View Equation) and (225View Equation) match with the temperatures defined at extremality Tϕ, RχTe and − 1 −1 −1 (p1Tϕ + p2(R χTe) ), respectively, where Tϕ and Te are defined in (74View Equation). This is consistent with the interpretation that states corresponding to extremal black holes in the CFT can be defined as a limit of states corresponding to non-extremal black holes.

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