3.2 DLCQ and chiral limit of CFTs

The role of the DLCQ of CFTs in the context of the Kerr/CFT correspondence was suggested in [30Jump To The Next Citation Point] (for closely related work see [252]). Here, we will review how a DLCQ is performed and how it leads to a chiral half of a CFT. A chiral half of a CFT is here defined as a sector of a 2d CFT defined on the cylinder, where the right-movers are set to the ground state after the limiting DLCQ procedure. We will use these considerations in Section 4.4.

Let us start with a CFT defined on a cylinder of radius R,

2 2 2 ds = − dt + dϕ = − dudv, u = t − ϕ, v = t + ϕ. (91 )
Here the coordinates are identified as (t,ϕ ) ∼ (t,ϕ + 2πR ), which amounts to
(u,v) ∼ (u − 2πR, v + 2πR ). (92 )
The momentum operators P v and P u along the u and v directions are L0 and ¯L0, respectively. They have a spectrum
v 1 P |O⟩ = L0 |O ⟩ = (h + n) --|O ⟩, (93 ) ( ) R P u|O ⟩ = ¯L0|O ⟩ = ¯h + ¯n 1-|O ⟩, (94 ) R
where the conformal dimensions obey h,¯h ≥ 0 and n, ¯n ⁄= 0 are quantized left and right momenta.

Following Seiberg [239], consider a boost with rapidity γ

′ γ ′ −γ u = e u, v = e v. (95 )
The boost leaves the flat metric invariant. The discrete light-cone quantization of the CFT is then defined as the limit γ → ∞ with R′ ≡ Re γ fixed. In that limit, the identification (92View Equation) becomes
′ ′ ′ ′ ′ (u ,v ) ∼ (u − 2πR ,v). (96 )
Therefore, the resulting theory is defined on a null cylinder. Because of the boosted kinematics, we have
v′ 1 P |O ⟩ = (h + n )Re-γ|O ⟩, (97 ) ( ) eγ P u′|O ⟩ = ¯h + ¯n --|O ⟩. (98 ) R
Keeping P u′ (the momentum along v′) finite in the γ → ∞ limit requires ¯h = 0 and ¯n = 0.

Therefore, the DLCQ limit requires one to freeze the right-moving sector to the vacuum state. The resulting theory admits an infinite energy gap in that sector. The left-moving sector still admits non-trivial states. All physical finite-energy states in this limit only carry momentum along the compact null direction u′. Therefore, the DLCQ limit defines a Hilbert space ℋ,

ℋ = {|anything ⟩ ⊗ |0⟩ } (99 ) L R
with left chiral excitations around the SL (2,ℝ ) × SL (2,ℝ ) invariant vacuum of the CFT |0⟩L ⊗ |0 ⟩R. As a consequence, the right-moving Virasoro algebra does not act on that Hilbert space. This is by definition a chiral half of a CFT.

In summary, the DLCQ of a 2d CFT leads to a chiral half of the CFT with central charge c = cL. The limiting procedure certainly removes most of the dynamics of the original CFT. How much dynamics is left in a chiral half of a CFT is an important question that is left to be examined in detail in the future.

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