### 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 [30]
(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 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 ,

Here the coordinates are identified as , which amounts to
The momentum operators and along the and directions are and , respectively.
They have a spectrum
where the conformal dimensions obey and are quantized left and right
momenta.
Following Seiberg [239], consider a boost with rapidity

The boost leaves the flat metric invariant. The discrete light-cone quantization of the CFT is then defined
as the limit with fixed. In that limit, the identification (92) becomes
Therefore, the resulting theory is defined on a null cylinder. Because of the boosted kinematics, we have
Keeping (the momentum along ) finite in the limit requires and
.
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 . Therefore, the DLCQ limit defines a Hilbert space ,

with left chiral excitations around the invariant vacuum of the CFT .
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 CFT leads to a chiral half of the CFT with central charge .
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.