String Theory and M-Theory

646 Gauge theory/string theory dualities
is introduced. In these coordinates the hypersurface is v 2 − τ 2 = −1, and
the metric on this surface is
ds 2 = dy 2 i − dt 2 j = dv2
1 + v 2 + v2 dΩ 2 p − (1 + v 2 )dθ 2 . (12.101)
Note that the time-like coordinate θ is periodic! This would imply that
the conjugate energy eigenvalues are quantized as multiples of a basic unit.
This is definitely not what type IIB superstring theory on AdS5 × S 5 gives.
The energy quantization does hold for the supergravity modes and their
Kaluza–Klein excitations, but it is not true for the stringy excitations.
CAdS/CFT correspondence
The solution to this problem is to replace the AdS space-time with its
covering space CAdS. Therefore, strictly speaking, one should speak of
CAdS/CFT duality, but that is not usually done. To describe the covering
space, let us replace the circle parametrized by θ by a real line parametrized
by t. This gives a global description of the desired space-time geometry.
Letting v = tan ρ, the metric becomes
ds 2 = 1
cos 2 ρ (dρ2 + sin 2 ρ dΩ 2 p − dt 2 ). (12.102)
This has topology Bp+1 ס which can be visualized as a solid cylinder. The
¡ factor, which is a real line, corresponds to the global time coordinate t, and
Bp+1 denotes a solid ball whose boundary is the sphere S p . The boundary of
the CAdS space-time at spatial infinity (ρ = π/2) is S p × ¡ . The Poincaré
coordinates cover a subspace of the global space-time, called the Poincaré
patch, as shown in Fig. 12.6. This diagram, which shows the global causal
structure of the geometry, is called a Penrose diagram. All light rays travel
at 45 degrees in a Penrose diagram.
When one uses the covering space CAdS to describe the bulk theory, the
spatial coordinates of the dual gauge theory are naturally taken to form
a sphere S p . This does have a significant technical advantage: when the
spatial coordinates form a sphere, the Hamiltonian has a discrete spectrum
rather than a continuous one. This can be traced to the fact that the
time coordinate in global coordinates differs from the time coordinate in the
Poincaré patch. Thus, if P0 denotes the Yang–Mills Hamiltonian appropriate
to the Poincaré patch time coordinate, then
H = 1
2 (P0 + K0), (12.103)
is the Hamiltonian appropriate to global time, and it has a discrete spec-