String Theory and M-Theory

12.5 Plane-wave space-times and their duals 681
a BMN operator to the light-cone gauge energy of the corresponding state
in the plane-wave string theory
∆a = ∆ − J ↔ P+ = Hℓc. (12.166)
Here, ∆ denotes the dimension and J is the U(1) R charge. Note that
both of these become infinite in the limit under consideration, but that
the difference remains finite for BMN operators. Viewed in terms of global
coordinates, so that the dual gauge theory is defined on S 3 rather than R 3 ,
∆a can be alternatively interpreted as an energy. For half-BPS states, which
correspond to short representations, the anomalous dimension ∆a vanishes.
The BMN operators, by contrast, are not BPS, but they are kept sufficiently
close to BPS operators so that the anomalous dimension remains finite in
the limit. These operators are characterized by having an R charge J that
scales like N 1/2 in the large N limit. For most operators the limit of ∆a
is infinite. Such operators are presumed to decouple in the Penrose/BMN
limit and are therefore not considered.
This duality can be tested perturbatively in three quantities
λ ′ ↔ 1/(µα ′ P−) 2 , (12.167)
g2 = J 2 /N ↔ 4πgs(µα ′ P−) 2 , (12.168)
1/J ↔ 1/(µR 2 P−). (12.169)
In each case, we have written dimensionless gauge-theory quantities on the
left and the corresponding string-theory quantities on the right. The λ ′
expansion is the loop expansion in the gauge theory (for correlation functions
of BMN operators), and the g2 expansion is the loop expansion of the stringtheory
description.
Since the plane-wave string theory is tractable, it is possible to obtain
results that are exact in their λ ′ dependence. In special cases these results
can be reproduced in the dual field theory. Thus, for example, Fock-space
states of the form a I†
n a J†
−n |0〉 correspond to certain single-trace two-impurity
operators in the gauge theory. To leading order in g2 and 1/J, but all orders
in λ ′ , it has been verified in the gauge theory that, for these operators,
∆a = 2 1 + λ ′ n 2 (12.170)
in agreement with expectations based on Eqs (12.162) and (12.163).
Some of the first-order corrections in g2 and 1/J have also been examined,
and agreement with the duality predictions has been found. The g2
corrections are obtained by using the vertex operator of the light-cone gauge