String Theory and M-Theory

12.3 The AdS/CFT correspondence 655
to placing N free fermions in a harmonic oscillator potential. That study
makes it clear how the energy eigenvalues of the fermions in the gauge theory
encode the geometry of the dual string theory. Generically, unless N is very
large and the eigenvalues are smoothly distributed, one obtains a turbulent
quantum-foam-like geometry. The term bubbling AdS has been introduced
in this context.
The modification of the bulk solution obtained by the addition of normalizable
modes, on the other hand, corresponds to changing the Hamiltonian
of the boundary CFT by the addition of relevant perturbations. Relevant
perturbations are defined to be ones with dimension less than four, which
are important in the IR and unimportant in the UV. For example, the addition
of a mass term for one (or more) of the six scalar fields is a relevant
perturbation of the gauge theory.
Chiral primary operators
An alternative to analyzing the gauge theory on S p and using the Hamiltonian
approach is to consider the gauge theory on ¡ p , as is natural in
the Poincaré patch description of the AdS space. In this case the physical
observables are correlation functions of gauge-invariant operators. The
gauge-invariant operators correspond to the various states in the Hamiltonian
description by a state–operator correspondence that is a higherdimensional
analog of that described for two-dimensional conformal field
theories in Chapter 3.
Testing the AdS/CFT correspondence in this set-up involves finding the
correspondence between gauge-invariant operators in the gauge theory and
particle states in the string theory. In each case these are classified by representations
of the superconformal symmetry algebra. Such representations
include three types: long, short, and ultrashort. As explained in Chapters 8
and 11, the N = 4 supersymmetry algebra provides lower bounds (BPS
bounds) on allowed masses or energies. If neither bound is saturated, the
representation is long and all 16 of the linearly realized supersymmetry generators
are effective in building up the multiplet structure. In this case the
allowed helicities cover a range of eight units, since each charge can shift the
helicity by one half. If one of the bounds is saturated and the other is not,
the representation is called short and eight of the supersymmetry generators
are effective. Then the helicities in the multiplet cover a range of four units.
In the ultrashort case, both bounds are saturated, and the helicities cover
a range of two units. The N = 4 super Yang–Mills fields themselves have
helicities ranging from −1 to +1 and form an ultrashort multiplet. However,
they are not gauge-invariant operators.