String Theory and M-Theory

12.3 The AdS/CFT correspondence 651
in the gauge theory is identified with the vacuum expectation value of the
complex scalar field of the string theory. Each of them is defined on a space
that is identical to the moduli space of complex structures of a torus, which
was described in Chapter 3 and utilized several times in Chapter 8.
A more precise correspondence
The tests of AdS/CFT duality described so far only required analyzing perturbations
of AdS5×S 5 . Another successful test in this framework, which we
have not explained, was to show that all the linearized supergravity states
correspond to states in the dual gauge theory. However, there is more than
this perturbative framework that needs to be understood to define a precise
map between a CFT and its AdS dual, since this set-up is only sensitive
to the supergravity states and their Kaluza–Klein excitations, but does not
probe the underlying string-theory structure of the theory.
A conformal field theory does not have particle states or an S-matrix. The
only physical observables, that is, well-defined and meaningful quantities, in
a CFT are correlation functions of gauge-invariant operators. Thus, what
is required is an explicit prescription for relating such correlation functions
to computable quantities in the AdS string-theory background. These are
very similar to ordinary S-matrix elements, with the definition suitably generalized
to AdS boundary conditions at infinity. Since the dimension of the
AdS exceeds that of the CFT by one, it is sensible that off-shell quantities
in the CFT should correspond to on-shell quantities in AdS.
The gauge-invariant operators are defined at a point, which corresponds
to perturbing the gauge theory in the ultraviolet. Therefore, according to
the holographic energy/radius correspondence, the gauge theory should be
considered to be at the AdS boundary.
It is technically easier to work with the Euclideanized conformal field theory
and to relate its correlation functions to quantities in the Euclideanized
anti-de Sitter geometry. So this is a good place to start. After that, we
discuss the case of Lorentzian-signature case. The prescription requires a
one-to-one correspondence of bulk fields φ and gauge-invariant operators O
of the boundary CFT.
The path integral
Schematically the correspondence works as follows. Denoting boundary values
of φ by φ0, one computes the bulk-theory path integral with these bound-