String Theory and M-Theory

12.3 The AdS/CFT correspondence 657
and remove all traces to make a traceless symmetric tensor. However, this
is not quite the whole story. These operators can be related to multi-trace
operators when n > N. By a multi-trace operator, we mean an operator
that is a product of operators of the form in Eq. (12.122). Thus, to state
the final conclusion, these operators provide a complete list of single-trace
chiral primary operators for n = 2, 3, . . . , N. This rule reflects the fact that
these are the orders of the independent Casimir invariants of SU(N). This
is explored further in Exercise 12.9.
These chiral primary operators form the (0, n, 0) representation of SU(4).
In the large-N limit this matches perfectly with what one finds from Kaluza–
Klein reduction on the five-sphere in the dual string-theory picture. It has
been shown that the masses of these bulk scalar fields match the conformal
dimensions of the chiral primary operators in the way required by the duality
that was described earlier. It is interesting, though, that for finite N the
Kaluza–Klein excitations with n > N seem to be missing in the dual gaugetheory
description. This is how it should be, however. The infinite tower
of Kaluza–Klein excitations actually is truncated at N. The reason will be
explained later.
Anomalies
In general, it is difficult to compare gauge theory and string theory correlation
functions, because the AdS/CFT correspondence relates weak coupling
to strong coupling. However, there are certain quantities that are controlled
by anomalies that can be computed exactly enabling the comparison to be
made. Let us describe an example.
The SU(4) R symmetry is a chiral symmetry of N = 4 super Yang–
Mills theory. This is evident because left-handed and right-handed fermions
belong to complex-conjugate representations (4 and ¯4). If one were to add
SU(4) gauge fields and make this symmetry into a local symmetry, one
would obtain an inconsistent quantum theory, because the SU(4) currents
would acquire an anomalous divergence
(∇ µ Jµ) a = N 2 − 1
384π2 i dabcɛ µνρλ F b µνF c ρλ . (12.123)
Such SU(4) gauge fields are not present in the super Yang–Mills theory,
so there is no inconsistency. However, they do exist in the bulk theory,
where they arise by the Kaluza–Klein mechanism as a consequence of the
isometry of the five-sphere. The anomaly means that if one turns on nonzero
field strengths for these gauge fields the bulk theory would no longer be
gauge invariant. The associated anomaly can be computed from the bulk