We consider the low energy effective action corresponding to the 1-loop,
planar, dilatation operator in the scalar sector of N=4 SU(N) SYM theory. For a
general class of non-holomorphic ``long'' operators, of bare dimension L>>1, it
is a sigma model action with 8-dimensional target space and agrees with a limit
of the phase-space string sigma model action describing generic fast-moving
strings in the S^5 part of AdS_5 x S^5. The limit of the string action is taken
in a way that allows for a systematic expansion to higher orders in the
effective coupling λ/L2. This extends previous work on rigid rotating
strings in S^5 (dual to operators in the SU(3) sector of the dilatation
operator) to the case when string oscillations or pulsations in S^5 are
allowed. We establish a map between the profile of the leading order string
solution and the structure of the corresponding coherent, ``locally BPS'', SYM
scalar operator. As an application, we explicitly determine the form of the
non-holomorphic operators dual to the pulsating strings. Using action--angle
variables, we also directly compute the energy of pulsating solutions,
simplifying previous treatments.Comment: LaTeX, 50 pages, 1 figure. v2: References added, minor corrections.
54 pages. v3: Few changes. One paragraph added at the end of section 3. 55
page