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 S5part of AdS5 × S5.
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 S5(dual to operators in the SU(3) sector of the dilatation operator) to the case when string
oscillations or pulsations in S5are 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.