  Effective magnetic $SU(N)$ gauge theory with classical $Z_N$ flux tubes of
intrinsic width $\frac{1}{M}$ is an effective field theory of the long distance
quark-antiquark interaction in $SU(N)$ Yang-Mills theory. Long wavelength
fluctuations of the $Z_N$ vortices of this theory lead to an effective string
theory. In this paper we clarify the connection between effective field theory
and effective string theory and we propose a new constraint on these vortices.
We first examine the impact of string fluctuations on the classical dual
superconductor description of confinement. At inter-quark distances $R\sim
\frac{1}{M}$ the classical action for a straight flux tube determines the heavy
quark potentials. At distances $R \gg \frac{1}{M}$ fluctuations of the flux
tube axis $\tilde{x}$ give rise to an effective string theory with an action
$S_{eff} (\tilde{x})$, the classical action for a curved flux tube, evaluated
in the limit $\frac{1}{M} \rightarrow 0~$. This action is equal to the
Nambu-Goto action. These conclusions are independent of the details of the
$Z_N$ flux tube. Further, we assume the QCD flux tube satisfies the additional
constraint: $$\int_0^\infty r dr \frac{T_{\theta \theta} (r)}{r^2} =0,$$ where
$\frac{T_{\theta \theta}(r)}{r^2}$ is the value of the $\theta\theta$ component
of the stress tensor at a distance $r$ from the axis of an infinite flux tube.
Under this constraint the string tension $\sigma$ equals the force on a quark
in the chromoelectric field $\vec{E}$ of an infinite straight flux tube, and
the Nambu-Goto action can be represented in terms of the chromodynamic fields
of effective magnetic $SU(N)$ gauge theory, yielding a field theory
interpretation of effective string theory.
