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: $$\int_0^\infty r dr \frac{T_{θθ} (r)}{r^2} =0,$$ where $\frac{T_{θθ}(r)}{r^2}$ is the value of the $θθ$ component of the stress tensor at a distance $r$ from the axis of an infinite flux tube. Under this constraint the string tension $σ$ 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.

29 pages, 3 figures, fixed typos and references