In this paper we shall show that the leading part of the fluctuations of large loops from (5) are transverse fluctuations of the minimal surface embedded by the curve x ( θ ∗ ). This we have shown for a rectangle and an ellipse, but we suspect the result to be more general. To explain our result we mention that in the Nambu-Goto action transverse fluctuations add a contribution to the area term for a large R × T rectangle with T ≫ R . These quantum fluctuations (6) are called the L¨ uscher term [7]. By lattice Monte Carlo calculations in three and four dimensions, this term has been found to occur [8] in quenched SU ( N ) for various N ′ s. Thus, for large distances the two leading terms from the Nambu-Goto action describe QCD quite well. Therefore, for a rectangular boundary curve the T/R term can be identified with transverse fluctuations of the minimal surface . With these remarks in mind we now give our main result: after integration over the fluctuations β in the functional integral we obtain as the leading contribution the L¨ uscher term corresponding to d= 26. In general, we obtain the d -dimensional contribution (plus the area term) from Here O is the operator which emerges in the semiclassical expansion of A [ x ( θ )] to quadratic order as exhibited in Eq. (5). This modification of Eq. (1) does not effect the classical limit leading to the area law, and has the meaning of a pre-exponential in the semiclassical approximation. Thus, this equation gives the leading effective QCD string behavior in terms of functional integration over reparametrizations. We mention that our considerations may have potential applications in condensed matter. For example, in the inverse square XY model one encounters expressions somewhat similar to Eq. (5), see for example Ref. [9]. We shall, however, not pursue this track in the present paper. The plan of this paper is the following: in Sect. II we discuss the general framework for the semiclassical approximation, and in Sect. III we carry out the functional integral over reparametrizations for a rectangle. A similar calculation for an ellipse is done in Sect. IV. ∫ D θ e − KA [ x ( θ )] (det O ) − ( $^{d}$− 26) / $^{48}$. (8) ∫ D β e − A 2 (7) − d − 2 2 tr log ( − ∂ $^{2}$) = ( d − 2) π 24 T R (6) 3