content_image/353867.jpg content_image/353866.jpg content_image/353865.jpg
and using Eq. (32), we write the contribution of this domain to A$_{2}$ as 

<img src='content_image/353867.jpg'>

with a positive constant C . A similar contribution appears if t$_{1}$ − R/ 2 T ∼ t$_{2}$ − R/ 2 T ∼ R $^{2}$/ (2 T ) 2 as is prescribed by Eq. (34) for both t$_{1}$ > R/ 2 T and t$_{2}$ > R/ 2 T . An analogous contribution (with possibly some powers of T/R ∝ ln(4 /µ )) emerges also when t$_{1}$ < R/ 2 T and t$_{2}$ > R/ 2 T or t$_{1}$ > R/ 2 T and t$_{2}$ < R/ 2 T . 

Therefore, the path integral over the quantum fluctuations around the minimizing func- tion t ∗ ( s ) gives, using the ζ -function regularization and Eq. (21) (and disregarding the logs to the order in consideration) 

where the 4th power is due to the four sets of modes $^{1}$. This remarkably reproduces the L¨ uscher term (6) in d = 26! 

<img src='content_image/353866.jpg'>

There was a subtlety in the derivation – the appearance of a logarithmic divergence at the corners of the rectangle if ˙ β (corners) ̸ = 0. It is seen from Eq. (40), where the region near y$_{1}$ = y$_{2}$ = 1 or y$_{1}$ = y$_{2}$ = − 1 (associated with t$_{1}$ = t$_{2}$ → R/ 2 T or t$_{1}$ = t$_{2}$ → − R/ 2 T ) produces the logarithmic divergence 

with the upper (lower) sign referring to y = 1 ( y = − 1). The coefficient is nonvanishing if ˙ β ( ± 1) ̸ = 0. Analogously, the integral is logarithmically divergent at the corner, when t$_{1}$, t$_{2}$ > R/ 2 T or t$_{1}$ < R/ 2 T , t$_{2}$ > R/ 2 T and vise versa. 

<img src='content_image/353865.jpg'>

The logarithmic divergence can be regularized by smoothing the corners like in Refs. [7, 14]. It is clear from such a regularization that the contribution of trajectories with ˙ β ( R/ 2 T ) ̸ = 0 to the path integral over β ( t ) will be suppressed as the smoothing is removed. Consequently, this corner divergence does not effect the result of this section. In the next section we repeat the consideration for the case of an ellipse, when there are no corners. 

If d ̸ = 26, the asymptotic ansatz for the Wilson loops has to be improved to get the correct factor ( d − 2) / 24 in the L¨ uscher term (6). This issue will be described in Sect. V. 

