with Differentiating, we get d z d u = √$_{ν}$ 2 K π (1 − ν $^{2}$) sn ( 2 K π u, ν ′ ) cn ( 2 K π u, ν ′ ) dn 2 ( 2 K π u, ν ′ ) . (D7) ν ′ = √$_{1}$ − ν 2 → 0 . (D6) Using the fact that Jacobi elliptic functions reduce to trigonometric functions as ν ′ → 0 and substituting K $^{ν}$→ 1 → π 2 8 a b , (D8) d z d u ∝ exp ( − i π 2 a b u ) = ⇒ ρ ∝ exp ( π a b σ ) (D9) L¨ uscher term = 1 96 π ∫ τ 0 ≈ b a 0 d τ ∫ 2 π 0 d σ ( π a b ) 2 = π 2 48 a b (D10) and that confirms the extra π/ 2 in Eq. (51) for an ellipse in comparison with a rectangle. we infer from Eq. (D7) 21 [7] M. L¨ uscher, K. Symanzik and P. Weisz, Anomalies of the free loop wave equation in the WKB approximation, Nucl. Phys. B 173 , 365 (1980); M. L¨ uscher, Symmetry breaking aspects of the roughening transition in gauge theories, Nucl. Phys. B 180 , 317 (1981). [6] Y. Makeenko and P. Olesen, Wilson loops and QCD/string scattering amplitudes , Phys. Rev. D 80 , 026002 (2009) [arXiv:0903.4114 [hep-th]]. [5] V. S. Rychkov, Wilson loops, D-branes, and reparametrization path integrals , JHEP 0212 , 068 (2002) [arXiv:hep-th/0204250]. [4] A. A. Migdal, Loop equation and area law in turbulence , Int. J. Mod. Phys. A 9 , 1197 (1994) [arXiv:hep-th/9310088]. [3] A. M. Polyakov, Talk at the Workshop “Particles, Fields and Strings”, Vancouver, July 1997, unpublished. [2] A. G. Cohen, G. W. Moore, P. C. Nelson, and J. Polchinski, An off-shell propagator for string theory, Nucl. Phys. B 267 , 143 (1986). [1] J. Douglas, Solution of the problem of Plateau , Trans. Am. Math. Soc. 33 , 263 (1931).