which is related to the boundary metric: Under the reparametrization s → f ( s ) (d f/ d s ≥ 0), the boundary value of the Liouville field ψ ( s ) transforms as that clarifies its relation to reparametrizations. The results of this Appendix make it possible to relate the ansatz (1) with Eq. (17) of Ref. [14], where the path integral over ψ is the same as the path integral over reparametriza- tion in view of Eq. (11) of [14]. They coincide in d = 26 provided S$_{cl}$ is the Douglas integral. The construction of the coordinates of the minimal surface, enclosed by a plane contour, is given by conformal mappings. Let us describe such a contour by two functions x$_{1}$ ( t ) and x$_{2}$ ( t ). Motivated by Appendix H of Ref. [4], we define the analytic functions The real and imaginary parts of Φ$_{µ}$ are Therefore, Douglas’ minimization equation is satisfied if at the real axis. For the circle we have 14 x$_{1}$ ( t ) + i x$_{2}$ ( t ) ≡ C ( t ) = i − t i + t (B5) ∑ µ Im Φ 2 $_{µ}$( s ) = 0 (B4) ∫ $^{+}$∞ −∞ d s ′ ˙ x ( t ∗ ( s )) · [ x ( t ∗ ( s )) − x ( t ∗ ( s ′ ))] ( s − s ′ ) 2 = 0 . (B3) Re Φ$_{µ}$( s ) = ∫ $^{+}$∞ −∞ d s ′ s ′ − s ˙ x$_{µ}$ ( t ∗ ( s ′ )) , (B2a) Im Φ$_{µ}$( s ) = d x$_{µ}$ ( t ∗ ( s )) d s = ˙ x$_{µ}$ ( t ∗ ( s )) d t ∗ ( s ) d s . (B2b) Φ$_{µ}$( z ) = ∫ $^{+}$∞ −∞ d s ′ π [ x$_{µ}$ ( t ∗ ( s ′ )) − x$_{µ}$ ( t ∗ ( z ))] ( s ′ z ) 2 . (B1) ψ ( s ) → ψ ( f ( s )) = ψ ( s ) − 2 ln d f ( s ) d s , (A10) Appendix B: Douglas’ algorithm for plane contours: conformal map √ ( d x $^{µ}$( s ) d s ) 2 = e ψ ( s ) / 2 . (A9)