content_image/351725.jpg content_image/351723.jpg content_image/351722.jpg content_image/351720.jpg content_image/351721.jpg content_image/351719.jpg content_image/351717.jpg content_image/351718.jpg
which is related to the boundary metric: 

<img src='content_image/351725.jpg'>

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. 

<img src='content_image/351723.jpg'>

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. 

## Appendix B: Douglas’ algorithm for plane contours: conformal map 

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 

<img src='content_image/351722.jpg'>

<img src='content_image/351720.jpg'>

<img src='content_image/351721.jpg'>

<img src='content_image/351719.jpg'>

<img src='content_image/351717.jpg'>

<img src='content_image/351718.jpg'>

