content_image/372903.jpg content_image/372902.jpg content_image/372899.jpg content_image/372900.jpg content_image/372901.jpg content_image/372898.jpg content_image/372897.jpg content_image/372895.jpg
## Acknowledgments 

We are indebted to Andrey Mironov and Niels Obers for useful discussions. 

## Appendix A: Measure for integrating over reparametrization 

Introducing 

<img src='content_image/372903.jpg'>

we rewrite the measure of [6] for the integration over reparametrizations as 

<img src='content_image/372902.jpg'>

The integration over v$_{i}$ ’s in Eq. (A2) can be represented through the integration over a scalar field as follows. Writing 

we have 

and 

This represents the continuous measure as 

<img src='content_image/372899.jpg'>

<img src='content_image/372900.jpg'>

<img src='content_image/372901.jpg'>

<img src='content_image/372898.jpg'>

where t is a certain parametrization of the contour (e.g. through the proper time) and D ψ is the usual measure s$_{f}$ 

The scalar field ψ , that appears in Eqs. (A1), (A3), is in fact a discretization of the boundary value of the Liouville field 

<img src='content_image/372897.jpg'>

<img src='content_image/372895.jpg'>

