  We study path-integrals over reparametrizations of the world-sheet boundary.
Such integrals arise when string propagates between fixed space-time contours.
In gauge/string duality they are needed to describe gauge theory Wilson loops.
We show that (1) in AdS/CFT, the integral is well defined and gives a finite
1-loop correction to the Wilson loop; (2) in critical string theory, the
integral is UV divergent, and fixed contour amplitudes are off shell. In the
second case, we show that the divergences can be removed by renormalizing the
contour. We calculate the 2-loop contour beta-function and explain how it is
related to the D0-brane effective action. We also apply this method to compute
the first alpha' correction to the effective action of higher dimensional
branes.
