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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 β-function and explain how it is related to the D0-brane effective action. We also apply this method to compute the first α correction to the effective action of higher dimensional branes.

It is shown that for a class of models having Goldstone modes, any single-site theory must predict unphysical results.