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We reconsider the calculation of the one-loop effective action for an openGreen-Schwarz superstring in the $AdS_5\times S^5$ background for a circularboundary loop. By an explicit computation of the ratio of relevantdeterminants, describing semi-classical fluctuations about the minimal surfacein AdS and flat spaces, we show that it does not depend upon the AdSregularizing parameter $\epsilon$. The only dependence upon $\epsilon$ residesin the reparametrization path integral of the exponential of the classicalboundary action. We analyze how the result depends on the choice of theboundary condition imposed on fluctuating fields and show that, despite thefact that the contribution of individual angular modes changes, the productover the modes remains unchanged.

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