From algebraic curve to minimal surface and back

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We derive the Lax operator for a very large family of classical minimal surface solutions in AdS₃ describing Wilson loops in N=4 SYM theory. These solutions, constructed by Ishizeki, Kruczenski and Ziama, are associated with a hyperellictic surface of odd genus. We verify that the algebraic curve derived from the Lax operator is indeed none-other than this hyperelliptic surface.