The Holomorphic Quadratic Differentials of Amplitudes for Strings with Boundaries

We show that the amplitudes for strings with boundaries can be written as complex structure forms. The C, curves used by Krichever and Novikov to parametrize compact Riemann surfaces satisfy the string boundary conditions. Thus they can be regarded as the initial and final states of the atrings. Using Krichever-Novikov bases, we explicitly construct a basis of holomorphic quadratic differentials for strings with boundaries.