On Genus Two Riemann Surfaces Formed from Sewn Tori

Abstract

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane ${\mathbb{H}}_2$. Equivariance of these maps under certain subgroups of $Sp(4,\mathbb{Z})$ is shown. The invertibility of both maps in a particular domain of ${\mathbb{H}}_2$ is also shown.

61 pages, 1 Figure

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• Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function
• Two-Loop Superstrings I, Main Formulas
• Two-loop superstrings and S-duality
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