On Genus Two Riemann Surfaces Formed from Sewn Tori

TL;DR

The paper develops two explicit sewing formalisms (ε for sewing two genus-1 surfaces and ρ for self-sewing a genus-1 surface) to construct genus-2 Riemann surfaces from genus-1 data and to describe the genus-2 period matrix as a holomorphic map into the Siegel upper half-space. It furnishes detailed analytic and combinatorial tools—weighted moment matrices, determinant holomorphy, and chequered necklace expansions—that encode Ω^(2) and reveal equivariance under subgroups of Sp(4,ℤ). The work also establishes local invertibility near degeneration points, builds covering-space frameworks to handle logarithmic terms, and provides a precise mapping between the ε- and ρ-parameterizations, thereby enabling rigorous, modularly well-behaved higher-genus constructions relevant to vertex operator algebras and their partition functions. The explicit expansions in the appendices offer practical inputs for computations of genus-two data in both formalisms.

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.

Theorems & Definitions (50)

• Example 3.1
• Theorem 3.2: Ref. Y, Theorem 1, Theorem 4
• Lemma 3.3: op.cite., Lemma 4
• Lemma 3.4
• Proposition 3.5
• Lemma 3.6
• Theorem 3.7
• Lemma 3.8
• Theorem 3.9: op. cite. Theorem 4.
• Example 3.10