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 H 2 . Equivariance of these maps under certain subgroups of Sp(4, Z) is shown. The invertibility of both maps in a particular domain of H 2 is also shown.

Torus n-Point Functions for $${\mathbb{R}}$$ -graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

“…One may define n-point functions at genus one following Zhu [Z], and use these functions together with various sewing procedures to define n-point functions at successively higher genera [T], [MT2], [MT3]. In this paper we consider the genus one n-point functions for a Vertex Operator Superalgebra (VOSA) V with a real grading (i.e.…”

“…bosonization) we may employ results of ref. [MT2] to find alternative expressions for the n-point functions. In particular the generating function is expressible in terms of theta functions and the genus one prime form and we thus recover Fay's generalized trisecant identity for elliptic functions.…”

On the Torus Degeneration of the Genus Two Partition Function

“…This is described in detail in [MT1] where the explicit form for Ω is obtained in terms of the infinite matrix A a = (A a (k, l, q a , )) for k, l ≥ 1 where…”

“…There is a well developed mathematical theory of partition and n-point correlation functions for a VOA associated with a genus one torus [Z]. More recently, a study has begun of VOA partition and correlation functions on a genus two Riemann surface formed from one or two sewn tori where the genus two partition and correlation functions are defined in terms of correlation function data on the genus one surface(s) [T,MT1,MT2,MT3]. In particular, in this paper we consider a genus two Riemann surface formed by sewing two tori and study the behavior of the genus two partition function for a general VOA V in the non-trivial torus degeneration limit where one of the sewn tori is pinched down to a Riemann sphere.…”

Numerical Analysis of the Measurement of Near-Beam Electron Cloud Density in Field-Free Region at KEK B-Factory Low-Energy Ring

“…To discuss the degeneration limit, we will use the formalism of [29,32]. The genus two surface can be constructed according to a "sewing" procedure [33], where two tori with modular parameters q 1,2 = e 2πiτ 1,2 are joined by excising a disc of radius |ǫ| from each torus (ǫ being is a complex "pinching" parameter) and making an appropriate identification of two annular regions around the excised discs.…”