The Szegő Kernel on a Sewn Riemann Surface

Abstract

We describe the Szegö kernel on a higher genus Riemann surface in terms of Szegö kernel data coming from lower genus surfaces via two explicit sewing procedures where either two Riemann surfaces are sewn together or a handle is sewn to a Riemann surface. We consider in detail the examples of the Szegö kernel on a genus two Riemann surface formed by either sewing together two punctured tori or by sewing a twice-punctured torus to itself. We also consider the modular properties of the Szegö kernel in these cases.

Ambitwistor strings and the scattering equations at one loop

“…The behavior of the Szego kernel depends on the spin structure, and is apparent from (4.21) or can be rigorously derived using the sewing formalism for Riemann surfaces [25,26]:…”

“…Every entry in M α falls into one of two classes: either both of its indices are on the same side of the separating cycle, or they are on different sides. If z, z ′ ∈ Σ L , then as s → 0 the Szego kernel S α (z − z ′ ; τ ) simply reduces to the Szego kernel on Σ L , and similarly for z, z ′ ∈ Σ R [25,26].…”

Twisted Frobenius Identities from Vertex Operator Superalgebras

“…we recall the construction of the Szegő kernel on a genus two Riemann surface in the -formalism [10]. The genus one prime form for , ∈ C and ∈ H 1 is ( , ) = ( − , )…”

“…for ̸ = −1/2 (with a different expression when = −1/2 given in [10]). We will assume ̸ = −1/2 throughout this paper.…”

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

“…In this paper (based on several conference talks of the author) we review our recent results [1][2][3][4][5] on construction and computation of correlation functions of vertex operator superalgebras with a formal parameter associated to local coordinates on a self-sewn Riemann surface of genus g which forms a genus 1 g  surface. In particular, we review result presented in the papers [1][2][3][4][5] accomplished in collaboration with M. P. Tuite (National University of Ireland, Galway, Ireland).…”