Worldsheet fermion correlators, modular tensors and higher genus integration kernels

TL;DR

This paper develops two complementary descent procedures to decompose cyclic products of Szeg"o kernels on higher-genus Riemann surfaces, separating point dependence from spin structure dependence, and generalizes these to linear chains.

Contribution

It introduces two new systematic descent methods for decomposing Szeg"o kernel products, linking point and spin structure dependencies with modular tensors and kernels, extending previous work.

Findings

Two distinct descent procedures are proven and shown to have similar combinatorial structures.

Dependence on points is expressed via Enriquez kernels and DHS kernels, respectively.

Decompositions are generalized to linear chain products of Szeg"o kernels.

Abstract

The cyclic product of an arbitrary number of Szeg\"o kernels for even spin structure $\delta$ on a compact higher-genus Riemann surface $\Sigma$ may be decomposed via a descent procedure which systematically separates the dependence on the points $z_i \in \Sigma$ from the dependence on the spin structure $\delta$. In this paper, we prove two different, but complementary, descent procedures to achieve this decomposition. In the first procedure, the dependence on the points $z_i \in \Sigma$ is expressed via the meromorphic multiple-valued Enriquez kernels of e-print 1112.0864 while the dependence on $\delta$ resides in multiplets of functions that are independent of $z_i$, locally holomorphic in the moduli of $\Sigma$ and generally do not have simple modular transformation properties. The $\delta$-dependent constants are expressed as multiple convolution integrals over homology cycles of $\Sigma$, thereby generalizing a similar representation of the individual Enriquez kernels. In the second procedure, which was proposed without proof in e-print 2308.05044, the dependence on $z_i$ is expressed in terms the single-valued, modular invariant, but non-meromorphic DHS kernels introduced in e-print 2306.08644 while the dependence on $\delta$ resides in modular tensors that are independent of $z_i$ and are generally non-holomorphic in the moduli of $\Sigma$. Although the individual building blocks of these decompositions have markedly different properties, we show that the combinatorial structure of the two decompositions is virtually identical, thereby extending the striking correspondence observed earlier between the roles played by Enriquez and DHS kernels. Both decompositions are further generalized to the case of linear chain products of Szeg\"o kernels.