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Worldsheet fermion correlators, modular tensors and higher genus integration kernels

by Eric D'Hoker, Oliver Schlotterer

Submission summary

Authors (as registered SciPost users): Oliver Schlotterer
Submission information
Preprint Link: scipost_202506_00025v1  (pdf)
Date submitted: June 11, 2025, 10:10 a.m.
Submitted by: Schlotterer, Oliver
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
  • Mathematical Physics
Approaches: Theoretical, Computational

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.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
In refereeing

Reports on this Submission

Report #1 by Anonymous (Referee 1) on 2025-7-22 (Invited Report)

Report

This paper is a tour de force that aims to understand to space of functions needed to describe the moduli space integrands of string amplitudes at higher genus. The immediate goal is to make progress on the integrands, but of course a wise representation of the integrands is a big step towards performing the moduli space integrals, at least in a low energy expansion.

The focus is on sums over spin structures of cyclic or chain products of Szego kernels, which appear from fermonic correlation functions in the RNS formalism. In fact, within this paper, no sum over spin structures is actually performed, but the simplifications obtained (and proven for any length of the products) go a long way because they factorise the puncture dependence from the spin structure dependence. The paper elegantly compares the alternative approaches of the Enriquez kernels and the D'Hoker-Schlotterer kernels (basically a choice between meromorphicity, appropriate for chiral splitting, or single valuedness), which extends a story that is well established at genus one. I did not try to follow the substantial technical details of the proofs.

I recommend the paper for publication. It is written in a very clear manner, which helps the reader find the essential information among the various proofs. It brings together many previous results and it will surely be useful for the future study of string amplitudes at higher genus. Although the authors do not comment on it, it will hopefully lead to solving the puzzles with the "chiral measure" that occur starting at genus 3. I don't have significant questions, and recommend that it is published as it stands, unless the authors wish to make minor changes.

Recommendation

Publish (surpasses expectations and criteria for this Journal; among top 10%)

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