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Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

Eric D'Hoker, Oliver Schlotterer

TL;DR

The paper establishes a deep link between flatness of multivariable flat connections on arbitrary Riemann surfaces and a complete set of algebraic identities among integration kernels. By expressing the DHS and Enriquez connections in terms of shared kernel data and exploiting a detailed Lie-algebra analysis of the non-freely generated $hat{\mathfrak{t}}_{h,n}$, it shows that flatness is equivalent to all interchange and Fay identities for DHS kernels (and that flatness of the Enriquez connection implies the analogous identities for Enriquez kernels, though the reverse implication remains conjectural). The results unify two complementary viewpoints on polylogarithms generated by iterated integrals, providing a rigorous backbone for the combinatorial relations among kernels across configurations with $n\ge 3$ variables. The work highlights both the power and the current limits of translating flatness into a full identity network, pointing to future progress in resolving coincident-limit structures and dihedral symmetries in the meromorphic domain and thereby achieving a full equivalence.

Abstract

A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.

Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

TL;DR

The paper establishes a deep link between flatness of multivariable flat connections on arbitrary Riemann surfaces and a complete set of algebraic identities among integration kernels. By expressing the DHS and Enriquez connections in terms of shared kernel data and exploiting a detailed Lie-algebra analysis of the non-freely generated , it shows that flatness is equivalent to all interchange and Fay identities for DHS kernels (and that flatness of the Enriquez connection implies the analogous identities for Enriquez kernels, though the reverse implication remains conjectural). The results unify two complementary viewpoints on polylogarithms generated by iterated integrals, providing a rigorous backbone for the combinatorial relations among kernels across configurations with variables. The work highlights both the power and the current limits of translating flatness into a full identity network, pointing to future progress in resolving coincident-limit structures and dihedral symmetries in the meromorphic domain and thereby achieving a full equivalence.

Abstract

A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.
Paper Structure (45 sections, 35 theorems, 229 equations)

This paper contains 45 sections, 35 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Review of the DHS connection and Fay identities
  3. The Lie algebra
  4. The DHS kernels
  5. The multivariable DHS connection
  6. The interchange identities for DHS kernels
  7. The Fay identities for DHS kernels
  8. Transformation under swapping $i$ and $j$
  9. Trace decomposition
  10. Coincident limits of DHS kernels and Fay identities
  11. Flatness of DHS connection $\Longleftrightarrow$ Fay identities
  12. Decomposition of the commutator
  13. Reducing the component ${\cal C}^{(0)}_{ij}$
  14. Reducing the component ${\cal C}^{(1)}_{ij}$
  15. Reducing the component ${\cal C}^{(2)}_{ij}$
  16. ...and 30 more sections

Key Result

Theorem 2.2

The interchange identities among DHS kernels correspond to the relations, for arbitrary word ${\vec{I}} \in {\cal W}_h$, arbitrary index $K=1,\cdots,h$ and distinct points $x_i, x_j \in \Sigma$. The interchange identities for DHS kernels were established for various special cases in DHoker:2020tcqDHoker:2020uid and were proven for the general case in DHoker:2024ozn.

Theorems & Definitions (64)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Proposition 2.8
  • ...and 54 more