Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

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.

💡 Research Summary

The paper investigates flat connections on arbitrary compact Riemann surfaces Σ of genus h ≥ 1 whose values lie in the infinite‑dimensional Lie algebra ˆ𝔱ₕ,ₙ. Two families of connections are considered: the single‑valued, modular‑invariant DHS connection (named J_DHS) and the multivariable Enriquez connection (named K_E). Both connections are defined on the configuration space C_fₙ(Σ)=Σⁿ{diagonals} and are expressed as one‑forms decomposed into (1,0) and (0,1) components.

The Lie algebra ˆ𝔱ₕ,ₙ is generated by elements a_{Ii}, b_{iI} (of bi‑degree (1,0) and (0,1) respectively) and t_{ij} (of bi‑degree (1,1)). For n ≥ 2 the algebra is not freely generated; its defining relations (equations (2.1)–(2.2) in the paper) produce non‑trivial commutators among the generators. These relations are the source of the “non‑trivial” part of the flatness conditions.

Both connections are built from families of integration kernels. The DHS kernels are defined recursively from the Arakelov Green function G(x,y), the holomorphic Abelian differentials ω_I, and their complex conjugates. They admit a trace/traceless decomposition f_{I₁…I_r J}(x,y)=∂xΦ{I₁…I_r J}(x)−∂xG{I₁…I_{r−1}}(x,y)δ_{I_rJ}. Generating functions J_K and G are introduced to encode all kernels simultaneously. The Enriquez kernels have an analogous recursive definition and share the same combinatorial structure.

Flatness of a connection is expressed by the Maurer–Cartan equations dJ−J∧J=0 and dK−K∧K=0. After expanding the exterior derivative on the configuration space, the equations split into three groups: (i) the “easy” relations involving the (0,1) components, which follow from the Massey structure of the differential system; (ii) the symmetry relations ∂_iJ^{(1,0)}_j−∂_jJ^{(1,0)}_i=0 (and similarly for K) that are satisfied by the basic properties of the kernels; and (iii) the commutator conditions