Relating flat connections and polylogarithms on higher genus Riemann surfaces

In this work, we relate two recent constructions that generalize classical (genus-zero) polylogarithms to higher-genus Riemann surfaces. A flat connection valued in a freely generated Lie algebra on a punctured Riemann surface of arbitrary genus produces an infinite family of homotopy-invariant iterated integrals associated to all possible words in the alphabet of the Lie algebra generators. Each iterated integral associated to a word is a higher-genus polylogarithm. Different flat connections taking values in the same Lie algebra on a given Riemann surface may be related to one another by the composition of a gauge transformation and an automorphism of the Lie algebra, thus producing closely related families of polylogarithms. In this paper we provide two methods to explicitly construct this correspondence between the meromorphic multiple-valued connection introduced by Enriquez in e-Print 1112.0864 and the non-meromorphic single-valued and modular-invariant connection introduced by D’Hoker, Hidding and Schlotterer, in e-Print 2306.08644.

💡 Research Summary

This paper establishes a precise correspondence between two recent constructions of higher‑genus polylogarithms, namely the meromorphic multiple‑valued flat connection introduced by Enriquez (denoted (d-\mathcal K_E)) and the non‑meromorphic single‑valued modular‑invariant flat connection introduced by D’Hoker, Hidding and Schlotterer (denoted (d-\mathcal J_{DHS})). Both connections are defined on a once‑punctured compact Riemann surface (\Sigma_p) of arbitrary genus (h\ge1) and take values in the completed free Lie algebra (\mathfrak g) generated by (2h) elements ({a_i,b_i}_{i=1}^h), which are dual to the homology cycles (A_i,B_i).

The authors first recall that a flat connection (d-J) satisfying the Maurer–Cartan equation (dJ-J\wedge J=0) yields a path‑ordered exponential (\Gamma(x,y)=P\exp\int_y^x J) taking values in the group (\exp(\mathfrak g)). Expanding (\Gamma) in the non‑commutative alphabet ({a_i,b_i}) produces coefficients that are homotopy‑invariant iterated integrals; each such coefficient is identified as a higher‑genus polylogarithm. The shuffle product of words translates into the multiplication law for these polylogarithms, mirroring the classical genus‑zero case.

The core of the work consists of two complementary constructions that relate (d-\mathcal K_E) and (d-\mathcal J_{DHS}) by a combination of a gauge transformation and a Lie‑algebra automorphism. In the first construction (Section 3), the authors build a gauge transformation (U_{DHS}) as a smooth, multiple‑valued map on (\Sigma\times\Sigma) with values in (\exp(\mathfrak g_{\text{b}})), where (\mathfrak g_{\text{b}}) is the completed free sub‑Lie algebra generated by the same set of letters. The construction proceeds recursively in the degree of the free Lie algebra: at each order they solve the flatness condition and the monodromy constraints to determine the new term of (U_{DHS}). Simultaneously they define an automorphism (\phi_{DHS}:\mathfrak g\to\mathfrak g) that linearly mixes the generators, essentially sending each (a_i) to a combination of the (a)’s and (b)’s while leaving the (b_i) unchanged. The coefficients of this linear map are fixed by matching the residues (the “kernel” data) of the Enriquez connection with the modular‑invariant structure of the DHS connection. The final relation reads \