Relating flat connections and polylogarithms on higher genus Riemann surfaces

Abstract

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.

Figures (2)

• Figure 1: The left panel represents a genus-two Riemann surface $\Sigma$ in terms of a fundamental domain $D \subset \tilde{\Sigma}$ for the action of $\mathrm{Aut}(\tilde{\Sigma} /\Sigma)\simeq \pi_1(\Sigma, y)$, which can be obtained by cutting $\Sigma$ along the cycles in the right panel. The surface $\Sigma$ may be reconstructed from $D$ by pairwise identifying inverse boundary components with one another under the dashed arrows; the projection $\pi: \tilde{\Sigma} \to \Sigma$ maps all the vertices $y$ and $y_i$ for $i=1,\cdots , 7$ of $D$ to the same point $y$ in $\Sigma$. The points $y_i \in \Tilde \Sigma$ are related to $y \in \tilde{\Sigma}$ by $y_1 =\mathfrak{A}_1 \cdot y$, $y_2=\mathfrak{B}_1 \cdot y_1$, $y_3 = \mathfrak{A}_1^{-1} \cdot y_2$, $y_4= \mathfrak{B}_1^{-1} \cdot y_3$, $y_5=\mathfrak{A}_2 \cdot y_4$, $y_6 = \mathfrak{B}_2 \cdot y_5$ and $y_7 = \mathfrak{A}_2^{-1} \cdot y_6$, the product of loops being understood here as a composition of the corresponding elements in $\mathrm{Aut}(\tilde{\Sigma} /\Sigma)$ with $\mathfrak{B}_2^{-1} \cdot y_7=y$ in view of (\ref{['2.ABrel']}).
• Figure 2: A genus-two Riemann surface $\Sigma_p$ with puncture $p$ can be represented in terms of a (punctured) fundamental domain $D_p \subset \tilde{\Sigma}\smallsetminus\pi^{-1}(p)$. The surface $\Sigma_p$ may be reconstructed from $D_p$ by pairwise identifying inverse cycles with one another under the dashed arrows. The points $y_i \in \tilde{\Sigma}$ are related to $y \in \tilde{\Sigma}$ as detailed in the caption of figure \ref{['fig:1']}. The curve $\mathfrak{C}_p$ is homotopic to the boundary curve $\partial D_p$.

Theorems & Definitions (61)

• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Corollary 2.4: See Enriquez:2011, Lemma 9
• Theorem 2.5
• proof
• Lemma 2.6
• proof : Proof of the lemma
• Remark 2.7