Equivalence of flat connections and Fay identities on arbitrary Riemann surfaces

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.

• The paper proves that the single-valued, modular DHS connection is flat exactly when its integration kernels satisfy all interchange and Fay identities for n≥3.
• The authors decompose Maurer–Cartan commutators in the non-free Lie algebra \hat t_{h,n}, showing linearly independent coefficients reproduce the kernel identities and providing a combinatorial proof of flatness.
• For the meromorphic Enriquez connection, flatness implies the corresponding identities, but the converse remains open because higher-rank cyclic symmetry of coincident-limit tensors is still conjectural.

Overview

This paper by D'Hoker and Schlotterer establishes a precise structural correspondence between two apparently distinct bodies of mathematics surrounding higher-genus polylogarithms on compact Riemann surfaces $\Sigma$ of arbitrary genus $h \geq 1$: the flatness conditions of multivariable flat connections valued in the Lie algebra $\hat t_{h,n}$, and the infinite families of interchange and Fay identities satisfied by the associated integration kernels. The central result is an equivalence theorem for the single-valued, modular, non-meromorphic DHS connection in $n \geq 3$ variables (D'Hoker et al., 1 Feb 2026): its Maurer–Cartan equations hold if and only if all interchange identities ($P=0$) and Fay identities ($F=0$) among DHS kernels are satisfied — the latter having been proven independently in (D'Hoker et al., 2024). For the meromorphic, multiple-valued Enriquez connection (Enriquez, 2011), flatness is shown to imply the analogous identities $Q=0$, $G=0$, but the converse is not established, owing to a conjectural ingredient concerning coincident limits of Enriquez kernels.

The significance of this result lies in unifying two previously separate proof strategies. The interchange and Fay identities were originally derived via analytic methods involving Arakelov Green functions, modular graph functions, and their degeneration limits; here they re-emerge as coefficients of a commutator condition $[J_i^{(1,0)}, J_j^{(1,0)}]=0$ decomposed onto linearly independent elements of a non-freely-generated Lie algebra. Conversely, the combinatorial machinery developed for the kernel identities yields a new route to flatness.

Background: connections, kernels, and the role of $\hat t_{h,n}$

Both constructions take values in the degree completion of the Lie algebra $h \geq 1$0 with generators $h \geq 1$1, $h \geq 1$2, and symmetric $h \geq 1$3, subject to structure relations including $h \geq 1$4 (for $h \geq 1$5), $h \geq 1$6, and vanishing of $h \geq 1$7 and $h \geq 1$8 for distinct indices. A crucial qualitative difference from the one-variable case is that $h \geq 1$9 is freely generated only for $\hat t_{h,n}$0; for $\hat t_{h,n}$1 it is not free, so the components of any connection built from independent kernels cannot be independent as Lie-algebra-valued objects. This failure of freeness is precisely what forces relations among bilinears of integration kernels.

The DHS kernels $\hat t_{h,n}$2 are defined recursively via convolution integrals against the Arakelov Green function and admit a trace/traceless decomposition $\hat t_{h,n}$3. The multivariable DHS connection is single-valued and modular invariant but not holomorphic:

$\hat t_{h,n}$4

while the Enriquez connection is meromorphic and holomorphic, $\hat t_{h,n}$5, built from Enriquez kernels $\hat t_{h,n}$6 with an analogous decomposition into a $\hat t_{h,n}$7-independent traceless part $\hat t_{h,n}$8 and trace part $\hat t_{h,n}$9. The companion paper (D'Hoker et al., 1 Feb 2026) showed that the two connections are related by a gauge transformation composed with a Lie algebra automorphism, which immediately transfers flatness between them.

Reduction of flatness to kernel identities

Among the Maurer–Cartan equations, most components are "easy": the $n \geq 3$0 equations follow from the Massey structure of the differential equations satisfied by the kernels, and the mixed conditions $n \geq 3$1 follow from reflection properties of the trace parts. The substantive content resides in the commutator conditions

$n \geq 3$2

which vanish trivially at $n \geq 3$3 and impose nontrivial bilinear constraints on kernels for $n \geq 3$4.

The paper's core technical achievement is the decomposition of $n \geq 3$5 onto four sequences of Lie algebra elements proven to be linearly independent for $n \geq 3$6 (and empty or trivially absent in special cases such as genus $n \geq 3$7). The decomposition proceeds by splitting the commutator according to the number of exposed $n \geq 3$8-generators, $n \geq 3$9, and reducing each piece using the structure relations together with an extensive combinatorial apparatus: shuffle products, antipodes, deconcatenation coproducts, and a generalized Leibniz rule for products $P=0$0 acting as nested adjoints of $P=0$1-generators. The resulting master formula expresses the commutator as a linear combination whose coefficients are exactly the Fay combination $P=0$2, the interchange combination $P=0$3, the traceless part $P=0$4 of $P=0$5, and a residual combination $P=0$6 built from coincident limits of Fay identities and dihedral symmetries of modular tensors $P=0$7.

Linear independence then forces each coefficient to vanish separately. A short induction on word length converts the shuffled sums over $P=0$8 into pointwise vanishing of $P=0$9, which combined with the trace part recovers the full Fay identity $F=0$0. The converse direction uses the fact that the dihedral reflection identities $F=0$1 imply the vanishing of the auxiliary combinations $F=0$2, so that $F=0$3 and $F=0$4 kill every term in the decomposition. Thus:

$F=0$5

An important caveat concerns the special case $F=0$6: the configuration space of two points does not accommodate three distinct points, so full Fay identities cannot be formulated there, and the analysis requires modifications supplied in remarks and footnotes. The equivalence theorem is stated for $F=0$7.

The Enriquez side: implication without equivalence

Because the DHS and Enriquez connections are gauge-equivalent up to a Lie algebra automorphism, the vanishing of $F=0$8 is equivalent to that of $F=0$9. Moreover, the entire combinatorial reduction of the commutator depends only on the structure relations of $Q=0$0 and on the $Q=0$1-independence of the traceless parts of the kernels — properties shared by both kernel families. Substituting $Q=0$2, $Q=0$3, $Q=0$4 throughout therefore yields the parallel decomposition of $Q=0$5 with coefficients $Q=0$6, $Q=0$7, $Q=0$8 (the traceless part of $Q=0$9), and $G=0$0. Flatness of $G=0$1 consequently implies all interchange identities $G=0$2 for Enriquez kernels (previously proven directly) and all Fay identities $G=0$3 (previously conjectured in (D'Hoker et al., 2024) and later proven by Baune et al. (Baune et al., 2024)), providing a unified alternative proof.

The reverse implication fails to go through for a specific, clearly identified reason. In the DHS case, the coincident limits $G=0$4 of the angularly singular kernels admit a decomposition in terms of modular tensors $G=0$5 whose cyclic symmetry follows from explicit convolution-integral representations; this symmetry underwrites the vanishing of the auxiliary quantities $G=0$6 needed for the converse. For Enriquez kernels, the meromorphic analogue of this decomposition is conjectural: consistency with $G=0$7 monodromies requires the cyclic symmetry $G=0$8 of the moduli-dependent tensors $G=0$9, which has been verified only at rank two. Without it, the vanishing of the counterparts $[J_i^{(1,0)}, J_j^{(1,0)}]=0$0 cannot be derived from the kernel identities alone. The paper notes that if the conjectural decomposition holds, flatness would additionally force the reflection parity $[J_i^{(1,0)}, J_j^{(1,0)}]=0$1, upgrading the tensors to full dihedral symmetry — but this conditional statement is explicitly left open.

Limitations and open questions

Three restrictions bound the results. First, the equivalence for DHS kernels and the implication for Enriquez kernels both require $[J_i^{(1,0)}, J_j^{(1,0)}]=0$2 variables; the $[J_i^{(1,0)}, J_j^{(1,0)}]=0$3 case admits only partial statements because Fay identities intrinsically involve three points. Second, the Enriquez correspondence is one-directional: promoting it to an equivalence hinges on proving the rank-$[J_i^{(1,0)}, J_j^{(1,0)}]=0$4 cyclic symmetry of the tensors $[J_i^{(1,0)}, J_j^{(1,0)}]=0$5 appearing in the conjectural decomposition of the coincident limits $[J_i^{(1,0)}, J_j^{(1,0)}]=0$6, together with the associated discrete symmetries left conjectural in section 9.4 of (D'Hoker et al., 2024). Third, the paper treats only the punctureless configuration spaces; the punctured variants of the multivariable DHS connection are deferred to the companion paper. Whether the flatness-based method can be extended to yield genuinely new identities beyond those already known — for instance at $[J_i^{(1,0)}, J_j^{(1,0)}]=0$7 or in degeneration limits — remains unaddressed.

Conclusion

The paper demonstrates that the flatness of the multivariable DHS connection is equivalent, coefficient by coefficient against a linearly independent basis of $[J_i^{(1,0)}, J_j^{(1,0)}]=0$8, to the complete system of interchange and Fay identities among DHS kernels, and that flatness of the Enriquez connection implies the corresponding identities for Enriquez kernels. The result reframes analytic functional identities among higher-genus polylogarithm kernels as consequences of Lie-theoretic structure relations, and conversely supplies a purely combinatorial route to flatness. The principal open problem left by the work is the status of the cyclic symmetry of the moduli-dependent tensors governing meromorphic coincident limits, whose resolution would upgrade the Enriquez statement from an implication to a full equivalence.