Higher-Genus Polylogarithms: Theory & Applications

• Higher-genus polylogarithms are iterated-integral functions defined on punctured compact Riemann surfaces, incorporating monodromy and complex-structure data.
• They are constructed via methods such as Enriquez's connection, modular tensor approaches, and family-theoretic gluing, each revealing distinct analytic and algebraic properties.
• These functions underlie advances in multiloop string theory and Feynman integrals, unifying KZ/KZB techniques with Arakelov geometry through explicit kernel evaluations.

Higher-genus polylogarithms are iterated-integral-type functions attached to punctured or varying compact Riemann surfaces of genus $h\ge 1$, defined by expanding horizontal sections of flat connections or, equivalently in several constructions, by iterating distinguished integration kernels adapted to the topology and complex structure of the surface. In current usage the term includes meromorphic multiple-valued theories based on Enriquez’ connection, single-valued non-meromorphic theories based on modular tensors and Arakelov geometry, family-theoretic constructions near stable degenerations, and, more recently, single-valued higher-genus polylogarithms on once-punctured compact surfaces obtained by trivializing monodromy (Baune et al., 16 Jun 2026, D'Hoker et al., 13 Jan 2025, Ichikawa, 2022, D'Hoker et al., 2023).

1. Geometric setting and basic notion

A standard setting is a compact Riemann surface $\Sigma$ of genus $h$ with a marked point $p$, together with the punctured surface

$\Sigma^\times=\Sigma\setminus\{p\}.$

For once-punctured surfaces, the topology is governed by

$$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$

where $A_i,B_i$ are handle cycles and $C$ is a small loop around the puncture. In this context, multivaluedness of iterated integrals is controlled by $\pi_1(\Sigma^\times)$, and “single-valued” means trivial monodromy with respect to that group, so that a function on the universal cover descends to a genuine function on $\Sigma^\times$ (Baune et al., 16 Jun 2026).

A general connection-theoretic formulation starts with a flat $\Sigma$0-form $\Sigma$1 on the universal cover of a punctured manifold, valued in the completion of a free Lie algebra generated by letters $\Sigma$2. The differential equation

$\Sigma$3

has a path-ordered exponential solution

$\Sigma$4

and expansion in words $\Sigma$5 in the alphabet $\Sigma$6,

$\Sigma$7

defines homotopy-invariant iterated integrals $\Sigma$8. In this sense, higher-genus polylogarithms are the word coefficients of holonomies of appropriate flat connections; because the generating series is group-like, the coefficients satisfy shuffle relations (D'Hoker et al., 13 Jan 2025).

The same idea appears in family form. For a family $\Sigma$9 of pointed compact Riemann surfaces with $h$0, a polylogarithm sheaf is a vector bundle with flat connection that is holomorphic on the punctured total space and has only simple poles along the marked sections. In this formulation, higher-genus polylogarithms are monodromy coefficients of a universal unipotent meromorphic connection varying over a neighborhood of a boundary point of moduli (Ichikawa, 2022).

2. Principal constructions and their mutual relation

Several constructions coexist in the literature, but they are not simply competing definitions. They emphasize different analytic properties.

Framework	Basic input	Character
Enriquez	Meromorphic flat connection $h$1	Multiple-valued, holomorphic/meromorphic
D’Hoker–Hidding–Schlotterer	Flat connection built from modular tensors and Arakelov Green function	Single-valued, non-meromorphic, modular-invariant
Ichikawa	Universal polylogarithm sheaf on families near stable curves	Local analytic over moduli, computable in plumbing coordinates
Single-valued higher-genus	Trivial-monodromy combination of higher-genus generating series	Honest functions on $h$2

The Enriquez connection is the higher-genus analogue of a KZB-type meromorphic system. Its coefficients are meromorphic kernels with prescribed simple poles, $h$3- and $h$4-cycle monodromies, and fixed $h$5-periods. By contrast, the D’Hoker–Hidding–Schlotterer construction uses a non-meromorphic but single-valued flat connection built from the Arakelov Green function and modular tensors; it generalizes the Brown–Levin philosophy from genus one to arbitrary genus by preferring single-valuedness over meromorphicity (D'Hoker et al., 2023).

A common misconception is that these constructions define unrelated function spaces. In fact, two recent comparison results make the opposite point. First, the paper on flat connections and higher-genus polylogarithms shows that the Enriquez connection and the D’Hoker–Hidding–Schlotterer connection are explicitly related by gauge transformation together with an automorphism of the free Lie algebra. At the level of function spaces this yields

$h$6

so the Enriquez polylogarithms are precisely the holomorphic slice of the larger DHS algebra on a fixed punctured surface (D'Hoker et al., 13 Jan 2025). Second, the family-theoretic construction of Takashi Ichikawa produces higher-genus polylogarithms on neighborhoods of boundary strata in $h$7 by gluing genus-zero KZ and genus-one reduced KZB blocks, thereby connecting the arbitrary-genus story directly to degeneration geometry and deformation theory (Ichikawa, 2022).

3. Kernels, generating series, and algebraic identities

In the meromorphic framework, the basic higher-genus kernels are Enriquez kernels

$h$8

defined on the universal cover of a genus-$h$9 surface with normalized holomorphic Abelian differentials $p$0 and period matrix $p$1. At rank $p$2,

$p$3

and at rank $p$4 they have a simple pole in $p$5 at $p$6 with residue $p$7. A central explicit formula is

$p$8

where $p$9 is the prime form. This realizes the first nontrivial kernels directly from standard geometric data of the surface (D'Hoker et al., 20 Feb 2025).

The higher-rank kernels admit a recursive construction by homotopy-invariant convolution over $\Sigma^\times=\Sigma\setminus\{p\}.$0-cycles. The defining formula expresses $\Sigma^\times=\Sigma\setminus\{p\}.$1 in terms of

$\Sigma^\times=\Sigma\setminus\{p\}.$2

together with explicit Bernoulli-number corrections. One of the main structural results is that the space of Enriquez kernels is closed under such convolutions and also under variation of the complex structure; the variational formula

$\Sigma^\times=\Sigma\setminus\{p\}.$3

is again bilinear in Enriquez kernels. This provides a workable kernel calculus rather than only an existence-and-uniqueness statement (D'Hoker et al., 20 Feb 2025).

The same meromorphic theory is controlled by higher-genus Fay-like identities. For Enriquez’ generating function and the induced integration kernels, a complete system of quadratic three-point identities has been proved, with a higher-genus Fay-like identity as the unique nontrivial quadratic identity modulo trivial linear ones. These identities perform the same role that Fay identities play in genus one: they reduce products of kernels with repeated integration variables to admissible combinations and thereby support the manipulation of iterated integrals (Baune et al., 2024).

An explicit Schottky-uniformized realization is also available. Schottky–Kronecker forms $\Sigma^\times=\Sigma\setminus\{p\}.$4 are defined as Möbius-invariant Schottky averages of genus-one Kronecker data. Expanding

$\Sigma^\times=\Sigma\setminus\{p\}.$5

produces higher-genus kernels as Poincaré series, and under convergence assumptions these kernels coincide with Enriquez’ differentials. For real hyperelliptic curves, where the relevant Schottky groups are classical and circle-decomposable, these Poincaré series can be evaluated numerically; the paper reports numerical evaluation of several genus-two polylogarithms (Baune et al., 2024).

4. Single-valued higher-genus polylogarithms

The most direct higher-genus notion of single-valuedness is formulated on a once-punctured compact Riemann surface. Starting from a group-like solution

$\Sigma^\times=\Sigma\setminus\{p\}.$6

of a flat KZB-type system, one expands

$\Sigma^\times=\Sigma\setminus\{p\}.$7

and the coefficients $\Sigma^\times=\Sigma\setminus\{p\}.$8 are higher-genus polylogarithms. Single-valuedness means that the monodromy representation of the modified generating series is trivial with respect to $\Sigma^\times=\Sigma\setminus\{p\}.$9, so the resulting series descends from the universal cover to a well-defined function on $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$0 (Baune et al., 16 Jun 2026).

The concrete higher-genus construction uses Enriquez’ connection $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$1, characterized by $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$2-invariance, $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$3-cycle conjugation by $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$4, and residue

$$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$5

at the puncture. The associated multivalued generating series $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$6 is normalized by

$$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$7

Its weight-one term is

$$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$8

so Abelian integrals provide the first layer of the theory (Baune et al., 16 Jun 2026).

The single-valued ansatz combines conjugation, an automorphism $$\pi_1(\Sigma^\times)= \bigg\langle A_i,B_i,C\,\Big|\,i\in[h],\,\prod_{j=h}^1 B^{-1}_jA^{-1}_jB_jA_j=C \bigg\rangle,$$9, and a non-holomorphic trivialization $A_i,B_i$0 that compensates the geometric monodromy. For Enriquez’ connection the single-valued generating series is

$A_i,B_i$1

The higher-genus single-valued conditions are the $A_i,B_i$2 equations

$A_i,B_i$3

and solvability is reduced to invertibility of the weight-one monodromy matrix

$A_i,B_i$4

This yields a generating series

$A_i,B_i$5

of single-valued higher-genus polylogarithms (Baune et al., 16 Jun 2026).

A second misconception is that single-valued higher-genus polylogarithms should remain holomorphic. The higher-genus construction shows the opposite: already at genus one a non-holomorphic correction is needed, and in higher genus the $A_i,B_i$6-sector acquires a correction involving $A_i,B_i$7. The genus-one predecessor made this mechanism explicit on the once-punctured torus through elliptic associators and a recursively determined second alphabet; the higher-genus paper is presented as its extension from genus one to compact surfaces of genus $A_i,B_i$8 (Baune et al., 19 Nov 2025).

A central application of the single-valued construction is the identification of the Arakelov Green’s function. If $A_i,B_i$9 is the Arakelov Green’s function and $C$0 is defined from the real part of the coefficient of $C$1 in the single-valued series, then the paper proves that $C$2 and $C$3 have the same local singularity and satisfy the same differential equation, hence differ by an additive constant fixed by normalization. In this sense the Arakelov Green’s function occurs inside the weight-two part of the single-valued higher-genus generating series (Baune et al., 16 Jun 2026).

5. Families, degeneration, and computability

The family-theoretic approach constructs higher-genus polylogarithms near maximally degenerate stable curves. One begins with a stable graph $C$4 of type $C$5, glues copies of $C$6 along plumbing relations

$C$7

and inserts on loop components the reduced elliptic KZB connection while keeping KZ connections on rational components. The resulting vector bundle with flat meromorphic connection $C$8 analytically continues to nearby smooth curves and is called the universal polylogarithm sheaf. Its monodromies are products of KZ associators, logarithmic annulus factors $C$9, and elliptic KZB monodromies; therefore the associated higher-genus polylogarithms are computable as power series in deformation parameters and logarithms, with coefficients “essentially expressed by multiple zeta values” and elliptic multiple zeta data (Ichikawa, 2022).

The degeneration of the Enriquez connection sharpens this picture. For a plumbing family $\pi_1(\Sigma^\times)$0 obtained by gluing elliptic curves $\pi_1(\Sigma^\times)$1 to a rational spine, the normalized Abelian differentials and third-kind differentials converge to stable differentials, and the basic Enriquez kernels satisfy

$\pi_1(\Sigma^\times)$2

Consequently, on elliptic components the connection degenerates to the elliptic KZB connection, while on the rational component it degenerates to the KZ connection with residues determined by the commutators $\pi_1(\Sigma^\times)$3. The paper concludes that the Enriquez connection becomes the connection constructed earlier for degenerating pointed curves, and therefore the associated higher-genus polylogarithms admit explicit power-series expansions in deformation parameters and their logarithms with coefficients expressed by multiple zeta values (Ichikawa, 1 Oct 2025).

This degeneration theory is not merely asymptotic bookkeeping. It clarifies why higher-genus polylogarithms simultaneously display genus-zero, genus-one, and genuinely higher-genus features: near the boundary of moduli, the geometry decomposes into KZ and KZB building blocks, but the global object remains a higher-genus connection or higher-genus polylogarithm sheaf (Ichikawa, 2022).

6. Applications, related structures, and conceptual significance

A principal application lies in multiloop string theory. Cyclic products of Szegő kernels on a compact genus-$\pi_1(\Sigma^\times)$4 surface admit decompositions

$\pi_1(\Sigma^\times)$5

where all spin-structure dependence is isolated in modular tensors $\pi_1(\Sigma^\times)$6, while the marked-point dependence sits in coefficients $\pi_1(\Sigma^\times)$7 built from the integration kernels of higher-genus polylogarithms. In this sense, higher-genus polylogarithm kernels generalize the role of Parke–Taylor factors at genus zero and Kronecker–Eisenstein kernels at genus one in organizing multiloop fermionic correlators (D'Hoker et al., 2023).

The subject also interfaces with Feynman integrals through its lower-genus precursors. Elliptic generalizations of polylogarithms already arise for the massive two-loop sunrise integral, where ordinary multiple polylogarithms fail because the underlying geometry is elliptic. This suggests that the enlargement of function spaces by the geometry of the relevant algebraic curve is not specific to genus one; a plausible implication is that genuinely higher-genus geometries should require higher-genus analogues built from higher period data and higher-genus kernels (Bogner, 2016).

More broadly, the field sits at the intersection of iterated integrals, free Lie algebras, period matrices, Arakelov geometry, and modular tensor structures. Meromorphic and single-valued constructions should not be conflated: one side is closer to KZ and regular-singular connections, the other to single-valued non-holomorphic geometry and modular invariance. The comparison theorem between the Enriquez and DHS connections shows that these are complementary realizations of one higher-genus iterated-integral theory rather than incompatible definitions (D'Hoker et al., 13 Jan 2025).

Taken together, the recent literature establishes a concrete function-theoretic framework for higher-genus polylogarithms. It now includes explicit meromorphic kernels from prime forms and Abelian differentials, exhaustive Fay-like identities, recursive closure under convolution and moduli variation, local analytic constructions on families of degenerating curves, explicit comparisons between meromorphic and single-valued flat connections, and a single-valued theory on once-punctured compact Riemann surfaces in which the Arakelov Green’s function appears as a weight-two coefficient. This does not yet collapse all approaches into a single formalism, but it shows that higher-genus polylogarithms have moved from heuristic analogy to a structured theory with explicit constructions, comparison theorems, and concrete applications (Baune et al., 16 Jun 2026).