Single-valued polylogarithms for higher genera

Abstract: We extend the construction of single-valued polylogarithms at genus one from arXiv:2511.15240 to once-punctured Riemann surfaces of higher genera. The resulting functions have a trivial monodromy representation with respect to the fundamental group, hence they descend to well-defined functions on the surface. Our construction of single-valued polylogarithms is based on Enriquez' connection and relates them to the polylogarithms from D'Hoker-Hidding-Schlotterer. Finally, we identify the Arakelov Green's function within our framework.

• The paper introduces an explicit construction of single-valued polylogarithms on once-punctured Riemann surfaces with arbitrary genus, unifying and generalizing previous polylogarithmic frameworks.
• It employs a generalized KZB equation and flat connection methods to convert multi-valued iterated integrals into globally single-valued functions.
• The results provide a robust analytic foundation for closed-string loop amplitude computations and link higher-genus structures to modular forms and associators.

Single-Valued Polylogarithms for Higher Genera: Summary and Implications

Introduction and Motivation

This work synthesizes an explicit construction of single-valued polylogarithms on once-punctured Riemann surfaces of arbitrary genus $h \geq 1$ (2606.17911). The motivation lies in generalizing the strong analytic and iterated integral structures that underlie tree-level and one-loop amplitude computations in quantum field theory and string theory, where single-valued (genus zero) and elliptic (genus one) polylogarithms play a critical role. The natural expectation for higher-loop closed-string amplitudes is a function space of single-valued, higher-genus polylogarithms (svhgPLs), whose construction has until now been incomplete.

Algebraic and Geometric Framework

The approach builds on the formulation of iterated integrals associated with flat connections on the universal cover of a genus-$h$ Riemann surface $\Sigma$. The fundamental group $\pi_1(\Sigma^\times)$ (for the punctured surface) is presented with generators $A_i, B_i$ (handle cycles) and $C$ (encircling the puncture), satisfying

$\prod_{j=h}^1 B_j^{-1}A_j^{-1}B_jA_j = C$

with a corresponding free Lie algebra $u$ on the symbols $a_i, b_i$.

The key analytic objects are group-like solutions to a generalized KZB equation

$(d - K(z,p;X))\, \genf(z,p;X) = 0$

where $h$0 is a flat $h$1-valued connection, with $h$2, and $h$3 is a generating function for multi-valued polylogarithms with a specified normalization at the puncture.

General Mechanism for Single-Valuedness

The construction adapts Brown's single-valued map on genus zero to higher genus by algebraic manipulation. Multi-valuedness under monodromies around fundamental cycles is encoded via a monodromy representation $h$4. The authors leverage an involutive automorphism $h$5 of the universal enveloping algebra, together with an explicit "trivializing" gauge function $h$6, to produce a single-valued generating function

$h$7

which, by design, has trivial monodromy, provided $h$8 satisfies a system of explicit functional equations tied to the cycles $h$9, $\Sigma$0.

Explicit Realization: Enriquez’ and D’Hoker–Hidding–Schlotterer Connections

The main technical advance is the realization of all ingredients for this construction using Enriquez’s connection as the starting point. The authors construct explicit solutions to the KZB equation on the universal cover in terms of iterated integrals, regularized with tangential basepoints, and assemble group-like generating functions $\Sigma$1. Monodromy under cycles is encoded in series $\Sigma$2, $\Sigma$3, whose explicit form is connected to higher-genus MZVs.

Single-valuedness is achieved by solving a coupled system for the automorphism $\Sigma$4 and for $\Sigma$5, with uniqueness following from invertibility properties of the cycle-period matrix. The explicit realization of the trivialization function $\Sigma$6 is tied to the genus-$\Sigma$7 analog of the gauge transformation constructed by D’Hoker, Hidding, and Schlotterer (DHS), yielding a transparent reduction to the elliptic and genus-zero cases.

Connection to Classical Single-Valued Objects

The coefficients of the generating function for svhgPLs contain as a special case the higher-genus Arakelov Green’s function. This generalizes the identification, in genus one, of the Arakelov Green's function as a coefficient of the single-valued generating series. The relation is established by comparison of asymptotics, functional equations, and, crucially, normalization integrals over the Riemann surface. The systematic identification of higher-genus analogues of dilogarithmic functions (e.g., Bloch–Wigner dilog) is presented as an open direction.

Open Questions and Theoretical Implications

The framework suggests several avenues for further exploration:

• Higher-Genus Associators: The monodromy series $\Sigma$8 may provide explicit realizations of higher-genus Drinfeld associators, linking the present construction to deformation quantization and the Grothendieck–Teichmüller group in higher genus.
• Special Values: The prospect of defining higher-genus analogs of single-valued MZVs and their connection with modular graph functions emerges as a natural development, relevant for the low-energy expansions in string theory.
• Scattering Amplitudes: On the practical side, the constructed svhgPLs provide the correct analytic structure for closed-string loop amplitudes of arbitrary genus, potentially enabling KLT-like or double-copy representations for higher-loop integrands.
• Automorphy and Modularity: The dependence on the period matrix places these functions in a direct relationship with modular forms on Siegel upper half-spaces, inviting further investigation into their automorphic structures.

Conclusion

This paper constructs an explicit, algebraically controlled family of single-valued polylogarithms on punctured Riemann surfaces of arbitrary genus, recasting multivalued iterated integrals into well-defined global objects. By unifying the genus-zero and genus-one cases and relating the analytic content to monodromy and gauge structure, the authors consolidate the analytic foundation for amplitudes in higher-loop closed string and field theory contexts (2606.17911). The formalism establishes a blueprint for further developments in both mathematics and mathematical physics, especially regarding the special values, modular properties, and application to string amplitude computations.