Source-linked AI summary
From polygons and symbols to polylogarithmic functions
Claude Duhr, Herbert Gangl, John R. Rhodes
TL;DR
The paper reviews the symbol map as a tool for simplifying functional equations among multiple polylogarithms and gives a diagrammatic construction from rooted decorated polygons. It also develops an approach for integrating symbols, applies it to harmonic polylogarithms through weight four, and identifies ambiguity from the symbol map’s kernel.
Problem
Functional equations among multiple polylogarithms are difficult to establish directly, while comparing their symbols provides a usually easier necessary condition for equality modulo such equations.
Method
The paper combines a review of symbol-map properties with a diagrammatic rule using maximal dissections of decorated polygons and a systematic projector-based approach to integrating integrable symbols.
Results
The paper gives a direct decorated-polygon construction of multiple-polylogarithm symbols, constructs candidate functions matching integrable tensors, and derives a spanning set of indecomposable harmonic polylogarithms up to weight four.
Takeaways & Limitations
Symbol calculations can simplify multiple-polylogarithm relations and support function reconstruction, but matching symbols does not by itself identify the original function because kernel terms remain.
Takeaways & Limitations
The integration approach is not a complete algorithmic proof, may fail in some scenarios, and recursive symbol formulas require regularization for non-generic arguments.
Abstract
from arXiv · showhide
We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.
1. Introduction
The paper reviews the symbol map as a way to simplify functional relations among multiple polylogarithms and extends its use through polygon-based symbol extraction and symbol integration.
- Motivation: The symbol map assigns a weight-n multiple polylogarithm an n-fold tensor that captures important combinatorial and analytical properties.Matching symbols provides a necessary condition for equality modulo functional equations.
- Motivation: The paper reviews symbol-map applications in physics, especially its use in simplifying analytic expressions for N = 4 SYM amplitudes.Earlier applications included the two-loop six-point remainder function and symbols for higher-point amplitudes.
- Contributions: The authors introduce a diagrammatic rule that reads a multiple polylogarithm’s symbol from triangulations of an associated decorated polygon.This complements the previously recursive definition through iterated differentials.
- Contributions: They also present an approach for constructing a candidate function whose symbol matches a given tensor.The paper illustrates this integration problem for harmonic polylogarithms and discusses higher-weight extensions.
- Scope: The paper is written to accommodate both physics and mathematics audiences, sometimes trading depth for breadth of exposition.The authors warn that some arguments may be too detailed for some readers and too sketchy for others.
2. Short review of multiple polylogarithms
This section reviews multiple polylogarithms as iterated integrals, their algebraic properties and representations, and their role in constructing spanning sets for applications such as Feynman integrals.
- Definitions: Multiple polylogarithms are defined recursively as iterated integrals, with the number of singularity entries determining their weight.The singularities are collected in a vector, counted with multiplicity.
- Algebraic properties: Iterated integrals obey shuffle-algebra relations, expressing products of weights n1 and n2 as integer combinations of weight n1 + n2 functions.The shuffle set preserves the internal order of each factor’s indices.
- Algebraic reductions: Algebraic relations reduce some multiple polylogarithms modulo lower-weight products to functions with restricted rightmost indices.An explicit weight-three identity illustrates separating a desired term from product terms.
- Representations: Multiple polylogarithms also admit nested-sum representations when convergence conditions such as |x_i| < 1 hold.The paper notes that summation conventions can differ by reversing the ordering of indices.
- Spanning sets: The paper’s algorithmic aim is to find a possibly minimal spanning set for multiple polylogarithms with chosen singularities.The method uses tensor calculus associated with iterated integrals, called symbol calculus.
- Harmonic polylogarithms: Harmonic polylogarithms specialize the singularities to {-1, 0, 1} and are important because many one- and two-loop Feynman integrals use them through weight four.Through weight three they reduce to classical polylogarithms, whereas weights four and above are expected to require more general functions.
3. Symbols and polygons
The paper gives a combinatorial construction of polylogarithm symbols from rooted decorated polygons and shows that it agrees with the recursive differential definition.
- Polygon representation: A weight-n multiple polylogarithm corresponds to a rooted decorated (n + 1)-gon whose symbol lies in an n-fold tensor power.The polygon encodes part of the function’s differential structure.
- Dissections: The construction begins by listing maximal sets of non-intersecting arrows, with n − 1 arrows for an (n + 1)-gon.For the weight-three example, each maximal set has two arrows and there are twelve such sets.
- Trees and shuffles: Each dissection yields a rooted dual tree of decorated 2-gons, and compatible linear orders contribute symbol terms.Branching trees contribute shuffle products of the tensors associated with their branches.
- Tensor factors and signs: The 2-gons are mapped to rational functions of the polygon decorations, while signs are fixed by the number of backward arrows.The rolled-out polygon makes forward and backward arrow orientations explicit.
- Equivalence: The polygon construction and recursive symbol definition are equivalent, up to rearrangement and symbol additivity.The equivalence is checked through a bijection of terms and explicit agreement in a weight-two case.
- Scope and caveat: The polygon method handles degenerate arguments combinatorially, whereas the recursive approach requires regularization for non-generic cases.Degeneracies can be identified through arrows ending on zero-decorated sides and discarded from the outset.
4. A simple example
A weight-two example shows how symbol calculus derives a functional equation for G(−1, 1; x), reconstructs a matching classical-polylogarithm expression, and fixes its undetected constant separately.
- Example setup: The example uses G(−1, 1; x), which coincides with the harmonic polylogarithm −H(−1, 1; x), to illustrate functional-equation detection.The paper deliberately derives the relation with tensor calculus rather than applying a known closed formula immediately.
- Symbol calculation: The resulting symbol of G(−1, 1; x) is (1 + x) ⊗ 2 + (1 − x) ⊗ (1 + x) − (1 − x) ⊗ 2.The recursive differential calculation produces three corresponding terms, using the refined d log prescription.
- Tensor decomposition: The symbol is decomposed into symmetric and antisymmetric tensor parts to distinguish dilogarithm contributions from products of logarithms.Products of logarithms contribute no antisymmetric component, enabling a bootstrap-style subtraction.
- Constant fixing: Matching symbols determines the functional expression only up to an additive constant independent of x.The constant is fixed by specializing to x = 0 and using the known value of Li2(1/2).
5. Integrating symbols: an algorithmic approach
The paper presents an algorithm for constructing polylogarithmic functions whose symbols match integrable tensors, using selected function types, rational-function arguments, and projectors that remove shuffle products. It also identifies important ambiguities and kernel elements that prevent symbol matching from uniquely determining the original function.
- Integrating the symbol: The algorithm seeks a linear combination of multiple polylogarithms and products whose symbol equals a given integrable tensor S.The target function may use rational functions of the input variables as arguments.
- Integrating the symbol: At higher weight, the main challenges are choosing arguments whose symbols span S and generalizing tensor decompositions beyond weight two.The paper discusses these issues through weight four and lists function types such as Li4 and Li2,2.
- Choosing arguments: The construction factors rational functions into irreducible polynomials, which become multiplicatively independent building blocks for the tensor algebra.This turns the symbol into an element of the weight-w grading over the formal basis generated by those factors.
- Choosing arguments: The candidate function space is organized from simple, indecomposable function types, while arguments are selected from rational-function sets generated by the symbol’s building blocks.The procedure uses classical polylogarithm candidates and extends the construction through higher weights.
- Finding the arguments: No new rational functions are needed for R(2)_S because its information is already contained in R(1)_S, with additional symmetry constraints on the arguments.The construction enlarges the relevant factor set using expressions such as π_i ± π_j and 1 ± π_i.
- Projectors: Projectors Π_w annihilate precisely shuffle products, are idempotent, and help isolate components that cannot be expressed as linear combinations of shuffles.This projector structure supports the decomposition and elimination steps used in the algorithm.
- Kernel ambiguity: Matching a symbol does not uniquely recover an analytic expression because the original function may differ from the constructed one by elements in the symbol’s kernel.Multiple zeta values and certain nontrivial linear combinations provide examples of zero-symbol contributions.
6. Application: a spanning set for harmonic polylogarithms
The section constructs a spanning set for harmonic polylogarithms by classifying admissible arguments and applying symbol-based linear algebra, reaching a complete expression up to weight four.
- Argument classification: HPL-associated polygons have root decoration x and other decorations 0 or ±1, yielding a tensor basis generated by [x], [1−x], [1+x], and.The resulting rational-function class is RHPL = {±2^δ x^α (1−x)^β (1+x)^γ}.
- Argument classification: The classical-polylogarithm sector requires 16 different arguments, although inversion and distribution identities generate relations among them.For x ∈ [0,1], inverted arguments can be reduced using the inversion formula, while the distribution formula relates selected arguments.
- Spanning set: The authors find a spanning set of indecomposable functions sufficient for all HPLs through weight three, then add three multiple-polylogarithm functions to cover weight four.The displayed basis uses weight-specific functions B_i^(j), with three supplementary functions needed from weight four onward.
- Scope and choices: The chosen spanning set is not unique and is motivated by manifest reality on x ∈ [0,1]; outside this interval, branching can be more complicated.The authors note that the branch structure outside the interval is addressed separately.
- Symbol calculation: For H(0,0,1,1;x), the symbol is (1−x) ⊗ (1−x) ⊗ x ⊗ x.The symbol can be obtained either from the polygon dissection or recursively from differential equations.
- Symbol integration: Projectors isolate weight partitions and produce linear systems for coefficients, which are solved recursively by subtracting matched symbol contributions.The procedure successively applies Π4, Π3 ⊗ Π1, and Π2 ⊗ Π2 before determining the remaining logarithmic products.
7. Conclusion
The conclusion presents the symbol map, a polygon-based construction of symbols, and a systematic but non-universal method for integrating them, illustrated through HPLs up to weight four.
- Contributions: The symbol map associates a weight-n multiple polylogarithm with an n-fold tensor while preserving many combinatorial properties and functional equations.The paper supplements the recursive definition with a direct weighted sum over maximal dissections of decorated polygons.
- Contributions: The proposed integration procedure constructs a candidate spanning set and uses projectors to find a function matching an integrable target tensor.Its validity assumes that the candidate functions suffice to express the integrated symbol.
- Scope and applications: The approach is not a complete algorithmic proof and is not adequate in every scenario, but it has been applied to compact analytic results for one-loop hexagon integrals.The cited applications concern one-loop hexagon integrals in D = 6 dimensions.
- Scope and applications: The paper derives a spanning set for harmonic polylogarithms up to weight 4, which was used for an efficient numerical implementation.The numerical implementation is reported as a subsequent use of the derived spanning set.
A. Review on shuffle algebras
This appendix reviews algebras, tensor algebras, and shuffle algebras, then relates words, concatenation, and shuffles to multiple polylogarithms and their symbols.
- Algebraic preliminaries: An algebra over a field is a vector space with associative and distributive multiplication, while a unital algebra additionally has a unit element.The appendix also defines graded algebras through decomposition by word or tensor length.
- Tensor algebras: The tensor algebra T(V) is built from tensor powers of V, with multiplication given by concatenation of elementary tensors.The empty tensor supplies the scalar component T0(V) = F.
- Ideals: The kernel of an algebra homomorphism is an ideal, because multiplication by any algebra element preserves membership in the kernel.This follows directly from φ(a · b) = φ(a) · 0 = 0 for b in the kernel.
- Shuffle algebras: A shuffle algebra consists of formal linear combinations of words equipped with a product summing over order-preserving interleavings of two words.The shuffle product is graded by word length and has the empty word as its unit.
- Polylogarithmic application: For multiple polylogarithms, letters are singularities, words are singularity vectors, concatenation joins vectors, and word length equals polylogarithm weight.This identifies the multiple-polylogarithm product structure with a shuffle algebra.
B. Selected examples of symbols
This appendix gives symbol formulas for common polylogarithms and explains how maximal dissections of decorated polygons generate symbols of generic multiple polylogarithms.
- Classical and Nielson polylogarithms: The classical polylogarithm has symbol S(Li_n(x)) = −(1−x) ⊗ x ⊗ ... ⊗ x, with n−1 copies of x.For n = 1, this agrees with Li1(x) = −log(1−x).
- Classical and Nielson polylogarithms: The Nielson polylogarithm has symbol (−1)^p (1−x)^⊗p ⊗ x^⊗n and contains classical polylogarithms as a special case.The special case is S_{n−1,1}(x) = Li_n(x).
- Harmonic polylogarithms: For HPLs with singularities in {0,1}, the symbol is (−1)^k (a_n−x) ⊗ ... ⊗ (a_1−x), where k counts entries equal to 1.The compact expression follows because only one maximal polygon dissection contributes nontrivially.
- Harmonic polylogarithms: Generic HPLs with singularities in {−1,0,1} lack a compact symbol formula, so their symbols are obtained from generic multiple-polylogarithm formulas.The appendix reviews generic constructions through weight four.
- Polygon constructions: For generic multiple polylogarithms, weight one uses a bigon, weight two a trigon with three maximal dissections, and weight three a tetragon with twelve.At weight four, a pentagon generates 55 maximal dissections.
- Polygon constructions: Each maximal dissection contributes a tensor term, with shuffle notation organizing the sums for higher-weight symbols.The weight-four construction explicitly uses three non-intersecting arrows and permutation sums over subsets of decorations.
C. Proof of Proposition 4
The proof translates polygon dissections into hook-arrow trees and uses their combinatorics to extract symbols. This perspective identifies the nonzero dissections needed for the proposition and establishes the stated zero-symbol cases.
- Colored multiple zeta values are first related to multiple polylogarithms so their symbols can be represented by decorated polygons.
- Hook-arrow trees: Each full polygon dissection uniquely corresponds to a rooted, non-interlaced spanning tree whose root is the vertex on the final polygon side.Tree edges join side-midpoint vertices without crossing dissection arrows and are oriented toward the root.
- Symbol extraction: The symbol term is obtained by ordering hook-arrow-tree edges as the corresponding polygon 2-gons, while the dissection sign is (−1)^α for α backward edges.
- Proof strategy: Hook-arrow trees provide an equivalent but easier-to-view representation of maximal dissections, while the proof uses the dual-tree formulation.
- Vanishing cases: Certain decorated polygons have zero symbol because every possible connection involving their labeled vertices forces a zero coefficient, leaving no nontrivial dissection.
D. Some considerations on the implementation of the algorithm
The implementation constructs candidate rational-function sets subject to factorization constraints, then reduces the search using finite truncation and fast necessary tests. Infinite search spaces and slow polynomial factorization remain practical challenges.
- The algorithm selects elements of R(k) according to factorization properties defined recursively from the set R.
- Search-space reduction: The infinite set R is handled by decomposing it into subsets R_n and truncating the construction at a finite exponent bound N.The paper notes that large exponent sums are not expected in practical applications.
- Implementation constraints: Polynomial factorization remains slow on many computer algebra systems, creating serious speed issues during implementation.
- Observed behavior: Empirically, the set R(1) appears to be finite in general.
- Fast filtering: A necessary membership test evaluates polynomial divisibility after substituting distinct prime numbers, converting polynomial division into faster integer division.The test is necessary but not necessarily sufficient, so surviving candidates require further checking.
E.2 Analytic representation outside the unit disc: inversion relations
The paper analytically continues spanning-set functions beyond the unit disc using inversion relations supplemented by lower-weight products. Branch choices require special care on the unit circle and for real arguments.
- Functions with representations inside the unit disc are continued outside it through inversion relations involving products of lower-weight functions.
- Formula coverage: The appendix supplies inversion formulas for the spanning set at weights three and four, while the main text displays formulas for weights one and two.
- Real arguments: For real x, σ(x) is ambiguous and is fixed using the physics convention x → x + iε; Schwarz reflection still requires care when applying inversion formulas.
- Unit-circle boundary: For |x| = 1, the inside-versus-outside representation is ambiguous, although numerical checks find agreement except for B(14).
- Branch choice: The exceptional unit-circle case has an ambiguity in the imaginary part that can be resolved by requiring continuity near the unit circle.
G.1 Results for weight two
The weight-two examples express harmonic polylogarithms in terms of logarithms and classical polylogarithms. These identities form part of a minimal set from which other cases follow through shuffle relations.
- H(−1, 1; x) is represented using log 2, logarithms of 1 − x and 1 + x, and a constant term.
- H(0, 1; x) equals the classical dilogarithm Li2(x).
G.2 Results for weight three
The weight-three results list explicit expressions involving logarithms, powers of π, and the trilogarithm, together with a Mathematica file containing harmonic polylogarithms through weight four.
- The section presents weight-three expressions built from log(1 + x), log(1 − x), their products, and powers of logarithms.
- Several displayed results combine logarithmic terms with π2 and Li3.
- A Mathematica distribution includes text files containing expressions for all HPLs up to weight four for x ∈ [0, 1].
G.3 Results for weight four
The weight-four results provide numerous explicit formulas combining logarithms, polylogarithms, zeta values, and powers of π, including named harmonic-polylogarithm identities.
- The displayed expressions also contain ζ3, π2, and powers of log 2 as constant factors or additive terms.
- Several formulas include Li2, Li3, or Li4 evaluated at x, −x, or 1 − x.
- Additional formulas mix logarithmic products with Li2, Li3, Li4, ζ3, and π2 across multiple weight-four combinations.
- The section lists weight-four expressions formed from products and powers of log(1 + x), log(1 − x), and log x.
- Named identities are given for H(0, 0, −1, −1; x) and H(0, 1, 0, 1; x) in terms of classical polylogarithms and logarithms.