Rudenko's Quadrangular Polylogarithms

• Rudenko’s quadrangular polylogarithms are defined by alternating sums of motivic correlators over even-sided polygon configurations, capturing key cluster polylogarithmic structures.
• They provide a canonical generating family for type A cluster varieties and precisely describe the intrinsic depth-2 components in the Lie coalgebra of multiple polylogarithms.
• Their interpretations span functional equations, geometric volume formulas, and applications in Feynman integrals and de Sitter cosmology.

Searching arXiv for the cited papers and related work on quadrangular polylogarithms. Rudenko’s quadrangular polylogarithms are motivic polylogarithmic functions attached to configurations of $2n+2$ points on $\mathbb{P}^1$, equivalently to even-sided polygons and their quadrangulations. In the formulation of Matveiakin–Rudenko, they are specific alternating sums of motivic correlators $\mathrm{Cor}^L$, indexed by sequences of vertices with a parity constraint, and they define cluster polylogarithms on the type $A$ cluster varieties $\mathfrak{M}_{0,2n+2}$ (Matveiakin et al., 2022). In weight $4$, the same sector appears in the Lie-coalgebraic model of multiple polylogarithms as the depth-$2$ generators $[x,y]_{3,1}$, identified with the Goncharov–Rudenko symbol $\{x,y\}_{3,1}$ up to depth-$1$ corrections, so that the “quadrangular” part is precisely the first genuinely new nonclassical layer beyond the Bloch groups (Greenberg et al., 2022).

1. Definition by alternating correlator sums

For integers $\mathbb{P}^1$0 and $\mathbb{P}^1$1, let $\mathbb{P}^1$2 be the set of all nondecreasing sequences

$\mathbb{P}^1$3

such that each even number $\mathbb{P}^1$4 appears at most once. Let $\mathbb{P}^1$5 consist of those sequences for which, for every pair $\mathbb{P}^1$6, at least one of the two indices appears. The sign is defined by

$\mathbb{P}^1$7

For $\mathbb{P}^1$8, the quadrangular polylogarithm of weight $\mathbb{P}^1$9 is

$\mathrm{Cor}^L$0

and the symmetrized quadrangular polylogarithm is obtained by summing over $\mathrm{Cor}^L$1 instead of $\mathrm{Cor}^L$2 (Matveiakin et al., 2022).

This definition places quadrangular polylogarithms directly in the Lie coalgebra $\mathrm{Cor}^L$3 of motivic multiple polylogarithms. In that setting, $\mathrm{Cor}^L$4 denotes the motivic correlator, and motivic iterated integrals are written as

$\mathrm{Cor}^L$5

The weight is $\mathrm{Cor}^L$6, while the polygon underlying $\mathrm{Cor}^L$7 has $\mathrm{Cor}^L$8 vertices, so the same construction simultaneously tracks transcendental weight and polygonal combinatorics (Matveiakin et al., 2022).

The term “quadrangular” refers to the quadrangulation model. For a convex $\mathrm{Cor}^L$9-gon with vertices $A$0, Rudenko’s quadrangulation formula states

$A$1

where $A$2 is the set of quadrangulations of the polygon, and $A$3 is the dual tree whose vertices correspond to quadrilaterals (Matveiakin et al., 2022). Each quadrilateral contributes a cross-ratio, and the full function is a sum over all noncrossing decompositions into quadrilaterals. That combinatorial origin is the source of both the terminology and the symmetry properties.

Quadrangular polylogarithms are projectively invariant. They depend on the configuration of points only through projective equivalence, so they naturally descend from $A$4 to $A$5 (Matveiakin et al., 2022). This projective invariance is essential: it distinguishes them from bare iterated integrals $A$6, which are only affine invariant.

2. Cluster-polylogarithmic interpretation on type $A$7 varieties

The ambient cluster varieties are the type $A$8 spaces $A$9 and their quotients $\mathfrak{M}_{0,2n+2}$0. In the polygon model, cluster variables are the Plücker coordinates $\mathfrak{M}_{0,2n+2}$1, with noncrossing chords corresponding to cluster-compatible variables. An element of the tensor algebra on cluster variables is called cluster adjacent if all letters in each tensor lie in a single cluster, and it is integrable if Chen’s condition is satisfied in the form

$\mathfrak{M}_{0,2n+2}$2

for every $\mathfrak{M}_{0,2n+2}$3 (Matveiakin et al., 2022).

Quadrangular polylogarithms are cluster polylogarithms in the precise sense that their symbols lie in the space $\mathfrak{M}_{0,2n+2}$4 of cluster integrable symbols. Proposition 2.10 in Matveiakin–Rudenko states that

$\mathfrak{M}_{0,2n+2}$5

so the symbol simultaneously satisfies cluster adjacency and Chen integrability (Matveiakin et al., 2022). In particular, the symbol is a linear combination of tensors in $\mathfrak{M}_{0,2n+2}$6 of Plücker coordinates, with only weakly separated variables appearing together.

The classification theorem for type $\mathfrak{M}_{0,2n+2}$7 gives quadrangular polylogarithms a canonical role. For $\mathfrak{M}_{0,2n+2}$8 and $\mathfrak{M}_{0,2n+2}$9,

$4$0

and $4$1 is generated by symbols of quadrangular polylogarithms

$4$2

(Matveiakin et al., 2022). Thus, in type $4$3, quadrangular polylogarithms span all cluster polylogarithms.

The same theme reappears in later work on de Sitter wavefunctions: the symbol of the chain-graph wavefunction is stated to satisfy total compatibility with the $4$4 cluster algebra, and Rudenko’s quadrangular polylogarithms provide, by construction, a complete basis for such functions (Ferro et al., 7 May 2026). In that paper, “total compatibility” is stronger than ordinary cluster adjacency: all letters in every symbol term belong simultaneously to a single cluster. Rudenko’s construction achieves this because each contribution is attached to one quadrangulation, hence to one maximal compatible family of chords.

A plausible implication is that quadrangular polylogarithms occupy for type $4$5 cluster geometry the role that classical polylogarithms occupy for the Bloch group: they are not merely examples of cluster polylogarithms, but the canonical generating family singled out by the geometry of $4$6.

3. Weight $4$7, depth $4$8, and the Lie coalgebra of multiple polylogarithms

In the Lie-coalgebraic approach to multiple polylogarithms, the graded Lie coalgebra $4$9 is generated by symbols

$2$0

with a cobracket $2$1 transplanted from Goncharov’s coproduct on motivic multiple polylogarithms (Greenberg et al., 2022). The resulting relations encode inversion, shuffle, and specialization identities. Within this structure, the weight-$2$2 depth-$2$3 sector is the first genuinely new one.

Goncharov–Rudenko’s concrete model in weights $2$4 uses generators $2$5 and $2$6, with a quadrangular relation $2$7 in weight $2$8. The comparison map

$2$9

satisfies

$[x,y]_{3,1}$0

and crucially

$[x,y]_{3,1}$1

Hence the Goncharov–Rudenko quadrangular symbol $[x,y]_{3,1}$2 is realized inside the Lie coalgebra as a depth-$[x,y]_{3,1}$3 symbol $[x,y]_{3,1}$4 plus depth-$[x,y]_{3,1}$5 corrections (Greenberg et al., 2022).

This identification is decisive for the meaning of “quadrangular polylogarithm” in weight $[x,y]_{3,1}$6. The weight-$[x,y]_{3,1}$7 piece $[x,y]_{3,1}$8 is generated, up to torsion, by $[x,y]_{3,1}$9 and $\{x,y\}_{3,1}$0. By contrast, in weights $\{x,y\}_{3,1}$1 and $\{x,y\}_{3,1}$2, depth-$\{x,y\}_{3,1}$3 symbols are reducible to depth $\{x,y\}_{3,1}$4. Example 2.2 gives

$\{x,y\}_{3,1}$5

and Proposition 2.8 expresses weight-$\{x,y\}_{3,1}$6 depth-$\{x,y\}_{3,1}$7 and depth-$\{x,y\}_{3,1}$8 classes in depth $\{x,y\}_{3,1}$9 (Greenberg et al., 2022). In weight $1$0, however, the relation

$1$1

shows that the depth-$1$2 generators $1$3 are intrinsic and cannot all be eliminated (Greenberg et al., 2022).

The same weight-$1$4 sector is central in Gangl’s reduction theory for multiple polylogarithms of weight $1$5. There the depth-$1$6 iterated integral $1$7 has symbol

$1$8

so $1$9 is the symbol-theoretic avatar of $\mathbb{P}^1$00 in $\mathbb{P}^1$01 (Gangl, 2016). This is the same quadratic dilogarithmic content that the quadrangular sector isolates motivically.

Accordingly, the paper on the Lie coalgebra of multiple polylogarithms states that all quadrangular functional equations satisfied by $\mathbb{P}^1$02 are realized inside the larger Lie coalgebra as consequences of the general relation system $\mathbb{P}^1$03, and that the comparison map sends the GR relation $\mathbb{P}^1$04 to an element of $\mathbb{P}^1$05 (Greenberg et al., 2022). Quadrangular polylogarithms are therefore not peripheral constructions; they are the canonical description of the first nonclassical piece of $\mathbb{P}^1$06.

4. Functional equations and polygonal generalizations

The central functional equation for quadrangular polylogarithms is the quadrangular relation

$\mathbb{P}^1$07

valid for $\mathbb{P}^1$08 (Matveiakin et al., 2022). Its proof uses a generating function

$\mathbb{P}^1$09

together with the identity

$\mathbb{P}^1$10

which implies the absence of monomials of total degree $\mathbb{P}^1$11 (Matveiakin et al., 2022).

Combined with the quadrangulation formula, this single identity recovers classical relations. For $\mathbb{P}^1$12 it gives Abel’s 5-term relation for the dilogarithm; for $\mathbb{P}^1$13 it yields Kummer’s 9-term and Goncharov’s 22-term relations for $\mathbb{P}^1$14; for $\mathbb{P}^1$15 it yields the $\mathbb{P}^1$16 relation from Goncharov–Rudenko and Gangl’s 931-term relation for $\mathbb{P}^1$17 (Matveiakin et al., 2022). In this sense, the quadrangular relation is a uniform generator of a large class of classical and higher functional equations.

Gangl’s weight-$\mathbb{P}^1$18 work gives a complementary description. Goncharov’s conjecture is formulated there as the statement that for any 5-term relation $\mathbb{P}^1$19,

$\mathbb{P}^1$20

modulo products. Theorem 4.2 proves this explicitly for the standard five-term relation $\mathbb{P}^1$21 with a sum of $\mathbb{P}^1$22 tetralogarithms (Gangl, 2016). That result supplies the weight-$\mathbb{P}^1$23 reduction theory behind the quadrangular relation.

The polygonal program extends beyond weight $\mathbb{P}^1$24. The paper on “Functional equations of polygonal type” presents identities in weights $\mathbb{P}^1$25 that generalize the crucial identity $\mathbb{P}^1$26. These identities are built from polyangulations of convex $\mathbb{P}^1$27-gons by even polygons and have highest-depth terms consisting of a single iterated integral $\mathbb{P}^1$28, evaluated at all possible full quadrangulations of a $\mathbb{P}^1$29-gon up to cyclic symmetry (Charlton et al., 2020). In particular, $\mathbb{P}^1$30 yields depth reduction of $\mathbb{P}^1$31, $\mathbb{P}^1$32 reduces $\mathbb{P}^1$33, and $\mathbb{P}^1$34 reduces $\mathbb{P}^1$35 (Charlton et al., 2020). This shows that the quadrangular identity is the first member of a broader polygonal hierarchy.

A plausible implication is that the quadrangular sector is not only the first nontrivial case of higher cluster polylogarithms, but also the template for systematically organized functional equations in all higher weights.

5. Geometric, analytic, and physical realizations

Rudenko’s quadrangular polylogarithms have a direct geometric realization in hyperbolic geometry. In the study of hyperbolic orthoschemes, a bijection is established between $\mathbb{P}^1$36-dimensional orthoschemes and configurations

$\mathbb{P}^1$37

with Gram matrix entries expressed through the $\mathbb{P}^1$38 (Ren et al., 2023). Rudenko’s volume formula states that for an odd-dimensional hyperbolic orthoscheme $\mathbb{P}^1$39,

$\mathbb{P}^1$40

where $\mathbb{P}^1$41 is an arborification word built recursively from generalized cross-ratios of even subsets, $\mathbb{P}^1$42 is an alternating multiple polylogarithm, and $\mathbb{P}^1$43 is Goncharov’s real period map (Ren et al., 2023).

In this context, the “quadrangular” structure is literal: the basic letters are four-point cross-ratios such as

$\mathbb{P}^1$44

and for six points the arborification produces explicit combinations of depth-$\mathbb{P}^1$45 words whose entries are products of four-point cross-ratios (Ren et al., 2023). For $\mathbb{P}^1$46, the orthoscheme volume becomes a combination of $\mathbb{P}^1$47; for $\mathbb{P}^1$48, of $\mathbb{P}^1$49 and $\mathbb{P}^1$50 (Ren et al., 2023). Thus, alternating quadrangular polylogarithms furnish explicit formulas for volumes.

The same construction leads to explicit one-loop scalar $\mathbb{P}^1$51-gon Feynman integrals in $\mathbb{P}^1$52 dimensions, for even $\mathbb{P}^1$53, through the relation

$\mathbb{P}^1$54

After triangulating a general simplex into orthoschemes, the resulting integral is written as a sum of real periods of Rudenko-type alternating quadrangular polylogarithms (Ren et al., 2023). For $\mathbb{P}^1$55, the paper states that the hexagon integral can be expressed in terms of $\mathbb{P}^1$56 with algebraic functions of cross-ratios as arguments (Ren et al., 2023).

A second physical realization appears in de Sitter space. For an alternating polygon $\mathbb{P}^1$57, Rudenko’s weighted symbol $\mathbb{P}^1$58 is built by summing rooted-tree expressions over all quadrangulations, using a parity-dependent cross-ratio

$\mathbb{P}^1$59

together with quasi-shuffle and dot products on weighted words (Ferro et al., 7 May 2026). Applying $\mathbb{P}^1$60 to $\mathbb{P}^1$61 produces $\mathbb{P}^1$62, and from this one defines even and odd quadrangular polylogarithms $\mathbb{P}^1$63 (Ferro et al., 7 May 2026).

These functions satisfy explicit coproduct recursions. For the basic polygon,

$\mathbb{P}^1$64

is expressed as a term proportional to $\mathbb{P}^1$65 plus a sum over admissible quadrilaterals $\mathbb{P}^1$66, with coefficients that are products of lower-weight $\mathbb{P}^1$67 on the three subpolygons (Ferro et al., 7 May 2026). The $\mathbb{P}^1$68-site chain-graph de Sitter wavefunction is then written explicitly in terms of these $\mathbb{P}^1$69, with a recursive structure matched to the coproduct recursion (Ferro et al., 7 May 2026). This establishes quadrangular polylogarithms as a complete basis for a nontrivial family of cosmological observables.

6. Algebraic reformulations and broader significance

The 2025 paper on multiple polylogarithms and the Steinberg module does not mention Rudenko’s quadrangular polylogarithms explicitly, but it provides a structural framework that illuminates them. It proves that every multiple polylogarithm of weight $\mathbb{P}^1$70 and depth $\mathbb{P}^1$71 can be expressed as a $\mathbb{P}^1$72-linear combination of

$\mathbb{P}^1$73

and products of lower-weight polylogarithms (Charlton et al., 4 May 2025). For the depth-$\mathbb{P}^1$74 world relevant to quadrangular polylogarithms, this means that the entire depth-$\mathbb{P}^1$75 content reduces to $\mathbb{P}^1$76 modulo products.

More conceptually, for a rank-$\mathbb{P}^1$77 torus $\mathbb{P}^1$78 with character space $\mathbb{P}^1$79, the depth-graded piece satisfies the $\mathbb{P}^1$80-equivariant isomorphism

$\mathbb{P}^1$81

This identifies pure depth-$\mathbb{P}^1$82 polylogarithmic information with a Steinberg-module tensor square (Charlton et al., 4 May 2025). The same paper relates $\mathbb{P}^1$83 to Milnor $\mathbb{P}^1$84-theory through

$\mathbb{P}^1$85

and develops a duality exchanging Steinberg polylogarithms and Steinberg iterated integrals (Charlton et al., 4 May 2025). For quadrangular polylogarithms, this suggests a precise interpretation of their depth-$\mathbb{P}^1$86 symbols as Steinberg-module classes attached to four-point configurations.

This does not replace the cluster and polygonal descriptions. Rather, it complements them. The cluster-polylogarithmic picture emphasizes admissible symbol alphabets and total compatibility on $\mathbb{P}^1$87; the Lie-coalgebraic picture identifies the quadrangular sector with $\mathbb{P}^1$88 and the relation $\mathbb{P}^1$89; the polygonal-functional-equation picture shows how quadrangulations generate higher-weight depth reductions; and the Steinberg-module picture gives an abstract homological model for the same depth-graded data (Matveiakin et al., 2022).

In summary, Rudenko’s quadrangular polylogarithms are a family of motivic and cluster polylogarithms defined by alternating correlator sums over configurations of $\mathbb{P}^1$90 points, computable via sums over quadrangulations, projectively invariant on $\mathbb{P}^1$91, and canonically spanning the type $\mathbb{P}^1$92 cluster-polylogarithmic sector (Matveiakin et al., 2022). In weight $\mathbb{P}^1$93 they coincide with the intrinsic depth-$\mathbb{P}^1$94 generators of the Lie coalgebra of multiple polylogarithms (Greenberg et al., 2022); in higher weights they generate polygonal functional equations and explicit depth reductions (Charlton et al., 2020); and in geometric and physical applications they appear in orthoscheme volumes, one-loop $\mathbb{P}^1$95-gon integrals, and de Sitter wavefunctions (Ren et al., 2023, Ferro et al., 7 May 2026).