Rogers Dilogarithm Identities

Updated 1 January 2026

Rogers dilogarithm identities are a set of functional equations built from a universal five-term relation, connecting number theory, algebraic geometry, and quantum field theory.
They underpin key concepts in algebraic K-theory, hyperbolic geometry, and cluster algebras, offering deep insights into topological and combinatorial structures.
These identities enable practical computation in models such as Chern–Simons theory and thermodynamic Bethe Ansatz, linking analytic summations with geometric frameworks.

The Rogers dilogarithm is a special function central to number theory, algebraic geometry, mathematical physics, and cluster algebras. Rogers dilogarithm identities provide a highly structured web of functional equations and summations, with far-reaching implications in K-theory, hyperbolic geometry, quantum field theory, and mathematical physics. At the heart of these identities lies a universal five-term relation, from which all other relations among Rogers dilogarithms are generated. Contemporary research elucidates the geometric, combinatorial, and algebraic underpinning of these identities, connecting them to moduli spaces, cluster algebras, quantum groups, and regulators.

1. Definition and Fundamental Properties

The classical Rogers dilogarithm for $x \in (0,1)$ is defined by

$L(x) = \operatorname{Li}_2(x) + \frac12 \log x \log(1-x),$

where $\operatorname{Li}_2(x) = \sum_{n=1}^\infty x^n/n^2$ is the classical dilogarithm. $L(x)$ extends by analytic continuation (with compatible branches) to $x \in \mathbb{C} \setminus ((-\infty,0] \cup [1,\infty))$ , satisfying $L(0) = 0$ and $L(1) = \pi^2/6$ (Jeu, 2020). The function appears naturally as the period of algebraic $K_2$ -theory (regulators), and as a generating function in multiple moduli and TQFT settings.

Two essential functional relations govern $L(x)$ :

Reflection: $L(x) + L(1-x) = \pi^2/6$
Inversion: $L(x) + L(1/x) = \pi^2/6$ for $x \neq 0,1$

The identities extend to multivariable linear combinations, but crucially all such relations are built, in a precise sense, from a single combinatorial archetype: the five-term relation (Jeu, 2020).

2. The Five-Term and Universal Relations

The five-term (Abel) relation takes the form:

$L(x) - L(y) + L\left( \frac{y}{x} \right) - L\left( \frac{1-y}{1-x} \right) + L\left( \frac{1-xy}{1-x} \right) = 0,$

for $x, y$ such that the arguments avoid $0,1,\infty$ (Jeu, 2020). This identity, together with the two-term inversion relation, generates the entire $\mathbb{Z}$ -module of functional equations for Rogers dilogarithm with rational function arguments.

A definitive classification theorem asserts: any integer-linear combination $\sum_i n_i L(\varphi_i(x_1,\ldots,x_k)) = C$ ( $C \in \mathbb{Q} \pi^2$ ) for rational functions $\varphi_i$ is a consequence of finitely many five-term and inversion relations, up to addition of torsion constants (Jeu, 2020).

This universality is further demonstrated by a family of multivariable functional equations on moduli spaces $M_{0,n}$ , where the so-called "dihedral coordinate" identities for $n \ge 4$ reduce entirely to combinations of five-term and reflection relations (Soudères, 2015). The deeper implication is that the structure of the Bloch group and all higher $K_2$ -regulator phenomena are governed by these universal templates.

3. Geometric and Topological Origin: Chern-Simons and Moduli Spaces

The Rogers dilogarithm arises canonically in the geometry of moduli spaces of flat $\mathrm{U}(1)$ -connections, especially in the context of abelian spin Chern-Simons theory. In this setting, the "enhanced Rogers dilogarithm"

$L(u_1, u_2) = \operatorname{Li}_2(e^{u_1}) + \frac{1}{2} u_1 u_2 \mod 4\pi^2$

appears as the holonomy of a canonical connection on a complex line bundle over the pair-of-pants moduli $\Sigma$ (Freed et al., 2020). Functional equations for $L$ (shift/monodromy, reflection, and, crucially, the 5-term relation) are derived purely from the functoriality of the Chern-Simons line bundle under gauge transformations and bordism. For example, the classical 5-term identity follows from gluing rules for spin 3-manifolds and the corresponding trivialization of boundary line bundles.

This geometric perspective avoids analytic continuation or contour arguments: all classical identities emerge uniformly as topologically forced constraints. The role of branch structure, spin data, and orientation is made explicit in this approach, and extensions to higher polylogarithms are natural in the context of non-abelian and higher-dimensional Chern-Simons theories (Freed et al., 2020).

4. Cluster Algebras, Y-Systems, and Scattering Diagrams

Rogers dilogarithm identities permeate the theory of cluster algebras and the combinatorics of $\mathcal{Y}$ -systems. In the Hamiltonian-Lagrangian formulation of cluster mutations, the Rogers dilogarithm appears as the Lagrangian function underlying mutation sequences, yielding families of identities:

$\sum_{s=0}^{T-1} \frac{\varepsilon_s}{d_{k_s}} L\left( \frac{\hat y_{k_s}[s]}{1+\hat y_{k_s}[s]} \right) = 0,$

where the $\hat y$ -variables encode the cluster algebraic mutation dynamics (Gekhtman et al., 2016). This formalism unifies the Rogers and Euler dilogarithms, linking them via Legendre transform and bridging classical and quantum settings.

Cluster scattering diagrams generalize this construction. Dilogarithm identities naturally correspond to loops in such diagrams, where the full infinite class of relations (finite or infinite sum identities) is constructed recursively by iterated application of the pentagon (five-term) relation (Nakanishi, 2021). All cluster algebra dilogarithm identities (finite types, affine types, and beyond) are, up to toric torsion, compositions of these basic moves.

5. Rogers Dilogarithm Identities in Mathematical Physics

The Rogers dilogarithm governs thermodynamic Bethe Ansatz (TBA), conformal field theory (CFT) central charges, and $q$ -series associated to affine Lie algebras and double affine Hecke algebras (DAHA). In Cherednik–Feigin's Nil-DAHA framework (Cherednik et al., 2012), Rogers–Ramanujan type $q$ -series are associated to a symmetrized matrix $A$ , with associated TBA systems

$1-Q_i = \prod_j Q_j^{a_{ij}},$

and the cyan identity

$L_A = \frac{6}{\pi^2} \sum_{i=1}^N L(Q_i) \in \mathbb{Q}.$

This generalizes both the Rogers–Ramanujan classical identities (rank one) and Nahm conjecture, encompassing all irreducible affine root systems, levels, and non-simply-laced types. The resulting dilogarithm sums encode effective central charges, and their rational values are forced by modularity and the structure of quantum cluster algebras, with Y-systems providing the linking analytic structure (Nakanishi, 2012).

Similarly, the connection to the spectrum of hyperbolic surfaces yields identities for values $L(\cosh^{-2}(\ell(\alpha)/2))$ over orthospectra, further relating them to special values, continued fraction expansions (Pell's equation), and the geometry of moduli space (Bridgeman, 2019). These constructions generalize to infinite series identities, Chebyshev polynomial parametrizations, and encompass all known classical value identities (e.g., Ramanujan’s).

6. Families of Series and Combinatorial Summations

The five-term relation not only generates functional equations but also underlies analytic summation identities for the Rogers dilogarithm. For example, by telescoping the pentagon relation along suitable recursions, two-parameter and Lucas-sequence series identities are obtained:

$\sum_{n=0}^{\infty} L(x_n y_n) = L(a) + L(b) - L\left(\frac{|a-b|}{1-\min\{a,b\}}\right),$

with $x_n, y_n$ constructed via rational recurrences, subsuming and generalizing numbers-theoretic families (Fibonacci, Pell, Chebyshev) and providing new analytic proofs for previously geometric results (Sanford, 2024).

Infinite family identities associated to moduli space $M_{0,n}$ dihedral coordinates likewise reduce combinatorially to the pentagon relation. Each such identity corresponds to a choice of marked points, branches, or combinatorial structures on the moduli space or cluster pattern (Soudères, 2015).

Table: Key Rogers Dilogarithm Identities and their Contexts

Type	Prototypical Formula	Context/Reference
Five-term (Abel) relation	$L(x)-L(y)+L(y/x)-L((1-y)/(1-x))+L((1-xy)/(1-x))=0$	(Jeu, 2020)
Reflection/dual	$L(x)+L(1-x)=\pi^2/6$	(Jeu, 2020, Soudères, 2015)
Cluster/Y-system periodicity	$\sum_s \frac{\varepsilon_s}{d_{k_s}}L( \frac{\hat y_{k_s}[s]}{1+\hat y_{k_s}[s]}) = 0$	(Gekhtman et al., 2016, Nakanishi, 2021)
Series/orthospectra	$L(e^{-L}) = \sum_{k=2}^{\infty} L\left( \frac{\sinh^2(L/2)}{\sinh^2(kL/2)} \right)$	(Bridgeman, 2019)
Modular/TBA sum	$\frac{6}{\pi^2}\sum_i L(Q_i) = \text{rational}$ for $(1-Q_i)=\prod_j Q_j^{a_{ij}}$	(Nakanishi, 2012, Cherednik et al., 2012)
$M_{0,n}$ dihedral family	$\sum_{{i,j}\in \chi_n} L(u_{i,j}) = \frac{(n-3)(n-2)}{2}L(1)$	(Soudères, 2015)

7. Implications and Outlook

Rogers dilogarithm identities serve as a universal organizing principle across a wide mathematical and physical landscape. The existence of a single, generative five-term relation (plus inversion/reflection) implies that all new multivariable or combinatorial functional equations, whether discovered in matrix models, scattering diagrams, moduli spaces, or summations, ultimately reduce—combinatorially and algebraically—to these archetypal templates (Jeu, 2020, Soudères, 2015).

The connections span numerous mathematical domains: algebraic $K$ -theory (via the Bloch group), moduli of geometric structures, cluster algebra dynamics, conformal field theory, and quantum integrable systems. The geometric and topological frameworks (Chern–Simons, moduli interpretation) not only provide conceptual clarity but also allow for explicit computation of identities without analytic continuation.

The methodology and foundational results indicate that this universality extends to higher polylogarithms and their associated functional equations, suggesting further generalizations in motivic cohomology, quantum groups, and beyond (Freed et al., 2020, Gekhtman et al., 2016).

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