Cyclic Quantum Dilogarithm Overview

• Cyclic quantum dilogarithm is a finite-product, root-of-unity specialization of the quantum dilogarithm that retains pentagon-type and quasi-periodic identities.
• It naturally appears in cluster mutations, cyclic quivers, quantum Teichmüller theory, and state-integral models, providing clear algebraic and arithmetic insights.
• Its diverse normalizations simplify complex identities, enhancing understandings in integrable quantum models and arithmetic K-theory frameworks.

The cyclic quantum dilogarithm is a root-of-unity specialization of the quantum dilogarithm, together with a family of closely related finite-product and operator-valued functions that retain pentagon-type, quasi-periodic, or mutation-theoretic structures at finite order. In the cited literature it appears in several non-equivalent normalizations: as a finite product $D_N(x;q)$, as the cyclic factor $d_N(x)$ extracted from the root-of-unity degeneration of the compact quantum dilogarithm, and as an operator $\Psi_p(X)$ on cyclic variables with $X^N=1$. These forms occur in cluster mutation identities, cyclic quivers, quantum Teichmüller theory, state-integral models, arithmetic $K$-theory, and Shintani-type invariants, while remaining tied to the broader compact/non-compact quantum dilogarithm framework (Garoufalidis et al., 2014, Ip et al., 2014, Ishibashi, 4 Jan 2025).

1. Definitions, normalizations, and scope

The cited papers use several distinct normalizations for the cyclic quantum dilogarithm and related root-of-unity objects.

Object	Formula	Setting
$D_N(x;q)$	$D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$	Rational-point state integrals (Garoufalidis et al., 2014)
$\slashed D_N(x;q)$	$\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$	Variant natural in rational-point formulas (Garoufalidis et al., 2014)
$d_N(x)$	finite-product cyclic factor extracted from $d_N(x)$0 at $d_N(x)$1 a root of unity	Root-of-unity degeneration of cluster identities (Ip et al., 2014)
$d_N(x)$2	$d_N(x)$3	Root-of-unity quantum Teichmüller theory (Ishibashi, 4 Jan 2025)
$d_N(x)$4	$d_N(x)$5	Shintani’s invariant (Yalkinoglu, 24 Aug 2025)

These objects are related by common root-of-unity and finite-product features, but the literature does not impose a single canonical normalization. In the state-integral setting, the cyclic quantum dilogarithm is the finite-product correction that survives when Faddeev’s quantum dilogarithm is specialized at rational $d_N(x)$6 (Garoufalidis et al., 2014). In the cluster-algebraic root-of-unity limit, the same phenomenon is packaged as a factorization of the compact quantum dilogarithm into a singular classical term and a genuinely cyclic term $d_N(x)$7 (Ip et al., 2014). In quantum Teichmüller theory, $d_N(x)$8 is formulated on the punctured Fermat curve $d_N(x)$9 and evaluated on cyclic operators with $\Psi_p(X)$0 (Ishibashi, 4 Jan 2025).

Two further cautions are standard in the literature. First, some papers use “cyclic” to describe a finite cyclic component such as $\Psi_p(X)$1, rather than a purely finite-product dilogarithm. In the axiomatic framework of quantum dilogarithms on Pontryagin self-dual groups, the Andersen–Kashaev dilogarithm on $\Psi_p(X)$2 is described as the closest object to a cyclic quantum dilogarithm, but not as a separate standalone purely finite cyclic function (Bazhanov et al., 29 Dec 2025). Second, survey treatments of $\Psi_p(X)$3, $\Psi_p(X)$4, $\Psi_p(X)$5, and $\Psi_p(X)$6 explicitly note that they do not construct a separate cyclic/root-of-unity finite-product object; the closest link there is the compact regime and the $\Psi_p(X)$7 limit (Ip, 2011).

2. Root-of-unity degeneration and finite-product structure

A central mechanism is the degeneration of the compact quantum dilogarithm $\Psi_p(X)$8 when $\Psi_p(X)$9 approaches a root of unity. For

$X^N=1$0

the compact dilogarithm has the asymptotic factorization

$X^N=1$1

Here $X^N=1$2 carries the singular classical contribution and $X^N=1$3 is the cyclic factor. Conjugation by $X^N=1$4 produces the root-of-unity mutation rule on the $X^N=1$5-th powers of quantum $X^N=1$6-variables, and after the $X^N=1$7-factors cancel one obtains a genuine cyclic dilogarithm identity (Ip et al., 2014).

An equivalent finite-product phenomenon appears in the rational-point evaluation of Faddeev’s non-compact quantum dilogarithm. At $X^N=1$8, the paper “Evaluation of state integrals at rational points” rewrites the specialized quantum dilogarithm as

$X^N=1$9

or equivalently with the slashed version $K$0. In this form the cyclic quantum dilogarithm is the exact finite product replacing the infinite $K$1-Pochhammer symbols at roots of unity, and it persists in the final closed formula for the state-integral together with a Rogers dilogarithm phase and a finite state-sum (Garoufalidis et al., 2014).

A related, but not identical, root-of-unity regime occurs in solvable discrete quantum mechanics with $K$2. There the function $K$3 satisfies

$K$4

and for rational $K$5, say $K$6, the paper records a residue-class decomposition reflecting periodic root-of-unity behavior. It explicitly remarks, however, that this is not a separate theory of a named cyclic quantum dilogarithm (Odake et al., 2014).

3. Mutation sequences, cyclic quivers, and Donaldson–Thomas factorizations

In quantum cluster theory, the cyclic quantum dilogarithm is tied to mutation sequences at roots of unity. For a $K$7-periodic mutation sequence $K$8, the cyclic dilogarithm identity proved in “Quantum Dilogarithm Identities at Root of Unity” is

$K$9

together with a standard universal form written in the reversed order. The same paper defines cyclic $D_N(x;q)$0-variables that transform formally like $D_N(x;q)$1-th roots of the dual variables, so that the pentagon and more general mutation identities survive the root-of-unity limit (Ip et al., 2014).

The cluster-theoretic background explains why “cyclic” often refers not only to finite products at roots of unity but also to periodic mutation behavior. The survey “On cluster theory and quantum dilogarithm identities” treats the basic pentagon

$D_N(x;q)$2

as the $D_N(x;q)$3 prototype, then interprets more general quantum dilogarithm products as consequences of mutation loops, wall-crossing, maximal green sequences, and Zamolodchikov periodicity in cluster categories. This suggests that, in cluster theory, cyclicity is as much a property of mutation-periodic factorizations as of any single special function (Keller, 2011).

For genuinely cyclic quivers, Hall-algebra methods produce order-$D_N(x;q)$4 cyclic identities. In the category $D_N(x;q)$5 of nilpotent representations of the cyclic quiver, any discrete stability function has a unique stable object of dimension vector $D_N(x;q)$6, and the ordered product

$D_N(x;q)$7

is independent of the chosen stability function. Moreover,

$D_N(x;q)$8

so the resulting quantum dilogarithm identity is cyclic of order $D_N(x;q)$9 in the sense of Bytsko–Volkov (Fu et al., 2013). For $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$0-cycle quivers with potential, a different construction yields the factorization

$D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$1

interpreted as a factorization of the refined Donaldson–Thomas invariant and conjecturally related to maximal green sequences (Allman, 2018).

4. Tetrahedron equations, quantum Teichmüller theory, and topological operators

A higher-dimensional source of cyclic quantum dilogarithm identities comes from the tetrahedron equation. For the triangular quivers $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$2, the paper “Tetrahedron equation and cyclic quantum dilogarithm identities” defines

$D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$3

where the product is over tetrahedral lattice points in lexicographic order. Its main theorem is

$D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$4

with $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$5 the order-3 automorphism induced by rotation of the triangular quiver. The tetrahedron equation acts as the local algebraic move that reorders the factors and produces the cyclic invariance (Bytsko et al., 2013).

At roots of unity, quantum Teichmüller theory furnishes an explicit operator-valued cyclic quantum dilogarithm. In “Cyclic quantum Teichmüller theory”, one fixes an odd $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$6, takes $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$7 to be a primitive $D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$8-th root of unity, and defines

$D_N(x;q)=\prod_{k=1}^{N-1}(1-q^k x)^{k/N}$9

for $\slashed D_N(x;q)$0 on the punctured Fermat curve

$\slashed D_N(x;q)$1

The key pentagon identity is

$\slashed D_N(x;q)$2

and the paper reinterprets the parameter constraints ensuring this identity as coefficient mutations in the cluster-algebraic sense. The flip operator

$\slashed D_N(x;q)$3

then generates a finite-dimensional projective representation of the dotted Ptolemy groupoid, reproduces the central charge of the $\slashed D_N(x;q)$4 Wess–Zumino–Witten model, and yields quantum intertwiners whose reduced form is stated to coincide with the transpose of the reduced quantum hyperbolic operator of Baseilhac–Benedetti (Ishibashi, 4 Jan 2025).

The cyclic quantum dilogarithm also has higher-rank analogues. In “$\slashed D_N(x;q)$5 Matrix Dilogarithm as a $\slashed D_N(x;q)$6-Symbol”, Kashaev’s $\slashed D_N(x;q)$7 matrix dilogarithm

$\slashed D_N(x;q)$8

is presented as the $\slashed D_N(x;q)$9 analogue of the cyclic quantum dilogarithm used in Kashaev’s invariants and in Baseilhac–Benedetti quantum hyperbolic invariants. It satisfies a pentagon-type relation,

$\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$0

functions as a $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$1-symbol for cyclic modules, and yields $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$2 state-sum invariants of 3-manifolds and links (Karemera, 2020).

5. Arithmetic and $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$3-theoretic realizations

The cyclic quantum dilogarithm also enters arithmetic $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$4-theory. For an odd integer $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$5, a primitive $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$6-th root of unity $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$7, and a field $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$8 with $\slashed D_N(x;q)=\prod_{k=1}^{N}(1-xq^k)^{k/N}$9, Calegari–Garoufalidis–Zagier define a homomorphism

$d_N(x)$0

using the cyclic quantum dilogarithm and the relation between $d_N(x)$1 and the Bloch group. Hutchinson proves that this map is the square of the Chern class map: $d_N(x)$2 The proof reduces the comparison to an explicit class $d_N(x)$3, uses the identity $d_N(x)$4 with the Bott element $d_N(x)$5, and combines the evaluations $d_N(x)$6 and $d_N(x)$7 (Hutchinson, 2021).

A distinct arithmetic appearance occurs in Shintani’s invariant for real quadratic fields. Under the simplifying assumption that the minus continued fraction expansion of the fundamental unit has length one, the paper “Shintani’s invariant via cyclic quantum dilogarithm” introduces

$d_N(x)$8

and proves that, for

$d_N(x)$9

Shintani’s invariant is expressed as a limit of ratios of cyclic quantum dilogarithms: $d_N(x)$00 The paper immediately draws the consequence that “Shintani’s invariant is approximated by Kummer extensions of cyclotomic fields” (Yalkinoglu, 24 Aug 2025).

These two applications display a common arithmetic pattern: a construction initially expressed in terms of Bloch-group data, double sine functions, or $d_N(x)$01-Pochhammer symbols becomes more transparent after passage to a cyclic quantum dilogarithm. A plausible implication is that the root-of-unity regime exposes arithmetic structures that are less visible in non-cyclic normalizations.

6. Relation to the broader quantum-dilogarithm ecosystem

The cyclic quantum dilogarithm is best understood as one stratum inside a larger quantum-dilogarithm ecosystem. In the general framework of “Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions”, a quantum dilogarithm is a function $d_N(x)$02 on a Pontryagin self-dual group satisfying inversion and pentagon identities, together with a Fourier self-duality condition. The paper gives three examples: the Faddeev modular quantum dilogarithm on $d_N(x)$03, the Andersen–Kashaev dilogarithm on $d_N(x)$04, and the Woronowicz dilogarithm on $d_N(x)$05. It explicitly identifies the Andersen–Kashaev case as the example that most clearly corresponds to a cyclic structure because of the finite cyclic factor $d_N(x)$06, while also emphasizing that the exact partition-function analysis is carried out only for the Faddeev case (Bazhanov et al., 29 Dec 2025).

Survey work on $d_N(x)$07 sharpens this context from the special-function side. “The Graphs of Quantum Dilogarithm” defines

$d_N(x)$08

records its functional equations, zeros, poles, asymptotics, and the $d_N(x)$09 limit to the gamma function, and translates between $d_N(x)$10, $d_N(x)$11, $d_N(x)$12, $d_N(x)$13, Volkov’s hyperbolic gamma, Faddeev’s original $d_N(x)$14, and other variants. The same paper states explicitly that it does not develop a separate cyclic quantum dilogarithm in the sense of a root-of-unity finite-product object; the closest link is the compact regime $d_N(x)$15 and the discussion of the classical limit (Ip, 2011).

A recent physical realization further enlarges the picture. “Heisenberg-Euler and the Quantum Dilogarithm” rewrites the Heisenberg–Euler one-loop QED effective Lagrangian so that Faddeev’s quantum dilogarithm becomes the natural resummation kernel, with the imaginary part expressed as a quantum dilogarithm and the real part as an integral transform involving the modular dual. The paper does not focus on the cyclic quantum dilogarithm in a narrow technical sense, but it explicitly links the Heisenberg–Euler effective action to the broader quantum-dilogarithm ecosystem of compact and non-compact forms, $d_N(x)$16-Pochhammer products, modular duals, and self-duality/quasi-periodicity identities, and describes this as a physically motivated realization of cyclic/quantum-dilogarithmic structure inside QED vacuum polarization (Dunne, 16 Dec 2025).

Taken together, these works place the cyclic quantum dilogarithm at the intersection of three limiting procedures: root-of-unity degeneration, finite cyclic reduction, and modular/cluster mutation. The resulting object is not unique in normalization, but its recurrent structural signatures are stable: finite-product behavior, pentagon-type identities, compatibility with cyclic or mutation-periodic symmetries, and persistence as the root-of-unity remnant of more general compact or non-compact quantum dilogarithms.