Removing the full-rank hypothesis from the cluster $a$-variable periodicity derivation of higher-degree Rogers dilogarithm identities

Nash and Rupel (2025) proved Nakanishi's dilogarithm identities of higher degree for generalized cluster algebras by deriving them from the classical Rogers dilogarithm identities attached to the same periodicity of mutations, using the periodicity of the $a$-variables of the Li–Rupel symplectic groupoid — but only under the assumption that the initial exchange matrix $B[1]$ is full rank. The truth of these identities for arbitrary exchange matrices is not in doubt: Nakanishi's original proof makes no rank restriction, and Remark 3.8 of the Nash–Rupel paper observes that the implications can be reversed. What is missing is a direct, self-contained derivation: a proof of the key integral vanishing identity with lower limit $0$ for rank-deficient $B[1]$ that does not presuppose the higher-degree identity, and consequently a non-circular reduction of the higher-degree identities to the classical right-companion identities in the non-full-rank regime. The problem asks to close this methodological gap, making the groupoid-theoretic derivation of Nakanishi's identities from the classical ones uniform in the rank of the exchange matrix.

Background

The Rogers dilogarithm is the single-valued function $\tilde{L}(x)=\frac{1}{2}\int_{0}^{x}\big(\frac{\log(1+y)}{y}-\frac{\log y}{1+y}\big)\,dy$ for $x>0$; it satisfies remarkable functional identities (most famously the five-term relation) and appears across conformal field theory, integrable systems, and geometry. A systematic source of such identities is cluster algebras, introduced by Fomin and Zelevinsky. A cluster algebra is generated from an initial seed by iterated mutations; a seed consists of an exchange matrix $B=(b_{ij})$ — an $n\times n$ skew-symmetrizable integer matrix, meaning that some diagonal matrix $R=\mathrm{diag}(r_{1},\dots,r_{n})$ with positive integer entries makes $RB$ skew-symmetric ($n$ is called the rank) — together with coefficient variables called $y$-variables taking values in a semifield (a set with commutative, associative addition and multiplication in which multiplication distributes over addition). Mutation at an index $k$ replaces $B$ by its standard exchange-matrix mutation and transforms the coefficient variables. A sequence of mutations returning to the initial data up to a permutation $\sigma$ of indices is a $\sigma$-periodicity; it is proper when it holds for formal (generic) initial data rather than by accident of a special choice of coefficients. The classical result connecting these notions, due to Nakanishi and to Kashaev–Nakanishi, is that every proper $\sigma$-periodicity of cluster mutations yields an identity for the Rogers dilogarithm: the sum $\sum_{\ell}\frac{\varepsilon_{\ell}}{r_{k_{\ell}}}\tilde{L}\big(\frac{(y_{k_{\ell}}[\ell])^{\varepsilon_{\ell}}}{1+(y_{k_{\ell}}[\ell])^{\varepsilon_{\ell}}}\big)$ vanishes, where $k_{\ell}$ are the mutation directions, $y_{j}[\ell]$ the $y$-variables along the sequence, and $\varepsilon_{\ell}=\pm 1$ the tropical signs — the signs of the $c$-vectors, integer vectors controlling the tropicalized mutation of the $y$-variables, whose sign-coherence (all entries of each $c$-vector share a sign) is a deep theorem of Gross–Hacking–Keel–Kontsevich.

Generalized cluster algebras, introduced by Chekhov and Shapiro to model Teichmüller spaces of surfaces with orbifold points, replace the binomial exchange rule by a polynomial one: each index $k$ carries a mutation degree $d_{k}\geq 1$ and a collection of $z$-variables $z_{k,1},\dots,z_{k,d_{k}-1}$, packaged in the exchange polynomial $P_{k}(\alpha)=1+z_{k,1}\alpha+\dots+z_{k,d_{k}-1}\alpha^{d_{k}-1}+\alpha^{d_{k}}$. The right-companion construction of Nakanishi–Rupel re-encodes such a generalized cluster algebra as an ordinary cluster algebra of larger rank, so the classical machinery applies. For a monic polynomial $P$ of degree $d$ with non-negative coefficients and constant term $1$, Nakanishi defined the Rogers dilogarithm of higher degree by

$$\tilde{L}_{P}(x)=\frac{1}{2}\int_{0}^{x}\Big(\frac{\log P(y)}{y}-\frac{\log y\,P'(y)}{P(y)}\Big)\,dy,$$

which reduces to the ordinary Rogers dilogarithm as $\tilde{L}_{P}(x)=\tilde{L}(x^{d})/d$ when $P(y)=1+y^{d}$. Nakanishi proved that every proper $\sigma$-periodicity of generalized seeds yields a higher-degree identity

$$\sum_{\ell=1}^{M}\frac{\varepsilon_{\ell}}{r_{k_{\ell}}}\,\tilde{L}_{P^{\circ}_{\ell}}\big((y_{k_{\ell}}[\ell])^{\varepsilon_{\ell}}\big)=0,\qquad P^{\circ}_{\ell}(\alpha)=P_{\ell}(\alpha^{\varepsilon_{\ell}})\,\alpha^{(1-\varepsilon_{\ell})d_{k_{\ell}}/2},$$

with no restriction on the rank of the exchange matrix; prior treatments of these identities worked with the initial $z$-variables held fixed.

Li and Rupel constructed symplectic groupoids — groupoids carrying a compatible symplectic structure that integrate the log-canonical Poisson structure on cluster manifolds — in which all choices of the $z$-variables are coordinates on one space. The groupoid charts carry fiber coordinates called $a$-variables, whose mutations are Hamiltonian flows generated by Euler dilogarithms of higher degree, together with conjugate $q$-variables; a shift of the base by the fiber is encoded in the map $\beta(y_{j})=y_{j}\exp\big(\sum_{i}r_{i}b_{ij}y_{i}q_{i}\big)$. Periodicity of groupoid mutations then yields, for each index $j$ and each $1\leq s\leq d_{j}-1$, an integral identity

$$\sum_{\ell:\,k_{\ell}=j}\frac{\varepsilon_{\ell}}{r_{j}}\int_{y_{j}[\ell]^{\varepsilon_{\ell}}}^{\beta(y_{j}[\ell])^{\varepsilon_{\ell}}}\frac{u^{\varepsilon_{\ell}s^{\circ}_{\ell}-1}}{P_{\ell}(u^{\varepsilon_{\ell}})}\,du=0,$$

where $s^{\circ}_{\ell}=s$ if an even number of earlier mutations occur at $j$ and $s^{\circ}_{\ell}=d_{j}-s$ if an odd number do. Nash and Rupel observed that the dilogarithm identity needs instead the corresponding identity with lower limit $0$:

$$\sum_{\ell:\,k_{\ell}=j}\frac{\varepsilon_{\ell}}{r_{j}}\int_{0}^{y_{j}[\ell]^{\varepsilon_{\ell}}}\frac{u^{\varepsilon_{\ell}s^{\circ}_{\ell}-1}}{P_{\ell}(u^{\varepsilon_{\ell}})}\,du=0,$$

They proved it (their Theorem 2.21) only when $B[1]$ is full rank (invertible): rewriting the groupoid identity shows one side is independent of the initial $q$-variables, and one may then degenerate all $\beta(y_{j})$ to $0$ simultaneously, which is possible exactly when the linear system $-RB\,\mathrm{diag}(\boldsymbol{y})\boldsymbol{q}=-\lambda(1,\dots,1)^{T}$ is solvable, i.e. when $B$ is invertible (their Lemma 2.20). Under full rank they then differentiate the higher-degree dilogarithm sum with respect to the initial $z$-variables, use the integral identity to see the derivative vanish — so the sum is independent of the $z$-variables, new information since earlier work fixed $z$ — specialize $z=0$, where the higher-degree dilogarithm collapses to the ordinary one, and conclude via the classical right-companion dilogarithm identity, which itself holds with no rank restriction. When $B[1]$ is rank-deficient, the image of $RB$ is a proper subspace that need not contain $(1,\dots,1)^{T}$, the simultaneous degeneration fails, and the paper's Remark 3.8 notes only that the implication can be reversed: using Nakanishi's rank-unrestricted theorem as an input one recovers the integral identity for all $B$, but that route presupposes the very identity the derivation is meant to establish.

Problem Statement

Let $(B[1],\boldsymbol{y}[1],\boldsymbol{z}[1])\longrightarrow\dots\longrightarrow(B[M+1],\boldsymbol{y}[M+1],\boldsymbol{z}[M+1])$ be a proper $\sigma$-periodicity of generalized cluster seeds over the positive reals, with skew-symmetrizer $R=\mathrm{diag}(r_{1},\dots,r_{n})$, mutation degrees $d_{i}$, exchange polynomials $P_{\ell}$, tropical signs $\varepsilon_{\ell}$, and initial exchange matrix $B[1]$ of possibly deficient rank ($\det B[1]=0$, so $B[1]$ has a nontrivial kernel). The problem is to remove the full-rank hypothesis from the Nash–Rupel derivation, specifically:

(1) Prove, by a direct argument that does not assume the higher-degree Rogers dilogarithm identity (Nakanishi's Theorem 4.8, restated as Theorem 3.7 of Nash–Rupel) or any of its consequences as an input — for instance by extending the groupoid $a$-variable periodicity method, or by any other self-contained technique — that for every $1\leq j\leq n$ and $1\leq s\leq d_{j}-1$:

$$\sum_{\ell:\,k_{\ell}=j}\frac{\varepsilon_{\ell}}{r_{j}}\int_{0}^{y_{j}[\ell]^{\varepsilon_{\ell}}}\frac{u^{\varepsilon_{\ell}s^{\circ}_{\ell}-1}}{P_{\ell}(u^{\varepsilon_{\ell}})}\,du=0,$$

where $y_{j}[\ell]$ are the $y$-variables along the periodicity, $s^{\circ}_{\ell}=s$ if an even number of $\ell'<\ell$ have $k_{\ell'}=j$ and $s^{\circ}_{\ell}=d_{j}-s$ otherwise.

(2) Deduce, as in the full-rank case, that the higher-degree Rogers dilogarithm sum $\sum_{\ell=1}^{M}\frac{\varepsilon_{\ell}}{r_{k_{\ell}}}\tilde{L}_{P^{\circ}_{\ell}}\big((y_{k_{\ell}}[\ell])^{\varepsilon_{\ell}}\big)$ is independent of the initial $z$-variables and vanishes, so that Nakanishi's higher-degree dilogarithm identities follow, uniformly in the rank of $B[1]$, from the classical right-companion Rogers dilogarithm identities obtained at $z=0$.

Permitted inputs are the rank-unrestricted results that precede the target: the $a$-variable periodicity (Nash–Rupel's Theorem 2.19, valid for all $B$), the classical right-companion dilogarithm identity (their Theorem 3.2, valid for all $B$), sign-coherence of $c$-vectors, separation of additions, and standard algebra and analysis. The argument must genuinely cover rank-deficient proper periodicities — for example the rank-one case $B=(0)$ with $d_{1}\geq 2$, where mutating twice is a proper $\sigma$-periodicity — and not merely re-derive the known full-rank case. The two requested items form one objective: establishing the integral identity by a non-circular route is the key intermediary from which the $z$-independence, the vanishing, and the reduction to the classical identities follow exactly as in the source paper's argument.

References

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