Nil-DAHA and quantum dilogarithm derivations of Rogers-Ramanujan type identities: the Cherednik-Feigin quantum lift and dilogarithm identities for untwisted nonsimply-laced affine types

The nilpotent double affine Hecke algebra (Nil-DAHA) of a root system produces, via inner products of q-Hermite polynomials against level-one theta functions, families of Rogers-Ramanujan type q-series attached to twisted affine root systems, and these series satisfy rational dilogarithm identities matching constant Y-systems of quantum affine Kac-Moody algebras (Nakanishi 2012). Two directions left open by that correspondence are formalized here. (a) Quantum lift: the matching dilogarithm identities are currently derived classically, from periodicities of cluster y-variables; derive the Cherednik-Feigin Rogers-Ramanujan type identities at the quantum level, from the quantum dilogarithm identities attached to periods of quantum y-variables, and exhibit the semiclassical-limit mechanism connecting them to the Nil-DAHA series. (b) Untwisted nonsimply-laced types: the known dilogarithm identities for the untwisted affine types $B_n^{(1)}$, $C_n^{(1)}$, $F_4^{(1)}$, $G_2^{(1)}$ are obtained by cluster-theoretic methods, while the Nil-DAHA series cover the Langlands-dual twisted types via folding; construct the untwisted nonsimply-laced identities directly from Nil-DAHA. A full answer consists of a Nil-DAHA (respectively quantum cluster) construction, identification of the resulting series, product-sum or modularity identities, correct semiclassical limits, and type-by-type verification.

Background

The Rogers dilogarithm is the function $L(x)=-\frac{1}{2}\int_0^x\left(\frac{\log(1-y)}{y}+\frac{\log y}{1-y}\right)dy$ on $[0,1]$, extended by continuity to $[0,\infty]$; its most remarkable feature is the family of rational identities it satisfies, beginning with $L(a)+L(1-a)=L(1)=\pi^2/6$. In the thermodynamic Bethe ansatz (TBA) for two-dimensional integrable models one meets functional relations called Y-systems; their constant version is an algebraic system for variables $Y_i$ indexed by the simple roots of an affine Kac-Moody algebra and by a positive integer level $\ell$, which has a unique solution in positive reals, and the accompanying dilogarithm identity (DI) states that $\frac{6}{\pi^2}\sum_i\nu_iL\left(\frac{Y_i}{1+Y_i}\right)$ (with root-length weights $\nu_i$, where $\nu_i=1$ for short and $2$ or $3$ for long simple roots in nonsimply-laced types) equals a prescribed rational number. "Nonsimply laced" refers to root systems whose Dynkin diagrams have multiple edges (types $B$, $C$, $F$, $G$); "untwisted" affine types $B_n^{(1)}$, $C_n^{(1)}$, $F_4^{(1)}$, $G_2^{(1)}$ are loop algebras of finite root systems, while "twisted" types are obtained by combining the loop construction with a diagram automorphism; Langlands duality exchanges the untwisted and twisted nonsimply-laced affine types, and folding (averaging over a diagram automorphism of a simply laced diagram) realizes the level-2 constant Y-systems of the twisted types inside those of simply laced types. These DIs were conjectured from conformal field theory in the 1990s and proved by cluster algebra methods: a cluster algebra is a commutative algebra generated by recursively mutating distinguished generating sets (seeds), its coefficient variables (y-variables) mutate alongside the generators, and every period of a y-variable pattern yields a DI by a sign-counting argument. The papers of Inoue-Iyama-Keller-Kuniba-Nakanishi established the periodicities and DIs for all finite types, including the untwisted nonsimply-laced affine types above. In the quantum setting the y-variables become noncommuting generators of a quantum cluster algebra, and each period carries a quantum dilogarithm identity: an ordered product of quantum dilogarithms (noncommutative analogues of exponentials of the Rogers dilogarithm, i.e. of $q$-series $\Phi(x)=\prod_{n\ge0}(1+q^{n+1/2}x)^{-1}\prod_{n<0}(1+q^{n+1/2}x)$ in a quantum torus) equal to a scalar, whose commutative ("semiclassical") limit recovers the classical DI.

On the other side, the double affine Hecke algebra (DAHA) is the algebra, attached to a root system, that affinizes and $q$-difference-izes the symmetry of the polynomial ring and diagonalizes Macdonald polynomials. Its nilpotent specialization (Nil-DAHA) is the degeneration in which the nonsymmetric Macdonald polynomials collapse to $q$-Hermite polynomials (Macdonald polynomials at $t=0$) and the difference Fourier transform of the algebra becomes exact. Cherednik and Feigin computed the inner products of $q$-Hermite polynomials against level-one theta functions and used them to expand products of theta functions, obtaining for arbitrary reduced twisted affine root systems families of $q$-series of Rogers-Ramanujan type (i.e. $q$-series equal to infinite products, or more generally modular functions, as in the classical Rogers-Ramanujan identities) of Nahm-sum form

$F_{A,B,C}(q)=\sum_{n\in\mathbb{Z}_{\ge0}^r}\frac{q^{\frac{1}{2}n^{T}An+B^{T}n+C}}{(q^{\nu_1})_{n_1}\cdots(q^{\nu_r})_{n_r}}$

with $(q)_n=\prod_{k=1}^{n}(1-q^k)$, with $A$ (in the simply laced case) the inverse Cartan matrix and in general its symmetrized analogue, and with $\nu_i$ the root-length multiplicities; they proved the modularity of these series. Following Nahm's conjecture, each such series controls an algebraic system $1-Q_i=\prod_jQ_j^{a_{ij}}$ (modified to $(1-Q_i)^{\nu_i}=\prod_jQ_j^{a_{ij}}$ in the nonsimply-laced case) for the matrix $(a_{ij})$ of the series, whose unique solution with $0<Q_i<1$ yields a rational value $\frac{6}{\pi^2}\sum_i\nu_iL(Q_i)$, rationality explained by torsion in the Bloch group (a quotient of the free abelian group on field elements measuring the failure of additivity of the dilogarithm). Thus each family of Nil-DAHA series produces a DI.

Nakanishi's note identified the DIs thus obtained with constant Y-system identities: for simply laced types, the Cherednik-Feigin systems match the level-2 and level-3 constant Y-systems of untwisted simply laced affine types, while for types $B_n$, $C_n$, $F_4$, $G_2$ they match the level-2 and level-3 constant Y-systems of the Langlands-dual twisted types, obtained by folding from simply laced types. The classical side of the correspondence is therefore completely understood. The note closes with the two unresolved directions that constitute this problem: deriving the Rogers-Ramanujan type identities from the quantum dilogarithm identities attached to periods of quantum y-variables, and producing the untwisted nonsimply-laced affine DIs themselves from Nil-DAHA.

Problem Statement

The problem is the two-direction question left open at the end of Nakanishi's note on the dilogarithm identities from Nil-DAHA.

(a) Quantum-dilogarithm derivation of the Cherednik-Feigin identities. For the twisted affine root systems treated by Cherednik and Feigin, the Rogers-Ramanujan type q-series come with dilogarithm identities that match constant Y-systems and are provable classically by cluster periodicity arguments. Give a derivation at the quantum level: starting from the period of the (quantum) y-variables associated with the relevant constant Y-system, from the corresponding quantum dilogarithm identity, and from its semiclassical limit, deduce the Cherednik-Feigin Rogers-Ramanujan type identities — the q-series themselves, their product-sum or modularity identities, and the accompanying rational dilogarithm values — exhibiting explicitly where the Nil-DAHA series enter the quantum construction (e.g. as graded partition q-series/state sums attached to the period).

(b) Nil-DAHA construction for the untwisted nonsimply-laced affine types. For each type $R^{(1)}\in\{B_n^{(1)},C_n^{(1)},F_4^{(1)},G_2^{(1)}\}$ and its known level-2 and level-3 dilogarithm identities (the levels matched by the Nil-DAHA correspondence; currently obtained by cluster-theoretic periodicity, and reached by Cherednik-Feigin only for the Langlands-dual twisted types via folding), construct from the nilpotent double affine Hecke algebra of the corresponding root system a family of Rogers-Ramanujan type q-series and/or dilogarithm identities that reproduce the untwisted-side identities directly, without passage to the Langlands dual.

Either direction may be settled independently; a complete answer to both yields a single algebraic procedure (Nil-DAHA at the quantum level) producing the modular q-series and the dilogarithm identities of both the twisted and the untwisted nonsimply-laced affine types.

References

1. T. Nakanishi, "Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras", Symmetry, Integrability and Geometry: Methods and Applications 8 (2012) 104. DOI: 10.3842/SIGMA.2012.104. URL: https://doi.org/10.3842/SIGMA.2012.104 (also arXiv:1210.0226)
2. I. Cherednik, B. Feigin, "Rogers-Ramanujan type identities and Nil-DAHA", Advances in Mathematics 248 (2013) 1050-1088. DOI: 10.1016/j.aim.2013.08.025. URL: https://doi.org/10.1016/j.aim.2013.08.025 (preprint arXiv:1209.1978)
3. R. Inoue, O. Iyama, B. Keller, A. Kuniba, T. Nakanishi, "Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r", Publications of the Research Institute for Mathematical Sciences 49 (2013) 1-42. DOI: 10.4171/prims/95. URL: https://doi.org/10.4171/prims/95 (preprint arXiv:1001.1880)
4. R. Inoue, O. Iyama, B. Keller, A. Kuniba, T. Nakanishi, "Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2", Publications of the Research Institute for Mathematical Sciences 49 (2013) 43-85. DOI: 10.4171/prims/96. URL: https://doi.org/10.4171/prims/96 (preprint arXiv:1001.1881)
5. K. Kato, Y. Terashima, "Quiver mutation loops and partition q-series", Communications in Mathematical Physics 336 (2015) 811-830. DOI: 10.1007/s00220-014-2224-5. URL: https://doi.org/10.1007/s00220-014-2224-5 (preprint arXiv:1403.6569)
6. K. Kato, Y. Terashima, "Quantum dilogarithms and partition q-series", Communications in Mathematical Physics 338 (2015) 457-481. DOI: 10.1007/s00220-015-2323-y. URL: https://doi.org/10.1007/s00220-015-2323-y (preprint arXiv:1408.0444)
7. A. Kato, Y. Mizuno, Y. Terashima, "Quiver mutation sequences and q-binomial identities", International Mathematics Research Notices 2018(23) (2018) 7335-7358. DOI: 10.1093/imrn/rnx108. URL: https://doi.org/10.1093/imrn/rnx108 (preprint arXiv:1611.05969)
8. Y. Mizuno, "Difference equations arising from cluster algebras", Journal of Algebraic Combinatorics 54 (2021) 295-351. DOI: 10.1007/s10801-020-00978-9. URL: https://doi.org/10.1007/s10801-020-00978-9 (preprint arXiv:1912.05710)
9. Y. Mizuno, "Periodic Y-systems and Nahm sums: the rank 2 case", Symmetry, Integrability and Geometry: Methods and Applications 21 (2025) 094. DOI: 10.3842/SIGMA.2025.094. URL: https://doi.org/10.3842/SIGMA.2025.094 (preprint arXiv:2301.13239)
10. Y. Mizuno, "Remarks on Nahm sums for symmetrizable matrices", The Ramanujan Journal 66(4) (2025), Article 62. DOI: 10.1007/s11139-025-01033-6. URL: https://doi.org/10.1007/s11139-025-01033-6 (preprint arXiv:2305.02267)
11. I. Cherednik, S. Kato, "Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules", Advances in Mathematics 374 (2020) 107335. DOI: 10.1016/j.aim.2020.107335. URL: https://doi.org/10.1016/j.aim.2020.107335 (preprint arXiv:1802.03819)
12. T. Nakanishi, "Cluster Algebras and Dilogarithm Identities", MSJ Memoirs 45, Mathematical Society of Japan, 2025. DOI: 10.1142/e078. URL: https://doi.org/10.1142/e078 (draft arXiv:2407.06668)
13. S. Cecotti, A. Neitzke, C. Vafa, "R-Twisting and 4d/2d Correspondences", preprint (2010). arXiv:1006.3435. URL: https://arxiv.org/abs/1006.3435
14. C.-H. Lee, "Nahm's conjecture and Y-systems", Communications in Number Theory and Physics 7 (2013) 1-14. DOI: 10.4310/CNTP.2013.v7.n1.a1. URL: https://doi.org/10.4310/CNTP.2013.v7.n1.a1 (preprint arXiv:1109.3667)
15. K. Sun, H. Wang, "Dynkin diagrams, generalized Nahm sums and 2d CFTs", preprint (2026). arXiv:2604.00847. URL: https://arxiv.org/abs/2604.00847
16. L. Wang, "Identities on Zagier's rank two examples for Nahm's problem", Research in the Mathematical Sciences 11 (2024) 49. DOI: 10.1007/s40687-024-00460-z. URL: https://doi.org/10.1007/s40687-024-00460-z (preprint arXiv:2210.10748)
17. R. Rimanyi, "Combinatorial and algebraic proofs of Keller's $A_2\square A_2$ q-dilogarithm identity", preprint (2026). arXiv:2608.05057. URL: https://arxiv.org/abs/2608.05057