Classification of modular partial Nahm sums: Nahm's problem for lattice-coset restrictions

Nahm's problem asks for all rational data $(A,B,C)$ such that the multisum $f_{A,B,C}(q)=\sum_{n\in\mathbb{Z}^r_{\ge0}} q^{\frac{1}{2}n^{\mathrm{T}}An+n^{\mathrm{T}}B+C}/\prod_i(q;q)_{n_i}$ is a modular form; Zagier solved the rank-one case completely, and the conjectural rank-two and rank-three examples have since been confirmed. Wang and Zeng (2025) extended the problem to lattice cosets, restricting the summation domain to a coset $v+L$ of a proper full-rank sublattice $L\subset\mathbb{Z}^r$, and posed the classification problem for all such partial Nahm sums. They found eight modular rank-one examples and fourteen families of rank-two matrices (for three specific lattices), the last of which was initially conjectural and was proved by Shi and Wang later in 2025. The classification itself remains open: it is unknown whether the rank-one list is exhaustive, whether the rank-two families are complete even for the three chosen lattices (several coset cases are explicitly undecided), which modular full Nahm sums admit modular lattice-coset restrictions at all, and which weights can occur. This record states the source-faithful classification problem and its concrete open components.

Background

Let $q=e^{2\pi i\tau}$ with $\tau$ in the complex upper half-plane, and write $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ for the finite $q$-Pochhammer symbol (so $(q;q)_n=(1-q)\cdots(1-q^n)$). The classical Rogers–Ramanujan identities express two $q$-hypergeometric series $\sum_{n\ge0}q^{n^2+kn}/(q;q)_n$, with $k\in\{0,1\}$, as infinite products, and these products are modular forms (eta-quotients, i.e. products of Dedekind eta functions $\eta(m\tau)=q^{m/24}\prod_{k\ge1}(1-q^{mk})$). Such sum-to-product identities are called Rogers–Ramanujan type identities.

Nahm's problem, arising from the study of characters of two-dimensional rational conformal field theories (CFTs) and of the dilogarithm, asks for all positive definite rational $r\times r$ matrices $A$, rational vectors $B\in\mathbb{Q}^r$ and rational scalars $C$ such that the Nahm sum $f_{A,B,C}(q)=\sum_{n\in\mathbb{Z}^r_{\ge0}}\frac{q^{\frac{1}{2}n^{\mathrm{T}}An+n^{\mathrm{T}}B+C}}{(q;q)_{n_1}\cdots(q;q)_{n_r}}$ is modular, meaning that as a function of $\tau$ it transforms like a modular form of some (possibly half-integral) weight on a congruence subgroup of $SL_2(\mathbb{Z})$ (a subgroup defined by congruence conditions on the matrix entries modulo a fixed integer), allowing a multiplier system (the root-of-unity-valued transformation factor that replaces the usual automorphy factor when half-integral weights occur). Zagier solved the rank-one case completely: exactly seven triples $(A,B,C)$ give modular sums, including the two Rogers–Ramanujan triples $(2,0,-1/60)$ and $(2,1,11/60)$. He also listed eleven rank-two and twelve rank-three candidate triples whose modularity has since been confirmed by Cherednik–Feigin (via nilpotent double affine Hecke algebras — algebraic structures from Cherednik's theory of double affine Hecke algebras, which arise in the study of q-difference operators), Vlasenko–Zwegers, Wang, and Cao–Rosengren–Wang, through explicit Rogers–Ramanujan type identities or representation-theoretic constructions. In the positive-definite full case, Zagier showed a modular Nahm sum must have weight zero.

Wang and Zeng (2025) extended the problem from the full lattice $\mathbb{Z}^r$ to lattice cosets. For a sublattice $L\subseteq\mathbb{Z}^r$, a vector $v\in\mathbb{Z}^r$ and the coset $v+L\in\mathbb{Z}^r/L$ they define $f_{A,B,C,v+L}(q)=\sum_{n\in(v+L)\cap\mathbb{Z}^r_{\ge0}}\frac{q^{\frac{1}{2}n^{\mathrm{T}}An+n^{\mathrm{T}}B+C}}{(q;q)_{n_1}\cdots(q;q)_{n_r}},$ now allowing $A$ to be any nonzero symmetric rational matrix (not necessarily positive definite) subject to absolute convergence. When $L$ has full rank and is a proper subset of $\mathbb{Z}^r$ this is called a rank-$r$ partial Nahm sum, and $(A,B,C,v+L)$ a quadruple. Their motivating Problem 1.1 asks for a classification of all quadruples for which $f_{A,B,C,v+L}(q)$ is modular — the lattice-coset analogue of Nahm's problem, natural from the CFT viewpoint where coset (parity) restrictions of characters are ubiquitous. Their results: in rank one, eight modular partial Nahm sums (the four triples $(1,0,-1/48)$, $(1,-1/2,1/24)$, $(1/2,0,-1/40)$, $(1/2,1/2,1/40)$, each on both cosets of $2\mathbb{Z}$), established in the paper via the classical single-sum identities of Euler and Rogers — notably, no modular parity restriction was found for the Rogers–Ramanujan triples with $A=2$; in rank two, for the three lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$ and $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$, fourteen families of matrices $A=M(a,b,c)$ (writing $A$ as a symmetric $2\times2$ matrix with diagonal $a,c$ and off-diagonal $b$) with modular partial sums, thirteen proved in the paper and the family $A=M(0,1/2,0)$ completed by the conjectured identities (3.12)–(3.13) that Shi and Wang later proved. A generalized Nahm sum is the full sum $f_{A,B,C}(q)$ with $A$ nonzero symmetric rational, not necessarily positive definite (subject to absolute convergence). Unlike the positive-definite full case, modular partial or generalized sums can have weights $-1/2$ or $1/2$. Several questions were left open by Wang and Zeng, and after the resolution of their conjecture the classification itself is the surviving target; they also ask, as a further open direction, whether the modular partial Nahm sums admit algebraic or physical interpretations — for instance as characters of vertex operator algebras (the algebraic structures underlying rational two-dimensional conformal field theories) subject to parity constraints — which is not part of the classification objective stated below.

Problem Statement

Following Problem 1.1 of Wang–Zeng: determine all modular quadruples $(A,B,C,v+L)$ — that is, all ranks $r\ge1$, nonzero symmetric rational matrices $A\in\mathbb{Q}^{r\times r}$, vectors $B\in\mathbb{Q}^r$, scalars $C\in\mathbb{Q}$, sublattices $L\subseteq\mathbb{Z}^r$ and cosets $v+L\in\mathbb{Z}^r/L$ — such that the lattice-coset Nahm sum $f_{A,B,C,v+L}(q)=\sum_{n\in(v+L)\cap\mathbb{Z}^r_{\ge0}}\frac{q^{\frac{1}{2}n^{\mathrm{T}}An+n^{\mathrm{T}}B+C}}{(q;q)_{n_1}\cdots(q;q)_{n_r}}$ converges absolutely and, as a function of $\tau$ with $q=e^{2\pi i\tau}$, is a modular form of some weight on some congruence subgroup of $SL_2(\mathbb{Z})$ (multiplier systems allowed; a rational power of $q$ is interpreted, as customary for eta-quotients, on the corresponding finite cover). The primary emphasis is the partial case where $L$ has full rank and is a proper subset of $\mathbb{Z}^r$; the full Nahm sums correspond to $L=\mathbb{Z}^r$. Concrete components of the same objective. Each of the components below is independently resolvable, and a complete resolution of any one of them is by itself a valid resolution of this task; the instances introduced below by "for example" or "e.g." are illustrative of the corresponding general demand rather than a reduced target.

(1) Rank-one exhaustiveness: prove or disprove that the eight modular rank-one partial Nahm sums of Wang–Zeng's Theorem 1.2 — the triples $(A,B,C)=(1,0,-1/48)$, $(1,-1/2,1/24)$, $(1/2,0,-1/40)$, $(1/2,1/2,1/40)$, each on both cosets $v+2\mathbb{Z}$, $v\in\{0,1\}$ — exhaust all modular rank-one partial Nahm sums, allowing all sublattices $m\mathbb{Z}\subset\mathbb{Z}$ ($m\ge2$), all cosets, and all rational $(A,B,C)$ with $A$ possibly indefinite (subject to absolute convergence). This is the lattice-coset analogue of Zagier's complete rank-one solution of Nahm's problem.

(2) Rank two and higher: decide whether the fourteen matrix families are complete for the three lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$, $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$; settle the coset cases left undecided by Wang–Zeng (for example, for $A=M(3/4,-1/4,3/4)$ with $B\in\{(0,1/2)^{\mathrm{T}},(1/2,0)^{\mathrm{T}}\}$ and $v\in\{(0,0)^{\mathrm{T}},(1,1)^{\mathrm{T}}\}$, and the analogous undecided cases for the families of their Tables 13–14); and extend the classification to other sublattices and higher ranks.

(3) Structure: characterize which modular full Nahm sums $f_{A,B,C}(q)$ admit nontrivial lattice-coset restrictions $f_{A,B,C,v+L}(q)$ that are modular (Wang–Zeng's evidence suggests this can fail, e.g. for the Rogers–Ramanujan triples $(2,0,-1/60)$ and $(2,1,11/60)$, but no nonexistence proof exists); and determine the set of possible weights of modular generalized or partial Nahm sums (Zagier's weight-zero theorem applies only to positive-definite full sums; the weights $-1/2$, $0$ and $1/2$ are currently known to occur).

References

1. Liuquan Wang and Wentao Zeng, Rogers–Ramanujan type identities for rank two partial Nahm sums, arXiv:2502.19309 (2025), DOI 10.48550/arXiv.2502.19309; published in Science China Mathematics (2026), DOI 10.1007/s11425-025-2605-1. https://arxiv.org/abs/2502.19309
2. Changsong Shi and Liuquan Wang, Proofs of Two Conjectural Identities on Partial Nahm Sums, arXiv:2507.20270 (2025), DOI 10.48550/arXiv.2507.20270. https://arxiv.org/abs/2507.20270
3. Don Zagier, The Dilogarithm Function, in Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, pp. 3–65, DOI 10.1007/978-3-540-30308-4_1. https://doi.org/10.1007/978-3-540-30308-4_1
4. Werner Nahm, Conformal Field Theory and Torsion Elements of the Bloch Group, in Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, pp. 67–132, DOI 10.1007/978-3-540-30308-4_2. https://doi.org/10.1007/978-3-540-30308-4_2
5. Masha Vlasenko and Sander Zwegers, Nahm's conjecture: asymptotic computations and counterexamples, Communications in Number Theory and Physics 5(3) (2011), 617–642, DOI 10.4310/cntp.2011.v5.n3.a2. https://doi.org/10.4310/cntp.2011.v5.n3.a2
6. Ivan Cherednik and Boris Feigin, Rogers–Ramanujan type identities and Nil-DAHA, Advances in Mathematics 248 (2013), 1050–1088, DOI 10.1016/j.aim.2013.08.025. https://doi.org/10.1016/j.aim.2013.08.025
7. Liuquan Wang, Identities on Zagier's rank two examples for Nahm's problem, Research in the Mathematical Sciences 11 (2024), Article 49, DOI 10.1007/s40687-024-00460-z. https://doi.org/10.1007/s40687-024-00460-z
8. Liuquan Wang, Explicit forms and proofs of Zagier's rank three examples for Nahm's problem, Advances in Mathematics 450 (2024), 109743, DOI 10.1016/j.aim.2024.109743. https://doi.org/10.1016/j.aim.2024.109743
9. Zhineng Cao, Hjalmar Rosengren, and Liuquan Wang, On some double Nahm sums of Zagier, Journal of Combinatorial Theory, Series A 202 (2024), 105819, DOI 10.1016/j.jcta.2023.105819. https://doi.org/10.1016/j.jcta.2023.105819
10. Zhineng Cao and Liuquan Wang, Multi-sum Rogers–Ramanujan type identities, Journal of Mathematical Analysis and Applications 522 (2023), 126960, DOI 10.1016/j.jmaa.2022.126960. https://doi.org/10.1016/j.jmaa.2022.126960
11. Kaiwen Sun and Haowu Wang, Dynkin diagrams, generalized Nahm sums and 2d CFTs, arXiv:2604.00847 (2026), DOI 10.48550/arXiv.2604.00847. https://arxiv.org/abs/2604.00847
12. Frank Calegari, Stavros Garoufalidis, and Don Zagier, Bloch groups, algebraic K-theory, units, and Nahm's conjecture, Ann. Sci. Éc. Norm. Supér. 56(2) (2023), 383–426, DOI 10.24033/asens.2537. https://doi.org/10.24033/asens.2537
13. Haijun Li, Sequences of odd length in strict partitions IV: the combinatorics of parameterized Rogers–Ramanujan type identities, arXiv:2509.24942 (2025), DOI 10.48550/arXiv.2509.24942. https://arxiv.org/abs/2509.24942
14. Su-Ping Cui and Julia Q. D. Du, Some multi-sum Rogers–Ramanujan type identities with one parameter, The Ramanujan Journal 68(1) (2025), Article 20, DOI 10.1007/s11139-025-01173-9. https://doi.org/10.1007/s11139-025-01173-9