Nahm's problem: classify the rational triples (A, B, C) for which the Nahm sum $f_{A,B,C}(q)$ is a modular function

Nahm's problem, arising from rational conformal field theory and reiterated as the central open question in Chen and Xu's 2026 study of bilateral Rogers-Ramanujan type identities, asks for a complete classification of the rational triples $(A,B,C)$ - with $A$ a rational positive definite symmetric $r\times r$ matrix, $B\in\mathbb{Q}^r$ and $C\in\mathbb{Q}$ - for which the $r$-fold $q$-hypergeometric Nahm sum $f_{A,B,C}(q)=\sum_{n\in\mathbb{Z}_{\ge 0}^r} q^{\frac{1}{2}n^tAn+n\cdot B+C}/((q;q)_{n_1}\cdots(q;q)_{n_r})$ is a modular function. The rank-1 case was completely solved by Zagier, who found exactly seven modular triples (the last two being the Rogers-Ramanujan sums); for every rank $r\ge 2$ the classification is open. The subsequent literature shows: all of Zagier's rank-2 and rank-3 candidate lists have now been proved modular, many new families have been discovered (showing the candidate lists incomplete), Nahm's conjectural Bloch-group criterion is refuted as a biconditional - Vlasenko and Zwegers (2011) found modular triples whose matrix parts fail its torsion condition - while its precise necessity direction, that modularity forces the vanishing of the Bloch-group element attached to the distinguished real solution of Nahm's equation, is a theorem (Calegari-Garoufalidis-Zagier, 2023). A complete characterization of modular triples in rank $\ge 2$, or a proven effective criterion replacing the refuted conjecture, remains open.

Background

The Rogers-Ramanujan identities state that $\sum_{n=0}^{\infty}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q,q^4;q^5)_\infty},\qquad \sum_{n=0}^{\infty}\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2,q^3;q^5)_\infty},$ where $q=e^{2\pi i\tau}$ (with $|q|<1$ for convergence), $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ is the $q$-shifted factorial (q-Pochhammer symbol), and $(a_1,\dots,a_m;q)_n$ denotes the product of $m$ such factors. Identities of this shape - a unilateral basic hypergeometric sum (a $q$-analogue of a hypergeometric series, in which the ratio of consecutive terms is a rational function of $q^n$) equal to an infinite product - are called Rogers-Ramanujan type identities, and they are surveyed in Warnaar and Zudilin's study of Dedekind's eta-function and such identities. After multiplication by a suitable rational power of $q$, the product sides above become modular functions: functions of the upper-half-plane variable $\tau$ transforming invariantly under a finite-index subgroup of the modular group $SL_2(\mathbb{Z})$ (a congruence subgroup, such as $\Gamma_1(N)$). The building blocks are products of Dedekind's eta-function $\eta(\tau)=q^{1/24}(q;q)_\infty$ and its generalizations. This modularity is invisible on the sum side, which raises the general question: which $q$-hypergeometric series are modular?

Nahm, motivated by two-dimensional rational conformal field theory (RCFT - a quantum field theory with finitely many irreducible representations, whose characters $\chi(\tau)$ are modular functions), singled out the following class of series. Fix a positive integer $r$ (the rank), a real positive definite symmetric $r\times r$ matrix $A$, a vector $B\in\mathbb{R}^r$, and a scalar $C\in\mathbb{R}$, and define the Nahm sum $f_{A,B,C}(q)=\sum_{n=(n_1,\dots,n_r)\in\mathbb{Z}_{\ge 0}^r}\frac{q^{\frac{1}{2}n^tAn+n\cdot B+C}}{(q;q)_{n_1}\cdots(q;q)_{n_r}}.$ The two Rogers-Ramanujan sums are the rank-1 Nahm sums for $(A,B,C)=(2,0,-1/60)$ and $(2,1,11/60)$. Nahm's problem asks to describe all triples $(A,B,C)$ with rational entries for which $f_{A,B,C}(e^{2\pi i\tau})$ is a modular function (of weight zero, on some congruence subgroup); such a triple is called a modular triple, and $A$, $B$, $C$ are its matrix, vector and scalar parts. Modular Nahm sums are expected, up to normalization, to be characters of RCFTs - the physical origin and significance of the problem. Nahm also conjectured a criterion on the matrix part $A$, formulated through the Bloch group (the group of five-term relations among values of the dilogarithm $\operatorname{Li}_2$, identifiable with the algebraic $K$-group $K_3$ of a number field): attach to $A$ the polynomial system $1-Q_i=\prod_{j=1}^{r}Q_j^{A_{ij}}$ for $i=1,\dots,r$, each solution $(Q_1,\dots,Q_r)$ of which determines an element $[Q_1]+\dots+[Q_r]$ of the Bloch group; the conjecture asserts that $A$ is the matrix part of a modular triple precisely when these elements are torsion (in the strongest form of the conjecture, for every solution), with a compatible central charge (the parameter $c$ of the Virasoro symmetry algebra of the conformal field theory).

The rank-1 case is completely solved: Zagier proved that there are exactly seven modular triples, namely $(1/2,0,-1/40)$, $(1/2,1/2,1/40)$, $(1,0,-1/48)$, $(1,1/2,1/24)$, $(1,-1/2,1/24)$, $(2,0,-1/60)$ and $(2,1,11/60)$, and confirmed Nahm's criterion in this case. For ranks 2 and 3, computer searches over restricted families of matrices produced candidate lists (eleven rank-2 example sets and twelve rank-3 sets, recorded by Zagier); all of these candidates have now been proved modular, but the searches covered only special matrix shapes with bounded entries, and modular triples outside the lists are known - so the lists are not classifications. Nahm's conjectural criterion is refuted as a biconditional: Vlasenko and Zwegers found rank-2 modular triples whose matrix parts fail the torsion condition at some non-real solution of the associated system. The precise necessity direction - modularity forces the Bloch-group element attached to the distinguished real solution to vanish - is a theorem (Calegari-Garoufalidis-Zagier), but no sufficiency direction is proven and no effective replacement criterion is known. For every rank $r\ge 2$ the classification is open.

Problem Statement

Nahm's problem: determine all triples $(A,B,C)$, where $A$ is a rational positive definite symmetric $r\times r$ matrix, $B\in\mathbb{Q}^r$ and $C\in\mathbb{Q}$, with $r\ge 1$, such that the Nahm sum $f_{A,B,C}(q)=\sum_{n\in\mathbb{Z}_{\ge 0}^r}\frac{q^{\frac{1}{2}n^tAn+n\cdot B+C}}{(q;q)_{n_1}\cdots(q;q)_{n_r}},\qquad q=e^{2\pi i\tau},$ where $(q;q)_n=\prod_{k=1}^{n}(1-q^k)$ is the $q$-shifted factorial, is a modular function of weight zero on some congruence subgroup of $SL_2(\mathbb{Z})$ - that is, it is meromorphic on the upper half-plane and at the cusps of the subgroup, and invariant, $f_{A,B,C}(e^{2\pi i\gamma\tau})=f_{A,B,C}(e^{2\pi i\tau})$ for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ in the subgroup, with $\gamma\tau=(a\tau+b)/(c\tau+d)$. The rank-1 case is settled (exactly seven modular triples), so the open content of the problem is the complete classification for every rank $r\ge 2$.

References

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