Partial Nahm Sums: Lattice Coset Modularity

• Partial Nahm sums are q-hypergeometric series defined by restricting the summation to lattice cosets, which alters both the combinatorial structure and modular behavior.
• They employ congruence and parity conditions to decompose full Nahm sums, leading to Rogers–Ramanujan type identities and explicit modular product formulas.
• Low-rank classifications and vector-valued approaches in partial Nahm sums open new avenues for applications in quantum topology and asymptotic arithmetic analysis.

Partial Nahm sums are $q$-hypergeometric series obtained by restricting the summation domain of a Nahm sum from all of $\mathbb Z_{\ge 0}^r$ to the nonnegative points of a lattice coset. In the formulation introduced by Wang and Zeng, one fixes a rational symmetric nonzero matrix $A$, a rational column vector $B$, a rational scalar $C$, a lattice $L\subseteq \mathbb Z^r$, and a coset $v+L\in \mathbb Z^r/L$, and defines

$$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$

with the convention that $1/(q;q)_n=0$ for $n<0$. When $\mathbb Z_{\ge 0}^r$0, this reduces to the ordinary Nahm sum; when $\mathbb Z_{\ge 0}^r$1 is a full-rank proper sublattice, it is called a rank-$\mathbb Z_{\ge 0}^r$2 partial Nahm sum. The central problem is a partial analogue of Nahm’s problem: determine which quadruples $\mathbb Z_{\ge 0}^r$3 give convergent and modular series (Wang et al., 26 Feb 2025).

1. Definition, terminology, and basic structure

The defining feature of a partial Nahm sum is the lattice-coset restriction. In the standard setting, the summation runs over all of $\mathbb Z_{\ge 0}^r$4; in the partial setting, one imposes congruence conditions encoded by $\mathbb Z_{\ge 0}^r$5. This restriction is arithmetically substantial rather than cosmetic, because it changes both the combinatorics of the denominator and the modular behavior of the resulting $\mathbb Z_{\ge 0}^r$6-series (Wang et al., 26 Feb 2025).

The terminology in the recent literature is not entirely uniform. In the strict sense used by Wang and Zeng and by Shi and Wang, “partial Nahm sum” means a lattice-coset sum of the form above (Wang et al., 26 Feb 2025, Shi et al., 27 Jul 2025). Closely related papers also use “partial” or “refined” for sums obtained by parity restrictions or by decomposing a full Nahm sum into pieces indexed by a finite abelian group, but those papers do not always adopt the lattice-coset definition as the primary one (Mizuno, 2023, Wang et al., 2024).

A common misconception is to identify partial Nahm sums with arbitrary truncations. The cited work does not support that identification. The lattice-coset restriction is intrinsic to the definition in (Wang et al., 26 Feb 2025), whereas cone restrictions, parity splittings, and seed decompositions appear as adjacent but distinct constructions in the broader Nahm-sum literature.

2. Position within the Nahm-sum landscape

The partial theory sits inside a larger ecosystem of Nahm-type series. The classical Nahm sum attached to symmetric positive definite $\mathbb Z_{\ge 0}^r$7 is

$\mathbb Z_{\ge 0}^r$8

while Mizuno’s symmetrizable generalization replaces the quadratic form and denominator by

$\mathbb Z_{\ge 0}^r$9

with $A$0 symmetric positive definite. This symmetrizable framework was developed partly because naturally occurring examples, such as the Kanade–Russell identities, do not fit the symmetric-denominator model (Mizuno, 2023).

Within cluster-algebraic rank-$A$1 Y-systems, refined pieces of an ordinary Nahm sum already appear. For a finite-type system one has

$A$2

where $A$3 is a finite abelian group of order $A$4. These $A$5 are described there as partial or refined Nahm sums, and it is conjectured that each partial sum is modular after an appropriate power of $A$6 (Mizuno, 2023).

A different neighboring generalization arises in quantum topology. Garoufalidis and Lê define generalized Nahm sums over a rational polyhedral cone,

$A$7

and explicitly note that the summation need not be over the full orthant. They do not formalize this as a theory of partial Nahm sums, but the stable series and $A$8-limits arising from colored Jones stability are described as naturally partial or truncated Nahm-type expansions (Garoufalidis et al., 2011).

This broader context matters because many modularity phenomena first appear in decomposed or restricted pieces before being assembled into a full modular object. Partial Nahm sums are therefore best viewed not as an isolated variation, but as one precise realization of a pervasive decompositional structure in Nahm-sum theory.

3. Low-rank classification and explicit modular families

The first systematic classification of modular partial Nahm sums is due to Wang and Zeng. In rank $A$9, they found eight modular partial Nahm sums using known identities. Equivalently, Theorem 1.2 identifies four modular quadruples

$B$0

$B$1

with $B$2 (Wang et al., 26 Feb 2025).

The rank-$B$3 theory is substantially richer. Wang and Zeng restricted attention to the three full-rank proper sublattices

$B$4

and searched for symmetric matrices

$B$5

for which some choice of $B$6 makes $B$7 modular. Their classification yields 14 types of symmetric matrices $B$8. The summary list given in the paper is

$B$9

$C$0

$C$1

For each listed matrix, the paper exhibits one or more specific choices of $C$2, $C$3, $C$4, and $C$5 producing a modular partial Nahm sum (Wang et al., 26 Feb 2025).

At the time of that work, one family remained open. Shi and Wang resolved it completely. The remaining family is

$C$6

with parameter choices

$C$7

$C$8

$C$9

Their proof completed the modularity verification for Wang–Zeng’s list of 14 modular families (Shi et al., 27 Jul 2025).

4. Rogers–Ramanujan type identities and proof technology

The decisive mechanism in the existing theory is explicit sum-to-product evaluation. For representative matrices $L\subseteq \mathbb Z^r$0, the modularity of a partial Nahm sum is established by a Rogers–Ramanujan type identity whose product side is manifestly modular. For example, for $L\subseteq \mathbb Z^r$1, Wang and Zeng prove

$L\subseteq \mathbb Z^r$2

and for $L\subseteq \mathbb Z^r$3 they derive four modular identities by parity dissection of an auxiliary two-variable series $L\subseteq \mathbb Z^r$4 (Wang et al., 26 Feb 2025).

Shi and Wang prove the two conjectural identities for the final open family: $L\subseteq \mathbb Z^r$5

$L\subseteq \mathbb Z^r$6

These identities are exactly the missing modular product formulas in Wang–Zeng’s classification (Shi et al., 27 Jul 2025).

Methodologically, the rank-$L\subseteq \mathbb Z^r$7 theory uses a dense blend of classical and modern $L\subseteq \mathbb Z^r$8-series tools. Wang and Zeng rely on the $L\subseteq \mathbb Z^r$9-binomial theorem, Euler’s exponential identities, Jacobi’s triple product, known Rogers–Ramanujan and Slater identities, Bailey pairs and Bailey’s lemma, and parity dissection formulas that encode coset restrictions through sign changes in auxiliary variables (Wang et al., 26 Feb 2025). Shi and Wang introduce a two-step proof strategy: first, Lovejoy’s transformation formula involving two Bailey pairs converts the partial Nahm sums into Hecke-type double series $v+L\in \mathbb Z^r/L$0; second, those Hecke-type series are converted to modular products either through Appell–Lerch expansions or through comparison with Kim–Lovejoy identities (Shi et al., 27 Jul 2025).

This proof architecture is notable because it does not treat modularity as an external black box. Instead, the modularity is exhibited constructively by exact product identities. In the present literature, that constructive route is the primary reason partial Nahm sums can be classified at all.

5. Parity restrictions, refined pieces, and vector-valued modularity

Partiality often appears not only through lattice cosets, but also through parity restrictions and decompositions of a full sum into modularly meaningful components. In the rank-$v+L\in \mathbb Z^r/L$1 periodic Y-system setting, ordinary Nahm sums arise from

$v+L\in \mathbb Z^r/L$2

and the paper states that

$v+L\in \mathbb Z^r/L$3

where the summands $v+L\in \mathbb Z^r/L$4 are partial or refined Nahm sums. The author conjectures that each such partial sum is modular after a suitable power of $v+L\in \mathbb Z^r/L$5 (Mizuno, 2023).

A more developed vector-valued theory appears in the rank-three generalized setting of index $v+L\in \mathbb Z^r/L$6. There, parity-restricted sums

$v+L\in \mathbb Z^r/L$7

are defined by restricting one summation index to a parity class, and are explicitly identified as the kind of partial Nahm sums under discussion. The resulting vector-valued functions $v+L\in \mathbb Z^r/L$8 and $v+L\in \mathbb Z^r/L$9 satisfy modular transformation formulas of the form

$$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$0

after a more precise decomposition through auxiliary functions $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$1. In the same paper, all 15 Mizuno examples are accounted for: 13 are modular in the usual sense, while two exceptional sums are shown to be sums of modular forms of weights $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$2 and $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$3 (Wang et al., 2024).

The symmetrizable matrix setting also produces partial sums naturally. In rank $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$4, one prominent numerical pattern is the appearance of “Langlands dual” pairs of matrices related by transposition, and in some cases by inverse or scaled inverse Cartan matrices. For a highlighted pair, the paper conjectures a modular $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$5-transformation relating vector-valued partial Nahm sums obtained by restricting $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$6 to a residue class mod $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$7; the conjectural $$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$8-matrix is written in terms of

$$f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^{\mathrm T}\in (v+L)\cap \mathbb Z_{\ge 0}^r} \frac{q^{\frac12 n^{\mathrm T}An+n^{\mathrm T}B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}},$$9

Here the partiality is again congruence-theoretic rather than merely truncational (Mizuno, 2023).

These results show that partial Nahm sums frequently organize themselves into vector-valued modular objects. The modular phenomenon is therefore often attached to a packet of restricted sums rather than to a single scalar series.

6. Asymptotics, arithmetic constraints, and open directions

The deepest arithmetic results currently concern full Nahm sums rather than lattice-coset partial sums. Garoufalidis and Zagier proved a full radial asymptotic expansion at primitive roots of unity for ordinary Nahm sums, with leading growth governed by the Rogers dilogarithm and all-order coefficients packaged by formal Gaussian integration. Their analysis proceeds through asymptotic expansion of the summand, decomposition into congruence classes, Poisson summation, and formal Gaussian integration. The same paper explicitly states that these techniques are highly relevant for any attempt to analyze partial Nahm sums, but also notes the main limitation: partial sums introduce boundary effects that are not treated there (Garoufalidis et al., 2018).

Mizuno’s symmetrizable extension shows that the asymptotic-dilogarithmic machinery survives beyond the symmetric case. In that setting, if the generalized Nahm sum $1/(q;q)_n=0$0 is modular, then the associated Bloch-group element

$1/(q;q)_n=0$1

is torsion. The proof follows the Calegari–Garoufalidis–Zagier strategy after replacing the symmetric asymptotic formula by its symmetrizable analogue. Although this theorem is not formulated for partial Nahm sums, the same paper’s numerical section suggests that partial sums are abundant in the symmetrizable world (Mizuno, 2023).

Several open problems are explicit. Wang and Zeng ask whether their rank-$1/(q;q)_n=0$2 theorem exhausts all modular rank-$1/(q;q)_n=0$3 partial Nahm sums, whether a complete classification in rank $1/(q;q)_n=0$4 is possible, and whether every modular full Nahm sum has a modular partial analogue; they also point out examples showing that the answer to the last question is likely no in general (Wang et al., 26 Feb 2025). This is an important corrective to any expectation that partial modularity should be inherited automatically from full modularity.

A related warning comes from the failure of duality in ordinary Nahm-sum theory. Counterexamples to Zagier’s duality conjecture show that a dual Nahm sum can fail to be modular even when the original sum is modular, and that the dual series may instead decompose as a sum of modular forms of different weights. That paper does not develop partial Nahm sums directly, but its decomposition arguments are described there as directly relevant to partial or decomposed $1/(q;q)_n=0$5-series, and it explicitly remarks that the correct formulation of the duality conjectures remains open (Wang, 2024).

Taken together, the current literature presents partial Nahm sums as a rapidly developing branch of Nahm-sum theory with three established pillars: a precise lattice-coset definition, an explicit low-rank modular classification proved by Rogers–Ramanujan type identities, and a growing vector-valued theory in which parity-restricted or refined pieces inherit modular transformation laws. What remains undeveloped is a general arithmetic or asymptotic framework comparable to that available for full Nahm sums. The existing papers strongly suggest that such a framework should involve congruence-class decomposition, cyclic-dilogarithm phenomena, and new control of boundary terms, but that implication remains conjectural rather than proved (Garoufalidis et al., 2018).