Nahm-Sum-Like Expressions in q-Series

• Nahm-sum-like expressions are q-hypergeometric multivariate series defined by parameters (A, B, C) that capture complex interactions in combinatorics, number theory, and mathematical physics.
• They extend to generalized, partial, and double-pole forms, providing concrete connections to Rogers–Ramanujan identities, modular phenomena, and vertex operator algebras.
• Advanced techniques such as Bailey pairs, constant term methods, and quantum dilogarithm reductions play a key role in classifying these sums and linking them to asymptotic invariants and quantum topology.

Nahm-Sum-Like Expressions

Nahm-sum-like expressions denote a broad class of $q$-hypergeometric multivariate series, typically of the form

$$f_{A,B,C}(q) = \sum_{n\in\mathbb{N}^r} \frac{q^{\frac12\,n^T A n + B^T n + C}}{(q)_{n_1} (q)_{n_2} \cdots (q)_{n_r}}$$

where $A$ is an $r\times r$ symmetric (often positive-definite) rational matrix, $B\in\mathbb{Q}^r$, $C\in\mathbb{Q}$, and $(q)_n = \prod_{j=1}^n (1-q^j)$. These sums, and generalizations thereof (incorporating symmetrizers, partial summations, or higher powers in the denominator), encapsulate deep phenomena in combinatorics, number theory, quantum topology, and representation theory of vertex operator algebras (VOA) and conformal field theory (CFT).

1. General Forms and Extensions

Beyond the "classical" Nahm sum described above, several important extensions arise:

• Generalized (Symmetrizable) Nahm Sums: Given a matrix $A\in\mathbb{Q}^{r\times r}$ symmetrizable by a diagonal $D=\mathrm{diag}(d_1,\dots,d_r)$—i.e., $A D$ is symmetric positive definite—and $B$, $C$ as above,

$$f_{A,B,C,D}(q) = \sum_{n\in\mathbb{Z}_{\ge0}^r} \frac{q^{\frac12 n^T A D n + B^T n + C}}{\prod_{i=1}^r (q^{d_i}; q^{d_i})_{n_i}}$$

This generality captures the modular examples associated to generalized Cartan matrices, including all simple Lie algebras and their twisted/affine types (Mizuno, 2023).

• Partial Nahm Sums: For a full-rank lattice $L\subset\mathbb{Z}^r$ and coset $v+L$,

$$f_{A,B,C; v+L}(q) = \sum_{n\in (v+L)\cap\mathbb{Z}_{\ge0}^r} \frac{q^{\frac12 n^T A n + B^T n + C}}{\prod_{i=1}^r (q;q)_{n_i}}$$

These "partial" sums admit modular families in low rank, sometimes yielding weight-$\frac12$ or $-\frac12$ modular forms outside the standard (weight-0) Nahm paradigm (Wang et al., 26 Feb 2025, Shi et al., 27 Jul 2025).

• Nahm-Type Sums with Double Poles: Introducing a denominator squared, as in

$$\sum_{n_1, ..., n_t\ge0} \frac{(-w)^{n_1+...+n_t} q^{Q(n)}}{(q)_{n_1}^2 \cdots (q)_{n_t}^2}$$

(with explicit $Q(n)$ depending on Anderson–Gordon/Bressoud parameters), naturally arises in Schur indices in 4d/2d dualities and in the explicit construction of both modular and mock/false-theta series (Kanade et al., 2021).

2. Modularity Phenomena, Rogers–Ramanujan Identities, and Transformation Properties

Nahm-sum-like expressions are deeply interwoven with modular representations:

• Modular Triple/Quadruple Data: The triple $(A, B, C)$ (or quadruple with $D$) is called modular if the corresponding series is (vector-valued) modular for some congruence subgroup $\Gamma_0(N)$. Examples include:
  • Andrews–Gordon and Bressoud identities, with explicit multivariate sums and infinite product modular forms as in (Wang et al., 2024).
  • Nahm sums attached to Cartan matrices of simply-laced Lie algebras, e.g., $D_k$-type identities of Warnaar–Flohr–Grabow–Koehn, which admit explicit expressions in terms of Rankin–Cohen brackets of theta series (Wang et al., 8 Dec 2025).
  • Tadpole Cartan matrices $C_r$ ($C_{rr}=1$, $C_{ii}=2$ for $i<r$), yielding Nahm sums modular on congruence subgroups of high level for $r\geq 3$ (Shi et al., 24 Apr 2025, Milas et al., 2023).
• Vector-Valued Modular Forms: For fixed $A$, multiple Nahm sums for varying $B$ (and $D$) often assemble into a finite-dimensional representation for $\Gamma_0(N)$, with explicit $S$- and $T$-matrices constructed via trigonometric (cosine) matrices or as combinations of theta functions and Weber functions (Wang et al., 2024, Wang et al., 2023).
• Symmetrizable and Index-Varying Nahm Sums: Generalized Nahm sums with nontrivial $D$ (e.g., $d = (1,1,2)$, $d = (1,2,2)$, $d = (2,2,1)$) and explicit parameters tabulated by Mizuno/Wang are modular in numerous cases, sometimes with exceptional behavior (e.g., non-pure weight or sums of modular forms of different weight) (Wang et al., 2024, Wang et al., 2024).

3. Explicit Families, Classifications and Computational Techniques

Extensive computational classification in low rank has been performed:

• Rank Two and Three Tables: Complete lists of modular quadruples for $r=2,3$ and admissible lattices have been compiled, linking the algebraic properties of $A$, $B$, $C$, $D$ to canonical Rogers–Ramanujan or (generalized) theta product forms (Wang et al., 26 Feb 2025, Wang et al., 2023, Mizuno, 2023).
• Partial/Non-integral Weight Examples: For partial sums (summed over sublattices), modular forms of weight $\pm 1/2$ arise, with 14 explicit families for $r=2$ and several for $r=1$. Each family corresponds to a Rogers–Ramanujan type identity, proven via $q$-hypergeometric techniques and the Bailey pair machinery (Wang et al., 26 Feb 2025, Shi et al., 27 Jul 2025).
• Rank Four and Five Lift-Dual Extensions and Obstructions: New modular rank-4 Nahm sums have been synthesized via lift-dual operations from modular rank-3 data, with many product identities proved and others conjectured (Cao et al., 17 Aug 2025). Yet, duality fails in higher rank: certain modular quadruples do not have modular duals, providing counterexamples to the duality conjecture (Wang, 2024).

4. $q$-Series Proofs: Bailey Pairs, Constant Term, and Hypergeometric Methods

Several interlocking $q$-series techniques underlie most identities:

• Bailey Pair/Machinery: Systematic use of Bailey pairs, their iterated transforms (Slater-type, S1/S3/S5), and base-change lemmas enable the reduction of multivariate Nahm sums to canonical single-sum theta, partial-theta, or Appell–Lerch forms (Wang et al., 8 Dec 2025, Wang et al., 2023, Wang et al., 2024).
• Constant Term and Quantum Dilogarithm Methods: Some sums naturally realize as constant terms in products of quantum dilogarithms; pentagon relations and quantum $q$-integral techniques reduce complex expressions to ordinary Nahm sums or known modular forms (Kanade et al., 2021).
• Hecke-Type or Appell–Lerch Reductions: For partial Nahm sums or double-pole types, transformation to Hecke-type double sums, subsequently to Appell–Lerch sums and finally to theta-quotients, enables matching with known modular forms (Shi et al., 27 Jul 2025).
• Direct $q$-Hypergeometric Summations/Reductions: q-binomial, $q$-Chu–Vandermonde, and elementary series manipulations are frequently used in explicit reductions, often combined with classical identities from Rogers, Ramanujan, and Slater catalogs (Wang et al., 2024).

5. Asymptotics, Modularity, and Bloch Group Connections

Nahm sums possess asymptotic expansions at roots of unity, governed by deep arithmetic invariants:

• Dilogarithmic Asymptotics and Nahm Equations: The leading exponential behavior of a Nahm sum as $q\to\zeta_m$ is controlled by the Rogers dilogarithm evaluated at solutions to $1-z_i = \prod_j z_j^{A_{ij}}$. The full expansion features Rogers (quantum) dilogarithms, Gaussian integrals, and correction series capturing the perturbative data (Garoufalidis et al., 2018, Mizuno, 2023).
• Bloch Group and Nahm's Conjecture: Modularity of a Nahm sum is linked—by the Calegari–Garoufalidis–Zagier theorem and more general results—to torsion classes in the (extended) Bloch group associated to the matrix $A$ and its "Nahm equation" solutions (Mizuno, 2023, Garoufalidis et al., 2018). This provides a criterion ("Bloch–Wigner torsion") for the possibility of modularity.
• Nahm Sums in Knot Theory and Quantum Topology: The tail of the colored Jones polynomial for alternating links is expressible as a generalized Nahm sum—with admissible region determined by the associated planar graph—which is directly related to the asymptotics of quantum invariants (Kashaev invariant, volume conjecture) via the aforementioned Bloch group data (Garoufalidis et al., 2011).

6. Physical Context and Applications: Conformal Field Theory, VOAs, and Topological Field Theory

Nahm-sum-like expressions permeate several physical and algebraic frameworks:

• Characters of Rational CFTs and VOAs: Many minimal model characters, $W$-algebra characters, and level-one affine or superconformal algebra characters admit explicit Nahm sum expansions (both single and multi-variable) (Gang et al., 2024, Garoufalidis et al., 2011, Milas et al., 2023).
• 3d $\mathcal{N}=2$ Gauge Theory and Half-Index: For abelian Chern–Simons–matter theories, Dirichlet half-indices realize Nahm sums with $A$ the Chern–Simons matrix and $b$ from boundary R-symmetry data; this bulk-boundary correspondence explains modularity in terms of unitarity, spin structure, and the modular $T$-matrix in the IR topological field theory (Gang et al., 2024).
• Defects and Quantum Dilogarithm Traces: In 4d/2d dualities and Schur/defect index computations, double-pole and non-standard Nahm sums appear as characters of generalized vertex algebras, often with explicit modular or mock-modular properties (Kanade et al., 2021).

7. Open Directions and Structural Observations

Active and challenging research avenues include:

• Full Classification of Modular Triples: Determining all modular $(A,B,C, D)$ and their vector-valued structures remains open beyond low rank, with modularity tied to intricate arithmetic constraints (Wang et al., 2024, Mizuno, 2023).
• Automorphic, Quantum, and Mock/False-Theta Behavior: Many Nahm-type or double-pole sums yield not genuine modular but (quantum or mock) modular forms; understanding the categorical or geometric meaning of such phenomena is ongoing (Kanade et al., 2021, Garoufalidis et al., 2011).
• Lift-Dual and Counterexamples: The presence of modular Nahm sums whose (lift-)duals are not modular demonstrates that naive duality is obstructed in higher rank, highlighting the subtlety and delicacy of modularity in these families (Wang, 2024).
• Combinatorial and Partition-Theoretic Interpretations: While certain sums enumerate partitions with specific difference conditions, developing bijective combinatorial proofs and understanding the full spectrum of possible partition-theoretic interpretations, especially under partial or non-conventional summation domains, remains incomplete.
• Higher Rank, Weight, and Non-pure Modularity: Emergent examples (especially with symmetrizers $D$ and partial sums) exhibit weight-splitting or vector-valued behaviors not captured by simple holomorphic modularity, pointing to a broader landscape of automorphic and representation-theoretic structures (Wang et al., 2024).

In summary, Nahm-sum-like expressions reveal a versatile, unifying structure at the intersection of $q$-series, modular forms, mathematical physics, and combinatorics, with a rapidly expanding theory encompassing both long-standing classical identities and a wealth of new, structurally intricate families and modular phenomena.