Precise generalized Nahm conjecture for half-indices of 3d Chern-Simons matter theories and the dynamical verification of the candidate modular Nahm sums

Characters of two-dimensional rational conformal field theories are often expressible as Nahm sums, multi-sum q-series built from a positive-definite matrix, and Nahm's conjecture ties their modularity to torsion in the Bloch group. Gang, Kim, Park and Stubbs (SciPost Phys. 19, 128 (2025)) showed that modified Nahm sums, carrying a sign factor fixed by the spin structure on the boundary torus and linear-logarithm corrections to the dilogarithm identities, arise as half-indices of three-dimensional abelian Chern-Simons matter theories flowing to unitary topological field theories or rank-zero superconformal fixed points, and found modular sums absent from Zagier's classical list. Two jointly linked questions remain open and form this problem: (i) a precise formulation, and a proof or refutation, of this physics-generalized Nahm conjecture — the source itself only 'comment[s] on a possible generalization of Nahm's conjecture' and states no criterion — meaning a criterion on the half-index data $(K,\alpha,\beta,\gamma)$ characterizing its modularity and supplying the signed replacement of Nahm's Bloch-group torsion condition; and (ii) the dynamical verification the source deferred for its candidate list: verifying the monopole-superpotential relevance conditions along the renormalization-group flow of the candidate theories of its $r\le3$, $|K_{ij}|\le17$ scan, resolving the borderline examples it flags, and determining whether $\tfrac{1}{2}\chi^{U(1)_2}_1(q)$ is the vacuum character of a rational VOA.

Background

A large class of characters of two-dimensional rational conformal field theories (RCFTs) — equivalently of rational, $C_2$-cofinite vertex operator algebras (VOAs), the algebraic axiomatizations of chiral CFTs — can be written as Nahm sums. For a positive-definite symmetric rational $r\times r$ matrix $A$, a rational vector $B$ of length $r$, and a constant $C$, the rank-$r$ Nahm sum is

$$\chi_{(A,B,C)}(q)=\sum_{m\in\mathbb{N}^{r}}\frac{q^{\frac{1}{2}m^{t}Am+B^{t}m+C}}{(q)_{m_{1}}\cdots(q)_{m_{r}}},$$

where $q=e^{2\pi i\tau}$, $\mathbb{N}$ is the set of non-negative integers and $(q)_n=\prod_{j=1}^{n}(1-q^{j})$. RCFT characters transform as weight-zero modular functions under $SL(2,\mathbb{Z})$ (vector-valued) or under a congruence subgroup (a subgroup of $SL(2,\mathbb{Z})$ containing the principal congruence subgroup $\Gamma(n)$), so it is natural to ask for which data $(A,B,C)$ the Nahm sum is modular. W. Nahm conjectured an answer in terms of the Bloch group, an algebraic $K$-theory invariant of a number field built from symbols $[x]$ modulo the five-term relation of the dilogarithm: writing $1-x_i=\prod_j x_j^{A_{ij}}$, the conjecture posits that the Bloch element $[x_1]+\cdots+[x_r]$ is torsion for every solution $x$ exactly when there exist rational $B,C$ for which the sum is modular (D. Zagier, 'The Dilogarithm Function', 2007). The rank-one case is proved (Zagier and Terhoeven, as described by Vlasenko and Zwegers). Vlasenko and Zwegers found rank-two and rank-four counterexamples to one direction — modular sums whose $A$ fails the all-solutions torsion condition — leaving the correct formulation open. Calegari, Garoufalidis and Zagier later proved the necessity direction in general: modularity forces the Nahm element to vanish in the Bloch group of $\overline{\mathbb{Q}}$, using the all-orders radial asymptotics of Nahm sums at roots of unity computed by Garoufalidis and Zagier. Sufficiency remains open in every rank $r\ge2$, and recent work (Liuquan Wang; Boxue Wang and Liuquan Wang; Milas and Wang; Mizuno; Hakimoglu-Brown) confirms modularity only family by family, through Rogers-Ramanujan-type identities (explicit $q$-series equalities), Bailey pairs (pairs of $q$-hypergeometric identities adapted to proving such equalities) and $Y$-system periodicity (the property that solutions of the algebraic recurrence relations associated with $A$ are periodic).

Gang, Kim, Park and Stubbs approached the question from quantum field theory. Three-dimensional $\mathcal{N}=2$ (four supercharges) abelian Chern-Simons matter theories $U(1)^r_K$ with level matrix $K$ and $r$ charge-one chiral multiplets, deformed by superpotentials built from monopole operators (disorder operators parameterized by magnetic charges), can flow in the infrared either to unitary topological field theories (TFTs) or to $\mathcal{N}=4$ rank-zero superconformal field theories (SCFTs; 'rank zero' meaning that the Coulomb branch, the moduli space of vacua of the abelian gauge multiplets, is trivial); in both cases the boundary is expected to support a rational VOA. The half-index — the supersymmetric partition function on $D^2\times S^1$ with a chosen boundary condition — then takes the modified Nahm form

$$I_{\rm half}=\sum_{m\in\mathbb{N}^{r}}(-1)^{\alpha^{t}m}\frac{q^{\frac{1}{2}m^{t}Km+\beta^{t}m+\gamma}}{(q)_{m_{1}}\cdots(q)_{m_{r}}},$$

where the sign factor $(-1)^{\alpha^t m}$ is fixed by the choice of spin structure on the boundary torus. The Bethe equations $1-x_a=\zeta_a\prod_{b=1}^{r}x_b^{K_{ab}}$ acquire phases $\zeta_a$, and the derivation of the modular $T$ matrix from the fibering operator (the supersymmetric partition-function operation that fibers the $S^1$ factor of the geometry over the base) shows that whenever the mixing parameter $\mu$ has non-even entries, the Rogers dilogarithm $L(x)$ and the Bloch-Wigner dilogarithm $D(x)$ (two standard single-valued modifications of the dilogarithm $\mathrm{Li}_2$) acquire additional terms linear in $\log x$. The authors call this a genuine generalization of Nahm's conjecture: the physically natural sums are signed and carry linear-log corrections.

Scanning all positive-definite integer level matrices with $r\le3$ and entries between $-17$ and $17$ (under simplifying symmetry assumptions), they found 27 candidate rank-zero theories (one of rank 1, three of rank 2 and twenty-three of rank 3, excluding direct products), together with infinite families and further candidates flowing to unitary TFTs, organized into 8 duality classes; several half-indices match known RCFT characters and yield modular sums not on Zagier's list. However, the paper itself flags what is missing. The relevance of each monopole superpotential along the renormalization-group (RG) chain — the condition $R^{(I-1)}_{0}(\mathcal{O}_I)<2$ on the F-maximized superconformal R-charge (F-maximization fixes the exact superconformal R-symmetry by extremizing the free energy $F=-\log|Z_{S^3}|$ of the theory on the three-sphere), needed for the chain of RG flows to make sense — was not verified and is called 'a challenging task, even numerically'. Two borderline examples (3-14-(2) and 3-20) show nonstandard features: both half-indices contain unexplained factors of $\chi_F^{-1}$ (an inverse free-fermion character), and both involve non-primitive monopole operators (monopoles whose magnetic charge vector is a non-unit multiple of another charge) — $V_{(-2,0,4)}$ in 3-14-(2) and $V_{(-2,2,2)}$ in 3-20 — which the authors suspect to be irrelevant; in addition, the A/B-twisted $S^2\times S^1$ partition functions (supersymmetric partition functions expected to equal 1 for a genuine unitary TFT) are not equal to 1 specifically in the 3-14-(2) case, while in the 3-20 case the half-index $I_A[W_{(1,1,0)}]$ is a sum of two characters modulo $\chi_F^{-1}$ even though $W_{(1,1,0)}$ lies in the class $W^{\rm sim}_A$ of UV Wilson lines expected to flow to simple IR lines. These features suggest some deformations are irrelevant and the candidate list is not yet provably correct. One of Zagier's rank-three modular sums, $\tfrac{1}{2}\chi^{U(1)_2}_1(q)$ — half the character of the charge-one module of the $U(1)$ current algebra at level 2 — is reproduced by a half-index but has no known interpretation as a rational-VOA vacuum character. And the search over theories flowing to unitary TFTs was explicitly not exhaustive, the $|K_{ij}|\le17$ bounds being, in the source's words, 'chosen primarily for technical reasons'. The source therefore leaves open both a precise formulation of the generalized Nahm conjecture on which it only 'comment[s]' and the dynamical validation of its candidate list, whose relevance checks are deferred to future work.

Problem Statement

The problem is to turn the half-index construction into established mathematics, along the directions the source paper itself leaves open: it only 'comment[s] on a possible generalization of Nahm's conjecture' without stating a precise criterion, and it defers to future work the dynamical verification of its candidate list. It consists of two jointly linked deliverables serving one objective, that of an established 3d realization of Nahm-type modular characters.

(1) Formulate precisely, and then prove or refute, a generalized Nahm conjecture for the modified Nahm sums arising as half-indices of 3d $\mathcal{N}=2$ abelian Chern-Simons matter theories:

$$I_{\rm half}=\sum_{m\in\mathbb{N}^{r}}(-1)^{\alpha^{t}m}\frac{q^{\frac{1}{2}m^{t}Km+\beta^{t}m+\gamma}}{(q)_{m_{1}}\cdots(q)_{m_{r}}}.$$

A complete answer supplies a criterion on the data $(K,\alpha,\beta,\gamma)$ — where $K$ is a positive-definite symmetric integer level matrix, while the spin-structure sign vector $\alpha$ and the exponent vector $\beta$, together with the constant $\gamma$, are half-integral or rational in general, with the integer case $\beta\in\mathbb{Z}^{r}$ recovered only when comparing to Zagier's list — incorporating the spin-structure sign factor $(-1)^{\alpha^t m}$, the Bethe-equation phases $\zeta_a=e^{i\pi\mu_a}$ in $1-x_a=\zeta_a\prod_b x_b^{K_{ab}}$, and the linear-logarithm modifications of the Rogers dilogarithm $L(x)$ and Bloch-Wigner function $D(x)$ for non-even mixing parameters, that characterizes when such a series is a modular function (or a component of a vector-valued modular form), together with the corresponding replacement of Nahm's Bloch-group torsion condition in this signed setting, and a proof or refutation of that criterion. The formulation is subject to a fidelity standard: the criterion must reduce to Nahm's classical Bloch-group torsion condition when $\alpha=0$ and the mixing parameter is even, and it must certify as modular the source's $r\le3$, $|K_{ij}|\le17$ candidate half-indices that the source identifies with known modular characters. Membership in the class of genuine half-indices is conditional on the relevance condition of deliverable (2), which the source leaves unverified; this is why the two deliverables are linked.

(2) Carry out the dynamical verification the source deferred for its candidate list, drawn from the scan over positive-definite integer level matrices $K$ with $r\le3$ and entries $|K_{ij}|\le17$ under the paper's simplifying symmetry assumptions; the source acknowledges the $|K_{ij}|\le17$ bounds as 'chosen primarily for technical reasons' and the unitary-TFT search as non-exhaustive, and these stand as acknowledged limitations of the candidate list rather than as a completed classification. Completing the verification requires the dynamical checks the source deferred: verifying the relevance condition $R^{(I-1)}_{0}(\mathcal{O}_I)<2$ for each monopole superpotential $\mathcal{O}_I$ along every proposed RG step (via F-maximization of the R-charge, including possible decoupling of UV flavor symmetries), and resolving the borderline cases — whether the $\chi_F^{-1}$-bearing examples 3-14-(2) (the second superpotential choice for candidate 3-14) and 3-20 are artifacts of irrelevant deformations (their non-primitive monopoles $V_{(-2,0,4)}$ and $V_{(-2,2,2)}$ being plausibly irrelevant) or genuine bulk-boundary constructions, accounting for the A/B-twisted $S^2\times S^1$ partition functions not equal to $1$ in the 3-14-(2) case and for $I_A[W_{(1,1,0)}]$ being a sum of two characters modulo $\chi_F^{-1}$ in the 3-20 case — and determining whether the modular sum $\tfrac{1}{2}\chi^{U(1)_2}_1(q)$ is the vacuum character of a rational VOA or a fractional combination thereof.

The two deliverables are jointly supporting: (1) provides the criterion that makes modularity decidable within the class and connects it to the Bloch-group framework, while (2) verifies the physical candidate list to which the criterion applies; a list without a criterion is a catalogue of conjectures, and a criterion without the verification does not upgrade the candidate list. Together they capture the open program the source leaves: a mathematically precise generalized Nahm conjecture for the signed half-index sums, and the dynamical validation of its candidate 3d realizations of Nahm-type modular characters.

Scope. The source's future-work discussion additionally lists two items that are recorded here as auxiliary context rather than completion criteria: the general criterion for producing every infrared simple object from explicit ultraviolet Wilson-line operators (the ultraviolet-infrared map for line operators), and a fully rigorous derivation of the modular $S$ and $T$ matrices — including overall phases and framing ambiguities — for all candidate examples. Both concern the line-operator sector and the extraction of modular data rather than the modularity and physical realizability of the Nahm-type sums; they are not required for the two deliverables above, although a solution would naturally illuminate them.

References

1. D. Gang, H. Kim, B. Park, S. Stubbs, 'Three Dimensional Topological Field Theories and Nahm Sum Formulas', SciPost Phys. 19, 128 (2025). DOI: 10.21468/SciPostPhys.19.5.128, arXiv:2411.06081, https://arxiv.org/abs/2411.06081
2. D. 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
3. F. Calegari, S. Garoufalidis, D. Zagier, 'Bloch groups, algebraic K-theory, units, and Nahm's conjecture', Ann. Sci. Ec. Norm. Super. (4) 56, 383-426 (2023). DOI: 10.24033/asens.2537, arXiv:1712.04887, https://arxiv.org/abs/1712.04887
4. S. Garoufalidis, D. Zagier, 'Asymptotics of Nahm sums at roots of unity', Ramanujan J. 55, 219-238 (2021). DOI: 10.1007/s11139-020-00266-x, arXiv:1812.07690, https://arxiv.org/abs/1812.07690
5. M. Vlasenko, S. Zwegers, 'Nahm's conjecture: asymptotic computations and counterexamples', Commun. Number Theory Phys. 5(3), 617-642 (2011). DOI: 10.4310/cntp.2011.v5.n3.a2, arXiv:1104.4008, https://arxiv.org/abs/1104.4008
6. K. Sun, H. Wang, 'Dynkin diagrams, generalized Nahm sums and 2d CFTs', arXiv:2604.00847 (2026), https://arxiv.org/abs/2604.00847
7. Y. Yoshida, '3d N=4 rank-zero mirror symmetry, TQFT interfaces, and Zagier duality of Nahm sums', arXiv:2608.08780 (2026), https://arxiv.org/abs/2608.08780
8. L. Wang, 'Counterexamples to some duality principles of modular Nahm sums', Proc. Amer. Math. Soc. 154, 2383-2396 (2026). DOI: 10.1090/proc/17624, arXiv:2411.09701, https://arxiv.org/abs/2411.09701 (preprint version entitled 'Counterexamples to Zagier's duality conjecture on Nahm sums')
9. B. Wang, L. Wang, 'Mizuno's rank three Nahm sums I: identities of index (1,1,2)', arXiv:2402.06253 (2024), https://arxiv.org/abs/2402.06253
10. Y. Mizuno, 'Periodic Y-systems and Nahm sums: the rank 2 case', SIGMA 21 (2025), 094. DOI: 10.3842/sigma.2025.094, arXiv:2301.13239, https://arxiv.org/abs/2301.13239
11. A. Milas, L. Wang, 'Modularity of Nahm sums for the tadpole diagram', Int. J. Number Theory 20(1), 73-101 (2024). DOI: 10.1142/S1793042124500040, https://doi.org/10.1142/S1793042124500040
12. L. Wang, 'Identities on Zagier's rank two examples for Nahm's problem', Res. Math. Sci. 11, Paper No. 49 (2024). DOI: 10.1007/s40687-024-00460-z, https://doi.org/10.1007/s40687-024-00460-z
13. C. Hakimoglu-Brown, 'On a conjecture due to Kanade related to Nahm sums', arXiv:2603.00836 (2026), https://arxiv.org/abs/2603.00836
14. D. Gang, H. Kim, S. Stubbs, 'Three-Dimensional Topological Field Theories and Nonunitary Minimal Models', Phys. Rev. Lett. 132, 131601 (2024). DOI: 10.1103/PhysRevLett.132.131601, arXiv:2310.09080, https://arxiv.org/abs/2310.09080
15. D. Gang, B. Park, H. Sohn, 'Torus Knots and Minimal Models Revisited: Rational VOA characters from non-hyperbolic knots', arXiv:2512.23122 (2025), https://arxiv.org/abs/2512.23122
16. C. Closset, A. Keyes, S. Kim, 'Three-dimensional TQFTs from Argyres-Douglas theories via the 3d/3d correspondence', arXiv:2607.20308 (2026), https://arxiv.org/abs/2607.20308
17. K. Jeong, H. Sohn, 'Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs', arXiv:2608.11946 (2026), https://arxiv.org/abs/2608.11946
18. Q. Jia, S. Kim, 'Classification of monopole deformed 3d N=2 Seiberg-like duality with an adjoint matter', JHEP 03 (2026), 247. DOI: 10.1007/JHEP03(2026)247, arXiv:2507.04950, https://arxiv.org/abs/2507.04950
19. H. Kim, H. Kim, J. Song, 'Macdonald index from 3d TQFT', JHEP 03 (2026), 213. DOI: 10.1007/JHEP03(2026)213, arXiv:2511.11186, https://arxiv.org/abs/2511.11186