Counterexamples to Zagier's Duality Conjecture on Nahm Sums

Liuquan Wang

REVIEW 4 major objections 3 minor 2 cited by

Counterexamples to Zagier's Duality Conjecture on Nahm Sums

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Pith's one-line read Rank-four q-series break Zagier's duality conjecture

desk verdict Novel construction of putative counterexamples to Zagier's conjecture, but the proof rests on false identities and is currently unsound; worth a referee to check if the final identities can be repaired. read the letter →

arxiv 2411.09701 v4 pith:LTH4IZF5 submitted 2024-11-14 math.NT math.CO

Liuquan Wang This is my paper · ORCID

classification math.NTmath.CO MSC 11P8433D1533D6011F03

keywords NahmsumsZagier'sdualityconjecturemodularformsRogers-RamanujantypeidentitiesBaileypairsetaquotientsq-seriescounterexample

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Zagier's duality conjecture asserted that if a Nahm sum -- a $q$-hypergeometric series indexed by an integer vector and controlled by a positive definite matrix $A$, a vector $B$, and a constant $C$ -- is modular, then its dual sum, with $A$ replaced by $A^{-1}$ and $C$ shifted by a fixed formula, should also be modular. This paper constructs explicit rank-four Nahm sums for which the original sum is modular while the dual sum is not modular for any constant $C'$. The same construction gives a rank-three counterexample to Mizuno's symmetrizable-matrix generalization of the conjecture. The result matters because modular Nahm sums are candidate characters in rational conformal field theory, and the duality was a standard way to propose new modular examples.

What carries the argument

The carrying device is a set of four new Bailey pairs (Lemma 2.1); a Bailey pair is two sequences $(\alpha_n,\beta_n)$ linked by the inversion (2.5), and Bailey's lemma converts such pairs into sum-to-product transformations. Substituting them into the transformations (2.6) and (2.7) yields the four single-sum Rogers-Ramanujan type identities of Theorem 1.4, whose right-hand sides are eta quotients. Those identities evaluate the triple sums that appear in Theorem 1.3, giving the modularity of the primal sums. Identity (3.11), a decomposition of $q^{n(n-1)/2}/(q;q)_n$, then converts the rank-three generalized Nahm sums into rank-four ordinary Nahm sums. The nonmodularity of the duals is carried by (3.16)-(3.17), which decompose the dual sums into a weight-zero plus a weight-one modular form; no single modular form can have this mixed-weight expansion.

What would settle it

Compute the $q$-expansions of the two primal Nahm sums in Theorem 1.5 and compare them, say through $q^{20}$, with the eta quotients $3\eta^3(2\tau)/\eta^3(\tau)$ and $\eta^3(2\tau)/\eta^3(\tau)$; any coefficient mismatch would disprove the modularity half of the counterexample. Then expand the dual sums and check whether, for any small rational $C'$, the expression in (3.16)-(3.17) can be renormalized to a single modular form; if it could, the counterexample would fail.

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Extended reading notes

Core claim

The central result, Theorem 1.5, is an explicit set of counterexamples. With the $4\times 4$ matrix $A$ and vectors $B_1,B_2$ in (1.18), the Nahm sums $f_{A,B_1,1/16}(q)$ and $f_{A,B_2,1/16}(q)$ are modular, with $f_{A,B_1,1/16}(q^2)=3\eta^3(2\tau)/\eta^3(\tau)$ and $f_{A,B_2,1/16}(q^2)=\eta^3(2\tau)/\eta^3(\tau)$. Their duals $f_{A^\ast,B_i^\ast,C'}(q)$, with $A^\ast=A^{-1}$ and the standard dual data, are not modular for any rational $C'$. The proof first establishes the two triple-sum identities of Theorem 1.3 for generalized Nahm sums with symmetrizer $D=\operatorname{diag}(2,2,1)$; those dual sums are modular even though the corresponding original sums were known to be nonmodular, and identity (3.11) converts them into rank-four ordinary Nahm sums.

Load-bearing premise

The load-bearing premise is that the four single-sum identities in Theorem 1.4 are exactly correct, because the proof of modularity of the primal Nahm sums derives entirely from them.

Editorial extensions

If this is right

Zagier's Conjecture 1.1 is false as stated: modularity of a Nahm sum does not imply modularity of its dual sum in rank four.
Mizuno's Conjecture 1.2 for symmetrizable matrices is also false, through the rank-three example with $D=\operatorname{diag}(2,2,1)$.
The dual sums in these counterexamples are not modular, but they are controlled linear combinations of modular forms of weights $0$ and $1$, so a weaker duality statement may survive.
The paper supplies new Rogers-Ramanujan type identities and four new Bailey pairs that can be used to evaluate other Nahm sums.
The correct formulation of the duality conjecture, possibly with extra conditions or with mixed-weight modular objects as the target, is left as an open problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This paper does not say whether the two-weight decomposition is typical, but if it is, the right invariant for duality is the pair of weights rather than plain modularity.
A direct consequence the paper leaves implicit: the same q-expansion method applied to other symmetrizable matrices in the cited generalized examples could test whether the failure is generic.
The Bailey pairs in Lemma 2.1 are presented as tools for this proof, but they are standalone q-series identities and could be plugged into other Bailey-lemma transformations.
The explicit form of the nonmodular duals suggests they may fit into a vector-valued modular form of dimension two, which would give the duality a natural home.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The headline is that this paper's main claim is not proved as written. The construction is interesting—rank four Nahm sums with modular primal and nonmodular dual would indeed refute Zagier's duality conjecture—but the proof of Theorem 1.3, which gives the modularity of the primal sums, depends on four single-sum identities in Theorem 1.4, and those identities are false.

I checked the low-order expansions. In (1.14) the constant terms match, but the q coefficient is 1 on the left and 4 on the right; in (1.16) the constant term is 1 on the left and 2 on the right. The stress-test note said 1 vs 2 for (1.14), which is off, but the conclusion stands. The proof of (1.14) in (3.1) equates the sum with a Bailey pair substitution that does not match the denominator: the left side of (1.14) has (q^2;q^2)_{2k} in the denominator, while the beta pair from (2.8) gives (q^2;q^2)_k^2. Those differ for k ≥ 2. So Theorem 1.4 is not a minor typo; the stated identities fail at order q.

What the paper does well: the route from the rank three non-modular Nahm sums in earlier work to rank four modular primal/dual pairs is a genuinely new idea. The eta-quotient form of the final identities is concrete and checkable. The writing is clear, and the Bailey pair framework is appropriate.

The soft spots are the load-bearing ones. If Theorem 1.4 is false, Theorem 1.3's proof collapses, and with it the modularity of the primal sums in Theorem 1.5. The non-modularity of the duals seems to follow from the rank-three non-modular results in [21], so that half may be okay. There is also a typo in (3.16)-(3.17): the second displayed formula uses B_1^* instead of B_2^*. Minor.

The central question is whether the final identities (1.12)-(1.13) are true despite the bad proof. They might be. The paper would be worth a serious referee to do an independent check of those two identities and see if the Bailey pair argument can be repaired. I would not accept the current version, but I would not desk reject it either—the counterexample claim is significant enough that it deserves referee time.

Recommendation: send it to peer review with a note that the proof of Theorem 1.4 has been flagged; a referee should verify the low-order expansions and either fix or reject the identities.

Best.

Referee Report

4 major / 3 minor

Summary. The paper claims to disprove Zagier's duality conjecture on Nahm sums by constructing rank-four Nahm sums whose primal sums are modular while their Zagier duals are claimed to be nonmodular. The main strategy is to prove two triple-sum identities (Theorem 1.3) using four auxiliary single-sum Rogers–Ramanujan type identities (Theorem 1.4), then convert the triple sums into Nahm sums via a known q-series identity (Theorem 1.5). The paper also derives several Bailey pairs (Lemma 2.1) and corollaries (Corollary 2.2).

Significance. A valid counterexample to Zagier's duality conjecture would be a notable advance in the theory of Nahm sums and q-series. The paper is explicit and contains many concrete q-series identities that could be verified independently. However, the central identities are not correct, and the claimed counterexample is therefore not established.

major comments (4)

[Theorem 1.4, Eq. (1.14)] Theorem 1.4 is false. For the k=1 term, the left side of (1.14) equals 4q/((1-q^2)(1-q^4)) = 4q + O(q^2), while the right side expands as 1/2[3(1+O(q^2)) - (1-2q+O(q^2))] = 1 + q + O(q^2). The q-coefficient is therefore 4 on the left and 1 on the right. This contradicts the identity and undermines every subsequent use of (1.14).
[Theorem 1.3, Eq. (1.12)] Theorem 1.3 is false already at the constant term. The left side at (i,j,k)=(0,0,0) equals 1, whereas the right side 3(q^2;q^2)_∞^3/(q;q)_∞^3 has constant term 3. This single coefficient contradiction invalidates (1.12).
[Section 3, proof of Theorem 1.3] The proof of Theorem 1.3 uses the four identities (1.14)-(1.17) via (3.6), (3.7), (3.9), and (3.10). Since (1.14) is false and (1.12) is false, the derivation cannot establish the claimed identities. The modularity of the primal Nahm sums in Theorem 1.5, which rests on (1.12)-(1.13), is therefore unsupported. The nonmodularity half of Theorem 1.5 depends on the author's earlier result [21] for (1.9)-(1.10), but the new modularity half fails.
[Theorem 1.5, Eqs. (3.16)-(3.17)] The assertion that the expressions in (3.16)-(3.17) cannot be modular for any C' is not proved. Each is of the form q^{2C'+α} times a sum of two eta-quotients of different weights; the text does not rule out the possibility that a suitable shift of C' could make the combination a modular form with a multiplier system. This would require a short modular-transformation argument, which is absent.

minor comments (3)

[Theorem 1.5, Eqs. (3.16)-(3.17)] In both (3.16) and (3.17) the same subscript B⋆_1 is used; the second equation should refer to B⋆_2.
[Section 3, Eq. (3.5)-(3.7)] The factorization of infinite products in (3.5)-(3.7) introduces negative powers of q for individual terms; the negative powers cancel in the combination, but the text should state this to avoid confusion.
[Abstract] There is a typographical error in the abstract: 'V agier' should be 'Zagier'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the counterexample derivation is independent of its inputs, though part of the nonmodularity relies on prior work by the same research group.

full rationale

The central derivation is not circular. The modularity of the primal Nahm sums in Theorem 1.5 is proved through the eta-quotient identities (1.12)-(1.13), which are derived from the single-sum identities (1.14)-(1.17) using Bailey pairs constructed in Lemma 2.1 from external identities of Andrews and standard Bailey's lemma. No fitted parameter is later renamed a prediction, and the transition from generalized sums to ordinary Nahm sums via (3.11)-(3.14) is a rewriting, not an equivalence imposed by the target claim. The nonmodularity of the dual sums is imported from Wang-Wang [21, Theorem 1.4], which is prior work by the same group; however, that theorem supplies explicit, parameter-free q-series identities (1.9)-(1.10) whose assumptions do not include the target counterexample, so it functions as independent evidence rather than as a circular premise. A reviewer's criticism that the identities (1.14)-(1.17) contain algebraic errors is a correctness concern about the proof, not a circularity of the kind where the conclusion is equivalent to an input. Thus the derivation chain is self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on standard q-series machinery and on two prior results from the author's own research. No free parameters are fitted, and no new entities are introduced. The main fragility is the erroneous single-sum identities in Theorem 1.4.

assumptions (3)

standard math Bailey pair theory and transformation formulas (2.6), (2.7), (2.22) from McLaughlin-Sills-Zimmer are valid.
Used in the proof of Theorem 1.4 to derive single-sum identities from Bailey pairs.
domain assumption Prior result [21, Theorem 1.4] that the sums (1.9) and (1.10) are not modular is correct.
The proof of non-modularity of the dual sums in Theorem 1.5 directly depends on this cited result from the author's own previous work.
standard math Identity (3.11) from [20, Lemma 2.2] is correct.
Used to transform rank three Nahm sums into rank four Nahm sums in the proof of Theorem 1.5.

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Cite this review

Pith. "Pith review of Counterexamples to Zagier's Duality Conjecture on Nahm Sums." pith.science (2026). https://pith.science/paper/LTH4IZF5

@misc{pith2026241109701,
  author       = {Pith},
  title        = {Pith review of: Counterexamples to Zagier's Duality Conjecture on Nahm Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LTH4IZF5}},
  note         = {Machine review of arXiv:2411.09701}
}

abstract

Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is modular. Around 2007, Zagier conjectured that if the rank $r$ Nahm sum for $(A,B,C)$ is modular, then so is the dual Nahm sum associated with $(A^{-1},A^{-1}B,B^\mathrm{T} A^{-1}B/2-{r}/{24}-C)$. We construct some explicit rank four Nahm sums which are modular while their duals are not modular. This provides counterexamples to Zagier's duality conjecture.

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Forward citations

Cited by 2 Pith papers

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hep-th 2026-08 conditional novelty 8.0 of 10

Zagier duality between the (E8,T1) and (T1,E8) Nahm systems is realized as 3d N=4 rank-zero mirror symmetry of two U(1)^8 Chern-Simons matter theories, with the duality interface generating the level-one E8 character.
Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation
math.NT 2024-12 conditional novelty 6.0 of 10

The paper proves modularity of new rank-three Nahm sums from a lift-dual construction and gives two new rank-three counterexamples to Zagier's duality conjecture.

Reference graph

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24 extracted references · 16 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Proofs of Mizuno's Conjectures on Rank Three Nahm Sums of Index $(1,2,2)$

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F. Calegari, S. Garoufalidis and D. Zagier, Bloch groups, algebraic K-theory, units, and Nahm’s conjecture, Ann. Sci. ´Ec. Norm. Sup´ er. (4) 56 (2023), no. 2, 383–426

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Calinescu, A

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Cao and L

Z. Cao and L. Wang, Multi-sum Rogers–Ramanujan type identities , J. Math. Anal. Appl. 522 (2023), Art.126960

work page 2023