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Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation

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Pith's one-line read The paper establishes that the lift-dual operation applied to three of the known rank-two Nahm sums yields new rank-three modular Nahm sums, with explicit Rogers-Ramanujan type identities, and exhibits two new rank-three counterexamples…

desk verdict A solid, workmanlike extension of the Nahm sums program that produces new modular rank-three sums and two counterexamples to Zagier's duality conjecture; the main caveat is a heavy reliance on one external identity from the authors' own earlier paper. read the letter →

arxiv 2412.15767 v1 pith:3E66MBUK submitted 2024-12-20 math.NT math.CO

Zhineng Cao , Liuquan Wang This is my paper · ORCID
classification math.NTmath.CO MSC 11P8433D1533D6011F03
keywords NahmsumsmodulartriplesRogers-RamanujantypeidentitiesBaileypairslift-dualoperationdualityconjectureq-seriesforms
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

Nahm sums are q-series built from a symmetric rational matrix and two rational vectors, indexed by triples of nonnegative integers; a sum is modular when a suitable q-power of it is a modular form. The paper tries to produce new modular rank-three Nahm sums by a lift-dual operation: lift a known rank-two sum to rank three with an embedding that leaves the sum unchanged, then apply the duality map $(A,B,C)\mapsto(A^{-1},A^{-1}B,\tfrac12 B^TA^{-1}B-\tfrac{r}{24}-C)$. For three of the known rank-two examples the lifted matrix is positive definite, and the paper proves that the dual sums are modular by writing them as finite combinations of the q-product functions $J_m$ and $J_{a,m}$. A byproduct is two new rank-three counterexamples to the duality conjecture, cases where a Nahm sum is not modular for any $C$ while its dual is modular. If the identities are correct, the known list of modular triples grows systematically and the boundary of the duality conjecture is drawn more sharply.

What carries the argument

The carrying mechanism is the two-step operation: the lifting operator sends a rank-two datum $(A,B,C)$ to a rank-three datum $(\tilde A,\tilde B,C)$ with $\tilde A$ and $\tilde B$ as displayed in (1.7), preserving the value of the Nahm sum with $C=0$, and the dual operator sends $(A,B,C)$ to $(A^{-1},A^{-1}B,\frac12 B^TA^{-1}B-\frac r{24}-C)$. The proof machinery is a set of q-series techniques: constant-term extraction and contour integration to reduce the triple sums, Bailey pairs and their change-of-base transformations to evaluate the reduced sums, and a catalogue of single-sum Rogers-Ramanujan type identities. A crucial auxiliary input is the 3-dissection of the Example 10 identities (5.1)-(5.2), which supplies the dissection of a double sum used in the proof of Theorem 1.1.

What would settle it

Expand both sides of identities (1.16), (1.17), (3.8)-(3.10), and (4.4)-(4.6) as power series in $q$ and compare coefficients through $q^{50}$; any mismatch at a single order would falsify the corresponding identity and hence the modularity claim for that Nahm sum. For the claimed counterexamples, the obstruction is visible directly from Theorem 6.2: the two summands have modular weights 0 and 1, so one can check by a modular-symbol computation that no shift $q^C$ kills the weight-1 part.

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Extended reading notes

Core claim

The central claim is that the dual Nahm sums obtained from the lift-dual operation applied to Zagier's rank-two Examples 1, 9, and 11 are modular. The proof is the explicit Rogers-Ramanujan type identities (3.8)-(3.10), (4.4)-(4.6), and (1.16)-(1.17), which express the sums as finite linear combinations of products of $J_m=(q^m;q^m)_\infty$ and $J_{a,m}=(q^a,q^{m-a},q^m;q^m)_\infty$; since these are eta-type products, modularity follows. The paper further claims that the same search produces two new rank-three counterexamples to the duality conjecture: for the matrix (6.1) with vectors $(1/2,1/2,0)^T$ and $(1,1,1)^T$, the Nahm sums are not modular for any $C$, while their dual sums with matrix (6.31) are modular (Theorem 6.3); Theorem 6.2 exposes the obstruction by writing the nonmodular sums as sums of a weight-zero and a weight-one modular form.

Load-bearing premise

The proof of Theorem 1.1 assumes the two identities (5.1) and (5.2) for Zagier's Example 10, which are taken from earlier work and not reproved here; if either identity is wrong, the product formulas (1.16)-(1.17) and the modularity of those two dual sums collapse.

Editorial extensions

If this is right

  • The dual Nahm sums of the lifted Examples 1, 9, and 11 are modular, with explicit product formulas (3.8)-(3.10), (4.4)-(4.6), and (1.16)-(1.17); modularity follows because each side is a finite combination of the functions $J_m$ and $J_{a,m}$.
  • Because the lifting identity preserves the sum, every modular rank-two triple whose lifted matrix is positive definite yields a modular rank-three triple; for the three treated examples the duals give genuinely new rank-three modular triples.
  • Two pairs (matrix, vector) found in the search are counterexamples to the duality conjecture: the lifted Nahm sums for $B=(1/2,1/2,0)^T$ and $B=(1,1,1)^T$ are not modular for any $C$, while their duals are modular.
  • Theorem 6.2 shows the failure is structural: the nonmodular sums split into a modular form of weight 0 plus one of weight 1, so no choice of q-power can make them modular.

Reading between the lines

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

  • The lift-dual recipe is not limited to the three examples; any modular rank-two triple with positive definite lift is a candidate, and the paper's methods should extend, although the required dissections may not always exist.
  • The weight-0-plus-weight-1 obstruction suggests a sharper form of the duality conjecture: the dual of a modular Nahm sum is modular exactly when all summands in its natural decomposition have the same weight.
  • The new counterexamples lower the rank at which the duality conjecture fails from four to three, placing the failure at the lowest rank currently known.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a 'lift-dual' operation for Nahm sums: starting from a rank-two Nahm sum with matrix A and vector B, it lifts to a rank-three sum via the matrix and vector in (1.7), then takes the dual via the standard duality operator D in (1.6). Applying this to Zagier's rank-two Examples 1, 9, and 11, the authors state explicit Rogers–Ramanujan type identities (Theorems 3.1, 4.1, 1.1) that express the resulting rank-three Nahm sums as products of generalized Dedekind eta functions, thereby proving modularity. Section 6 contains a specialization a=3/2 with additional modular triples, and two new counterexamples to Zagier's duality conjecture are claimed. The proofs use constant-term extraction, contour integrals, and Bailey-pair techniques.

Significance. If the results are correct, the paper provides several new explicit modular rank-three Nahm sums, a useful contribution to the ongoing classification in Nahm's problem. The proofs are detailed and rely on standard q-series methods, and the identities are explicit enough to be checked numerically. The claimed counterexamples to Zagier's duality conjecture are potentially significant for the broader understanding of that conjecture. However, the paper's motivating identity (1.8), which underlies the lift-dual construction, appears to be false as stated; this does not necessarily invalidate the modularity theorems (which are proved independently), but it undermines the paper's central narrative and requires substantial revision.

major comments (4)
  1. [§1, Eq. (1.8)] The identity f_{A,B,0}(q) = f_{\tilde A,\tilde B,0}(q) is stated for all A,B, but it is false for the lift defined in (1.7) even in the basic case a=2 of Example 1. Taking A = [[2,-1],[-1,2]], B=(0,0), the left side has q^2-coefficient 4 (from the triples (1,0), (0,1), (1,1)), while the right side, with \tilde A = [[2,0,1],[0,2,1],[1,1,2]], has q^2-coefficient 3 (from (1,0,0), (0,1,0), (0,0,1)). Thus the claim that every rank-two modular triple lifts to a rank-three modular triple 'for free' is not correct as stated, and the presentation of the sums in Sections 3–5 as 'lift-dual' of Zagier's examples is not justified for all parameters. The authors must correct the identity, impose the conditions under which it actually holds, or reframe the construction without claiming a lift relation.
  2. [§5, Lemma 5.1 and Theorem 1.1] The proof of Theorem 1.1 relies entirely on the Vlasenko–Zwegers identities (5.1)–(5.2), which are neither proved nor derived in this paper. These identities are the sole input to Lemma 5.1, and a single index or coefficient error in them would propagate directly into the modular product formulas (1.16)–(1.17). Because this is the only route to the modularity claim for the Example-11 dual sums, the authors should state precisely where in [5] these identities are proved, and ideally include an independent verification (for example, a numerical check of (1.16)–(1.17) to high order) so that readers can rule out transcription errors.
  3. [§6.2 and §6.3, Theorem 6.2] The nonmodularity conclusions in Section 6.3 depend on the assertion that the expressions in (6.9) and (6.10) are sums of modular forms of weights 0 and 1, but no weight computation or modular transformation law is provided. The authors should show explicitly that the first term in each expression transforms as a weight-0 modular form and the second as a weight-1 modular form (after multiplying by the appropriate power of q), and that the weight-1 piece does not vanish identically. Without this, the counterexamples to Zagier's duality conjecture are not fully substantiated.
  4. [§3, Theorem 3.1] The parameters m and ν in Theorem 3.1 are rational in the intended application (for instance, m = 1/(4(a-1))), but the right-hand sides involve J_{a,m} with indices such as 4(4m+1) and 4(4m+ν+1) that are not necessarily integers. Since the modularity interpretation of J_{a,m} in (1.10) generally requires integer indices, the authors should specify the domain of m and ν and, where necessary, the level of the modular group for which the products are modular forms. This is a technical point, but it is load-bearing for the modularity claim.
minor comments (5)
  1. [§1, first paragraph] The word 'sated' should be 'stated' in the sentence 'Nahm’s conjecture, sated explicitly by Zagier'.
  2. [§6.3, heading] The word 'counterexmaples' is a typo for 'counterexamples'.
  3. [§2, Eq. (2.2)] The notation in (2.2) introduces a new function but does not clearly distinguish it from J_{a,m} defined in (1.10); please use a different symbol (for example, \overline{J}_{a,m} or \mathcal{J}_{a,m}) to avoid ambiguity.
  4. [§4, proof of Theorem 4.1] The application of the contour integral formula (2.26) does not state the conditions on the contour (poles inside/outside) or the convergence of the interchanges; a remark that all manipulations are valid as formal power series or for |q| sufficiently small would make the proof more rigorous.
  5. [§5, Eq. (5.13)–(5.16)] The substitution q replaced by q^{4/3} is essential but may confuse readers because it introduces fractional powers; a brief note that the identities are formal and that the substitution is performed on the q-series coefficients would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new modular Nahm-sum identities are derived from external q-series results, not from the conclusions they prove.

full rationale

The derivation chain is not circular. The new modularity results (3.8)–(3.10), (4.4)–(4.6), and (1.16)–(1.17) are established by reducing rank-three Nahm sums to known single-sum and double-sum identities (Slater, Gasper–Rahman, Ramanujan's Lost Notebook, Vlasenko–Zwegers) via constant-term extraction, contour integrals, and Bailey pairs. The lift identity (1.8), attributed to Zwegers and Lee, is an external fact that does not presume the modularity of the dual sums. The only load-bearing input with author overlap is (5.1)–(5.2), proved in the authors' earlier paper [5]; those are prior published identities for Zagier's Example 10, originally conjectured by Vlasenko–Zwegers [15]. They are not equivalent to the target theorems, and the present paper applies them by substituting q^{4/3} and dissecting modulo 3, which is a legitimate derivation step. The Maple search mentioned in Section 6.2 only generates candidate triples; their modularity is then proven independently by explicit identities. Identity (6.29) and the identities of [16] used in Section 6 are parameter-free and externally checkable, and they do not assume the non-modularity or modularity conclusions being drawn. No fitted parameter is renamed as a prediction, no conclusion is assumed in its own proof, and no uniqueness or ansatz is smuggled in through self-citation. Accordingly, the paper's central derivations are self-contained against external q-series results, and the self-citation to [5] is real supporting evidence rather than circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted: the parameters m, nu, c, a are variables in identities proven for all admissible values. The Maple search in Section 6.2 generates candidate B-vectors, but the modular identities are then proven unconditionally. All other substantive inputs are prior published q-series theorems.

assumptions (8)
  • standard math Jacobi triple product identity (2.1).
    Used throughout Sections 3-5 to evaluate constant terms and dissections into infinite products.
  • standard math q-binomial theorem and Euler identities (2.4)-(2.5).
    Expand infinite products in the constant term and contour integral computations.
  • standard math q-Gauss summation (2.6), 1-phi-1 sum (2.7), and q-analogues (2.8)-(2.9).
    Used to sum the basic hypergeometric series appearing in the proofs.
  • standard math Bailey lemma and its variants (Lemma 2.1, (2.31)-(2.38), Lemmas 2.2-2.3).
    The core mechanism for transforming single sums in the proofs of Theorems 3.1, 6.2, and 6.3.
  • standard math Contour integral evaluation formula (2.26) from Gasper-Rahman.
    Used to evaluate the constant terms that appear in the proof of Theorem 4.1.
  • domain assumption Lift identity (1.8): f_A,B,0(q) = f_tilde A,tilde B,0(q) for the lifting operator L.
    Cited to [11] and [5]; transfers rank-two modularity to rank three and is load-bearing for all new triples.
  • domain assumption Vlasenko-Zwegers identities (5.1)-(5.2) for Zagier's Example 10, proved in [5].
    Used in Lemma 5.1 to prove Theorem 1.1; not reproved in the present paper.
  • standard math Slater's list identities (2.10)-(2.19) and Andrews-Berndt entries (2.20)-(2.23).
    Reduces the q-hypergeometric sums to eta products in the proofs of Theorems 4.1 and 6.3.

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Pith. "Pith review of Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation." pith.science (2026). https://pith.science/paper/3E66MBUK

@misc{pith2026241215767,
  author       = {Pith},
  title        = {Pith review of: Some New Modular Rank Three Nahm Sums from a Lift-Dual Operation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3E66MBUK}},
  note         = {Machine review of arXiv:2412.15767}
}
read the original abstract

Around 2007, Zagier discovered some rank two and rank three Nahm sums, and their modularity have now all been confirmed. Zagier also observed that the dual of a modular Nahm sum is likely to be modular. This duality observation motivates us to discover some new modular rank three Nahm sums by a lift-dual operation. We first lift Zagier's rank two Nahm sums to rank three and then calculate their dual, and we show that these dual Nahm sums are indeed modular. We achieve this by establishing the corresponding Rogers--Ramanujan type identities, which express these Nahm sums as modular infinite products.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Proofs of five conjectural identities on modular rank four Nahm sums

    math.NT 2026-06 unverdicted novelty 7.0 of 10

    Four previously conjectural modular rank-four Nahm-sum identities are proved by q-series reductions, and two further conjectures are shown to imply each other.

  2. Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities

    math.NT 2026-07 conditional novelty 6.0 of 10

    Duals of Zagier's rank-three modular Nahm sums are proved modular for Examples 7, 8, 10, and 11; Example 9 is conditional on Conjecture 3.4, and Example 12 remains conjectural.

  3. Modular Nahm Sums for the Inverse Cartan Matrix of Type $D_r$

    math.NT 2026-07 accept novelty 6.0 of 10

    Bailey-pair methods yield floor((r+4)/2) modular Nahm sums for C(D_r)^{-1}, confirming Sun–Wang’s zero-vector identity and partial companions.

Reference graph

Works this paper leans on

21 extracted references · 19 canonical work pages · cited by 3 Pith papers

  1. [5]

    Z. Cao, H. Rosengren and L. Wang, On some double Nahm sums of Z agier, J. Combin. Theory Ser. A 202 (2024), Paper No. 105819

    work page 2024

  2. [1]

    Andrews, The Theory of Partitions, Addison–Wesley, 1976; Reissued Cambridge, 1998

    G.E. Andrews, The Theory of Partitions, Addison–Wesley, 1976; Reissued Cambridge, 1998

    work page 1976

  3. [2]

    Andrews and B.C

    G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Par t II, Springer 2009

    work page 2009

  4. [3]

    Bressoud, M.E.H

    D.M. Bressoud, M.E.H. Ismail and D. Stanton, Change of base in Ba iley pairs, Ramanujan J. 4 (2000), 435–453

    work page 2000

  5. [4]

    Calegari, S

    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

    work page 2023

  6. [6]

    Cherednik and B

    I. Cherednik and B. Feigin, Rogers–Ramanujan type identities an d Nil-DAHA, Adv. Math. 248 (2013), 1050–1088

    work page 2013

  7. [7]

    D. Gang, H. Kim, B. Park and S. Stubbs, Three dimensional topolo gical field theories and Nahm sum formulas, arXiv: 2411.06081

    arXiv

  8. [8]

    Gasper and M

    G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd Edit ion, Encyclopedia of Math- ematics and Its Applications, Vol. 96, Cambridge University Press, 2 004

    work page

Show all 21 references
  1. [9]

    Kac and M

    V. Kac and M. Wakimoto, Modular invariant representations of infi nite dimensional Lie alge- bras and superalgebras, Proc. Nat. Acad. Sci. 85 (1988), 4956– 4960

    1988

  2. [10]

    Mc Laughlin, Topics and methods in q-series, Monographs in Number Theory, 8, World Scientific Publishing Co

    J. Mc Laughlin, Topics and methods in q-series, Monographs in Number Theory, 8, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018

    2018

  3. [11]

    Lee, Algebraic structures in modular q-hypergeometric series, PhD Thesis, University of California, Berkeley, 2012

    C.-H. Lee, Algebraic structures in modular q-hypergeometric series, PhD Thesis, University of California, Berkeley, 2012. 32 ZHINENG CAO AND LIUQUAN W ANG

    2012

  4. [12]

    Lovejoy, A Bailey lattice, Proc

    J. Lovejoy, A Bailey lattice, Proc. Am. Math. Soc. 132 (2004), 1507–1516

    2004

  5. [13]

    Rogers, Second memoir on the expansion of certain infinite p roducts, Proc

    L.J. Rogers, Second memoir on the expansion of certain infinite p roducts, Proc. London Math. Soc. 25 (1894), 318–343

  6. [14]

    Slater, Further identities of the Rogers–Ramanujan type , Proc

    L.J. Slater, Further identities of the Rogers–Ramanujan type , Proc. Lond. Math. Soc. (2) 54 (1) (1952), 147–167

    1952

  7. [15]

    Vlasenko and S

    M. Vlasenko and S. Zwegers, Nahm’s conjecture: asymptotic c omputations and counterexam- ples, Commu. Number Theory Phys. 5(3) (2011), 617–642

    2011

  8. [16]

    Xia and O.X.M

    E.X.W. Xia and O.X.M. Yao, Analogues of Ramanujan’s partition ident ities, Ramanujan J. 31 (2013), 373–396

    2013

  9. [17]

    Wang, Identities on Zagier’s rank two examples for Nahm’s pro blem, Res

    L. Wang, Identities on Zagier’s rank two examples for Nahm’s pro blem, Res. Math. Sci. (2024) 11:49

    2024

  10. [18]

    Wang, Explicit forms and proofs of Zagier’s rank three examp les for Nahm’s problem, Adv

    L. Wang, Explicit forms and proofs of Zagier’s rank three examp les for Nahm’s problem, Adv. Math. 450 (2024), 109743

    2024

  11. [19]

    Wang, Counterexamples to Zagier’s duality conjecture on Na hm sums, arXiv:2411.09701v3

    L. Wang, Counterexamples to Zagier’s duality conjecture on Na hm sums, arXiv:2411.09701v3

    arXiv

  12. [20]

    Zagier, The dilogarithm function, in Frontiers in Number Theor y, Physics and Geometry, II, Springer, 2007, 3–65

    D. Zagier, The dilogarithm function, in Frontiers in Number Theor y, Physics and Geometry, II, Springer, 2007, 3–65

    2007

  13. [21]

    Zwegers, presentation

    S. Zwegers, presentation. In: Workshop on Mock Modular For ms in Combinatorics and Arith- metic Geometry, American Institute of Mathematics, Palo Alto, Calif ornia Mar. 8–12. 2010. (Z. Cao) School of Mathematics and Statistics, Wuhan University, Wu han 430072, Hubei, People’s ...

    2010

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