Triple Hodge integrals and constant Poisson brackets for rank-one Dubrovin-Zhang hierarchies

Dimitrios Makris; Guido Carlet; Sergey Shadrin; Xavier Blot

REVIEW 1 major objections 4 minor 53 references

Triple Hodge integrals and constant Poisson brackets for rank-one Dubrovin-Zhang hierarchies

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The Poisson bracket of a rank-one Dubrovin-Zhang hierarchy is constant exactly for the Calabi-Yau triple Hodge class $\Lambda(r_1)\Lambda(r_2)\Lambda(r_3)$, with an explicit bracket formula.

desk verdict The converse of the DLYZ conjecture is proven by explicit DR/DZ computations; the only real question is whether the imported FVH transfer in Prop. 3.2 is solid enough to carry the base point. read the letter →

arxiv 2608.00462 v1 pith:NMC2ETIM submitted 2026-08-01 math.AG hep-thmath-phmath.MPnlin.SI

Xavier Blot , Guido Carlet , Dimitrios Makris , Sergey Shadrin This is my paper · ORCID

classification math.AGhep-thmath-phmath.MPnlin.SI MSC 14H1014N3537K10

keywords Dubrovin-ZhanghierarchiescohomologicalfieldtheoriesHodgeintegralsPoissonbracketstripleclassesCalabi-YauconditionfractionalVolterrahierarchymoduliofcurves

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

This paper proves a conjecture about when the Dubrovin-Zhang integrable hierarchy attached to a rank-one cohomological field theory has a Poisson bracket with constant coefficients. The answer is that this happens exactly for the triple Hodge classes $\Lambda(r_1)\Lambda(r_2)\Lambda(r_3)$ whose parameters satisfy the Calabi-Yau condition $r_1r_2+r_1r_3+r_2r_3=0$, and in that case the bracket has an explicit closed form. This matters because constant brackets are structurally special: they are the cases tied to the fractional Volterra hierarchy and to KP integrability, and the explicit formula makes the whole Poisson structure computable. The proof writes the bracket through a Miura transformation from the double-ramification hierarchy and shows that the first non-constant coefficients force every higher odd Hodge parameter to vanish.

What carries the argument

The load-bearing mechanism is the DR/DZ equivalence. For a rank-one theory the Dubrovin-Zhang hierarchy is obtained from the double-ramification hierarchy, whose Poisson bracket is simply $\partial_x$, by a Miura transformation $L$, so the bracket is $P=L^{-1}\partial_x(L^{-1})^*$ (Eq. (24)). The transformation is expressed through intersection numbers of the CohFT class with tautological classes $B^m_{g,n}$ on moduli spaces of stable rooted trees. The proof then reads off the top-derivative coefficients of $P$ in normal form: for odd $h$, the coefficient of $\epsilon^{2h}w_1^{h-1}\partial_x^{h+2}$; for even $h$, the coefficient of $\epsilon^{2h}w_1^{h-2}w_2\partial_x^{h+1}$. Lemma 5.5 evaluates these as nonzero multiples of Bernoulli numbers, producing a triangular system that forces each $e_{2h-1}$, $h\ge3$, to vanish.

What would settle it

For a rank-one theory with $e_5\neq0$ and all $e_7,e_9,\ldots$ zero, compute the coefficient of $\epsilon^6 w_1^2\partial_x^5$ in its Dubrovin-Zhang bracket; Example 5.2 reduces it to $120e_5\int_{M_{3,3}} ch_5\,\mathrm{Coeff}[a_1a_2a_3^2]B^0_{3,3}$, which Lemma 5.5 says is nonzero. A theory in which that coefficient vanishes while $e_5\neq0$ would refute Theorem 5.1.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 5.1: if the Poisson bracket $P$ of the Dubrovin-Zhang hierarchy associated with a rank-one cohomological field theory is constant in the dependent variables $w_i$, $i\ge1$, then $e_{2h-1}=0$ for all $h\ge3$. Equivalently, the theory is $C_{g,n}=\Lambda(r_1)\Lambda(r_2)\Lambda(r_3)$ with $r_1r_2+r_1r_3+r_2r_3=0$. Combined with Proposition 3.2, which exhibits the explicit constant bracket (4) for exactly this family, this proves Conjecture 3.1 of [DLYZ16] in full. The constancy condition leaves only two free parameters, $e_1$ and $e_3$.

Load-bearing premise

The proof that the triple Hodge Calabi-Yau hierarchy really has the constant bracket (4) is inherited from earlier work through a quasi-Miura transformation built from a tau function, and that earlier work uses infinite linear combinations of flows and a formally weaker system of Virasoro equations; if that identification is not exactly the Dubrovin-Zhang hierarchy, the step $Q(e_1,e_3)=0$ in the proof of Theorem 5.1 has no basis.

Editorial extensions

If this is right

Every rank-one theory outside the family $\Lambda(r_1)\Lambda(r_2)\Lambda(r_3)$ with $r_1r_2+r_1r_3+r_2r_3=0$ has a non-constant Dubrovin-Zhang bracket, with the first obstruction given by a computable Bernoulli-number coefficient.
For the surviving family the bracket is $P=\partial_x/[S(\sqrt{r_1r_2/r_3}\,\epsilon\partial_x)S(\sqrt{r_2r_3/r_1}\,\epsilon\partial_x)S(\sqrt{r_3r_1/r_2}\,\epsilon\partial_x)]$, so the full Poisson structure is explicit.
The limiting cases recover known constant brackets: $\Lambda(-r)$ gives $\partial_x/S(\sqrt r\,\epsilon\partial_x)^2$, and $r\to0$ gives the KdV bracket $\partial_x$.
Constancy of the Poisson bracket is therefore a sharp criterion that singles out the Calabi-Yau triple Hodge class among all rank-one cohomological field theories.

Reading between the lines

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

A testable extension is to ask whether requiring the bracket to be constant only up to a fixed differential order still forces the same vanishing; the $e_5$ obstruction suggests the effect already appears at the first non-vanishing odd Chern character.
The Bernoulli-number coefficients supply a quantitative obstruction spectrum, so one could measure how far a given hierarchy is from constant-bracket form by the size of its first surviving coefficient.
The same B-class intersection machinery may apply to partial cohomological field theories or to constancy of higher Poisson brackets, where the triangular structure would recur.
If the result is read together with the known KP integrability of the Calabi-Yau triple Hodge family, it suggests that within rank-one theories 'constant Poisson bracket' and 'KP-integrable Dubrovin-Zhang hierarchy' coincide.

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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

Short version: this is the converse half of the DLYZ16 conjecture, and it is a real result. The forward direction (triple Hodge Calabi-Yau gives constant bracket) was already in LYZZ21/LYZZ22; what's new is Theorem 5.1, which shows constancy forces e_{2h-1}=0 for h>=3. The proof is a serious computation: write the DZ bracket via the DR/DZ Miura transformation, expand in epsilon, and show the first non-vanishing higher e gives a nonzero coefficient. The use of Proposition 4.8 and the L operator is clean, and Lemma 5.5 reduces the key coefficients to Faber socle numbers with explicit Bernoulli factors. I did not find a gap in the coefficient extraction; the triangular structure works.

The soft spot is Proposition 3.2. The constant bracket for the triple Hodge case is imported from the fractional Volterra hierarchy via a non-invertible change of time variables, and the proof is explicitly a short guide to LYZZ21/LYZZ22. The step Q(e1,e3)=0 in Theorem 5.1 depends on that import being the actual DZ bracket on the nose. If the uniqueness in LYZZ22, Prop. 3.9 does not cover the needed generality, the base point of the induction is not verified. The authors know this and give reasons to trust it, but they do not provide an independent check, such as a direct evaluation of Eq. (36) or a machine-checked computation. This is a provenance issue, not an internal inconsistency. I would want a referee to look carefully at Prop. 3.2 and the cited uniqueness theorem.

Overall, the paper deserves serious refereeing. It resolves a conjecture, the new part is definitely new, and the main computation is explicit and cross-checked on Lambda(-r). The soft spot is real but narrow and clearly flagged; it is a fixable or verifiable gap, not a conceptual one. I would send it to a competent referee rather than desk reject.

Referee Report

1 major / 4 minor

Summary. The paper proves Conjecture 3.1 of [DLYZ16] for rank-one cohomological field theories. After recalling that every rank-one CohFT with a flat unit is written as Cl_{g,n}=exp(Σ(-1)^{i-1}(i-1)!p_i ch_i) and parametrized by e_{2h-1}, the authors use the DR/DZ equivalence to write the Poisson bracket P of the associated Dubrovin-Zhang hierarchy as P=L^{-1}∂x(L^{-1})*, where L is the linearization of the Miura map defined by B-class intersection numbers. The main theorem states that if P is constant in w_i for i≥1, then e_{2h-1}=0 for all h≥3, so Cl is a triple Hodge class Λ(r1)Λ(r2)Λ(r3) with r1r2+r1r3+r2r3=0. The proof isolates for every h≥3 a specific coefficient of P (ϵ^{2h}w1^{h-1}∂x^{h+2} for odd h, and ϵ^{2h}w1^{h-2}w2∂x^{h+1} for even h), shows via Lemma 5.5 that its leading term in e_{2h-1} is nonzero, and uses Proposition 3.2 to subtract all terms depending only on e1,e3. The converse direction, including the explicit formula (4), is imported from the fractional Volterra hierarchy literature and stated as Proposition 3.2.

Significance. If the main theorem and Proposition 3.2 are both fully justified, the result settles a conjecture that has been open for a decade and identifies the triple Hodge Calabi-Yau theories as the unique rank-one source of constant Poisson brackets. The paper also gives an explicit operator formula and connects the computation to Faber-type intersection numbers with explicit Bernoulli-number values. Strengths of the manuscript include the concrete coefficient computations in Lemma 5.5, the cross-check against the Λ(-r) example, and the transparent reduction of the constancy condition to a triangular system. The main concern is that the reverse implication rests on Proposition 3.2, whose proof in this paper is a guide to prior work rather than a complete verification.

major comments (1)

[§3, Proposition 3.2 and §5.3, Eq. (69)] Proposition 3.2 is load-bearing for the proof of Theorem 5.1: Eq. (69) uses it to conclude Q(e1,e3)=0, i.e. that the coefficient of ϵ^{2h}w1^{h-1}∂x^{h+2} vanishes in the triple Hodge Calabi-Yau specialization. The proposition is not proved in this paper; its proof is explicitly 'a short guide' to [LYZZ21, Lemma 3.9] and [LYZZ22, Prop. 3.9], and the transfer from the fractional Volterra hierarchy is via a non-invertible change of time variables and formally weaker Virasoro constraints. A referee cannot verify that the unique solution identified in [LYZZ22, Prop. 3.9] is the Dubrovin-Zhang hierarchy of the triple Hodge CohFT in the sense of Section 4. I request a self-contained proof of Proposition 3.2, or a precise statement of a theorem in the cited papers that applies directly to the DZ hierarchy together with a verification of its hypotheses. Without this, the base point of the triangular system in Theorem 5.1 is not established.

minor comments (4)

[Eq. (4) and abstract] In the arXiv text the formula is typeset as a product of S-factors times ∂x; the intended expression is a quotient with ∂x in the numerator and the three S-factors in the denominator. Please correct the typesetting.
[§3, proof of Proposition 3.2] The rescaling ϵ̃=√(r1r2r3)ϵ is used, but when e3=r1r2r3 vanishes this rescaling degenerates. Remark 3.3 treats the limit, but a sentence stating that the degenerate cases follow by continuity or by the direct computation in Section 4.4 would make the proposition unambiguous.
[§4.2, Lemma 4.4] The formula π_*∏_{i=1}^{n+1}(1-b_iψ_i)^{-1}=Σ_{i=1}^{n+1}b_i∏(1-b_iψ_i)^{-1} is stated without justification; adding a one-line derivation or a reference would improve readability.
[§5.2.2, Eq. (45)] The simplification of the two displayed integrals to -B_{2h}h!(h-2)/(3·2^{h-1}·(2h)!) is not shown; including the intermediate algebra would make the even-h nonvanishing easier to verify.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the converse theorem uses the already-proved constant triple-Hodge bracket as an external input, and the new obstruction equations are genuine computations.

full rationale

The paper's central claim is Theorem 5.1: if the Dubrovin-Zhang Poisson bracket P is constant, then e_{2h-1}=0 for all h>=3. The proof does not assume this conclusion. It computes explicit coefficients of P via the Miura formula (24), itself cited from the authors' prior DR/DZ correspondence work [BLS26; BS24; BSS25a]. That cited work is a general, parameter-free statement about rank-one CohFTs and does not contain the target conjecture, so it is independent support rather than circular input. In the key step around Eq. (69), the coefficient of epsilon^{2h} w_1^{h-1} partial_x^{h+2} is written as (2h-1)! e_{2h-1} times a nonvanishing intersection number plus an unknown polynomial Q(e1,e3) involving only e1 and e3. The paper then invokes Proposition 3.2 to conclude Q(e1,e3)=0, because for the triple Hodge specialization e_{2k-1}=0 (k>=3) the bracket is already known to be constant. This is a legitimate subtraction of a previously established special case, not a fit or a renaming of the conclusion: the constancy of the bracket is not being used to prove the constancy of the bracket. Proposition 3.2 itself is imported from [LYZZ21; LYZZ22] through a non-invertible change of time variables, and the proof explicitly concedes that those papers use infinite linear combinations of flows and a formally weaker Virasoro system, with uniqueness delegated to [LYZZ22, Prop. 3.9]. That is a provenance or correctness risk, not a circular step: if the identification failed, the proof would lose its base point, but the argument would still not be deriving its conclusion from itself. Similarly, Faber's socle formula (Lemma 5.11) is cited to [Fab99; GP98; Giv01; BSS25b], with BSS25b by overlapping authors, but it is an external known result not equivalent to the conjecture. No parameter is fitted to force e_{2h-1}=0; the coefficients are computed explicitly and the nonvanishing of the relevant intersection numbers is proved in Lemma 5.5. Therefore the derivation is self-contained in its new parts and the only reliance on prior work is as external input, so the circularity score is minimal.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All inputs are external theorems or structural assumptions of the DR/DZ framework; no parameter is fitted to data and no new entity is postulated. The most fragile input is the identification of the fractional Volterra hierarchy bracket with the Dubrovin-Zhang bracket in Proposition 3.2, which is inherited from [LYZZ21, LYZZ22] and involves a non-invertible change of time variables.

assumptions (5)

standard math Teleman classification: every rank-one cohomological field theory with a flat unit of norm one has the form exp(sum_{i>=1} (-1)^{i-1}(i-1)! p_i ch_i), as stated in Eq. (1), Section 2.
This is the external classification theorem that justifies the parametrization of all rank-one CohFTs used throughout the paper.
standard math Mumford's relations on the moduli space of curves, including Lambda(a)Lambda(-a)=1 and vanishing of even Chern characters ch_{2i} for i>=1, Section 2.
These relations support the rewriting of the CohFT class in terms of Lambda factors and the specialization of Newton identities.
standard math The DR/DZ equivalence and the Miura transformation formula u = w - ... with B^0 classes, Proposition 4.2, and the bracket relation L P L* = d_x, Proposition 4.8, taken from [BS24; BLS26].
This is the load-bearing bridge between the DR hierarchy, whose bracket is simply d_x, and the Dubrovin-Zhang hierarchy. It is cited from recent literature, partly by overlapping authors, and is not reproved in this paper.
standard math Faber's socle intersection number formula, Lemma 5.11, for integrals of ch_{2g-1} with psi classes on moduli spaces of curves.
This formula is used to evaluate the nontrivial coefficients in Lemma 5.5 and to prove their non-vanishing.
standard math The already-established forward direction of the conjecture: for the triple Hodge class with the Calabi-Yau condition, the Dubrovin-Zhang bracket is constant and given by Eq. (4), as proved in [LYZZ21] and summarized in Proposition 3.2.
The proof of Theorem 5.1 uses this known constant bracket to subtract the contributions of e1 and e3, reducing the coefficient equations to pure constraints on the higher e_{2h-1}. This is an external benchmark, not a restatement of the converse being proved.

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Pith. "Pith review of Triple Hodge integrals and constant Poisson brackets for rank-one Dubrovin-Zhang hierarchies." pith.science (2026). https://pith.science/paper/NMC2ETIM

@misc{pith2026260800462,
  author       = {Pith},
  title        = {Pith review of: Triple Hodge integrals and constant Poisson brackets for rank-one Dubrovin-Zhang hierarchies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMC2ETIM}},
  note         = {Machine review of arXiv:2608.00462}
}

read the original abstract

We prove a conjecture of Dubrovin, Liu, Yang, and Zhang that states that the Poisson bracket of the Dubrovin-Zhang hierarchy of a rank-one cohomological field theory is constant if and only if it is given by the triple Hodge class with the parameters satisfying the Calabi-Yau condition.

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Works this paper leans on

53 extracted references · 27 canonical work pages

[1]

Arnold, V. I. and Gusein-Zade, S. M. and Varchenko, A. N. , TITLE =. 1985 , PAGES =. doi:10.1007/978-1-4612-5154-5 , URL =

doi:10.1007/978-1-4612-5154-5 1985
[2]

2024 , eprint=

On the strong DR/DZ equivalence conjecture , author=. 2024 , eprint=

2024
[3]

2026 , eprint=

Beyond descendants: integrable observables for cohomological field theories , author=. 2026 , eprint=

work page 2026
[4]

2024 , eprint=

Rooted trees with level structures, -classes and double ramification cycles , author=. 2024 , eprint=

2024
[5]

Stable tree expressions with omega-classes and double ramification cycles , JOURNAL =

Blot, Xavier and Lewa\'. Stable tree expressions with omega-classes and double ramification cycles , JOURNAL =. 2025 , PAGES =. doi:10.1016/j.geomphys.2024.105391 , URL =

arXiv 2025
[6]

Blot, Xavier and Sauvaget, Adrien and Shadrin, Sergey , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/blms.13215 , URL =

doi:10.1112/blms.13215 2025
[7]

, TITLE =

Buryak, A. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00220-014-2235-2 , URL =

doi:10.1007/s00220-014-2235-2 2015
[8]

and Dubrovin, B

Buryak, A. and Dubrovin, B. and Gu. Tau-structure for the double ramification hierarchies , fjournal =. Comm. Math. Phys. , issn =. 2018 , language =. doi:10.1007/s00220-018-3235-4 , keywords =

doi:10.1007/s00220-018-3235-4 2018

Show all 53 references

[9]

and Dubrovin, B

Buryak, A. and Dubrovin, B. and Gu. Integrable systems of double ramification type , fjournal =. Int. Math. Res. Not. , issn =. 2020 , language =. doi:10.1093/imrn/rnz029 , keywords =

2020 doi
[10]

and Posthuma, H

Buryak, A. and Posthuma, H. and Shadrin, S. , title =. J. Differ. Geom. , issn =. 2012 , language =. doi:10.4310/jdg/1352211225 , keywords =

2012
[11]

and Posthuma, H

Buryak, A. and Posthuma, H. and Shadrin, S. , TITLE =. J. Geom. Phys. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.geomphys.2012.03.006 , URL =

2012 doi
[12]

Buryak, Alexandr and Rossi, Paolo and Shadrin, Sergey , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s11005-020-01341-6 , URL =

2021 doi
[13]

and Rossi, P

Buryak, A. and Rossi, P. , title =. Adv. Math. , issn =. 2021 , language =. doi:10.1016/j.aim.2021.107794 , keywords =

2021
[14]

Buryak, Alexandr and Rossi, Paolo , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00220-015-2535-1 , URL =

2016 doi
[15]

Buryak, Alexandr and Rossi, Paolo , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00220-025-05351-0 , URL =

2025 doi
[16]

and Shadrin, S

Buryak, A. and Shadrin, S. , title =. 2024 , language =. doi:10.46298/epiga.2024.10382 , keywords =

2024
[17]

Differential Geom

Carlet, Guido and Casati, Matteo , TITLE =. Differential Geom. Appl. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.difgeo.2025.102276 , URL =

2025
[18]

Nonlinearity , FJOURNAL =

Della Vedova, Alberto and Lorenzoni, Paolo and Savoldi, Andrea , TITLE =. Nonlinearity , FJOURNAL =. 2016 , NUMBER =. doi:10.1088/0951-7715/29/9/2715 , URL =

2016 doi
[19]

, TITLE =

Dubrovin, B. , TITLE =. Integrable systems and quantum groups (. 1996 , ISBN =. doi:10.1007/BFb0094793 , URL =

1996 doi
[20]

and Zhang, Y

Dubrovin, B. and Zhang, Y. , year=. Normal forms of hierarchies of integrable. math/0108160 , archivePrefix=

arXiv
[21]

Dubrovin, B. A. and Novikov, S. P. , TITLE =. Dokl. Akad. Nauk SSSR , FJOURNAL =. 1984 , NUMBER =

1984
[22]

2004 , PAGES =

Kock, Joachim , TITLE =. 2004 , PAGES =

2004
[23]

and Manin, Yu

Kontsevich, M. and Manin, Yu. , TITLE =. Invent. Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/s002220050055 , URL =

1996 doi
[24]

and Manin, Yu

Kontsevich, M. and Manin, Yu. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1994 , NUMBER =

1994
[25]

and Ruan, Y

Liu, S.-Q. and Ruan, Y. and Zhang, Y. , TITLE =. Invent. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00222-014-0559-3 , URL =

2015 doi
[26]

and Wang, Z

Liu, S.-Q. and Wang, Z. and Zhang, Y. , TITLE =. Adv. Math. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.aim.2024.110046 , URL =

2025
[27]

and Wang, Z

Lorenzoni, P. and Wang, Z. , year=. Generalised (bi-). 2604.12819 , archivePrefix=

arXiv
[28]

Teleman, Constantin , TITLE =. Invent. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1007/s00222-011-0352-5 , URL =

2012 doi
[29]

SIGMA Symmetry Integrability Geom

Liu, Si-Qi and Wang, Zhe and Zhang, Youjin , TITLE =. SIGMA Symmetry Integrability Geom. Methods Appl. , FJOURNAL =. 2022 , PAGES =. doi:10.3842/SIGMA.2022.037 , URL =

2022 doi
[30]

Liu, Si-Qi and Yang, Di and Zhang, Youjin and Zhou, Chunhui , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1515/crelle-2020-0051 , URL =

2021 doi
[31]

Dubrovin, Boris and Liu, Si-Qi and Yang, Di and Zhang, Youjin , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00220-020-03846-6 , URL =

2020 doi
[32]

Liu, Si-Qi and Zhang, Youjin and Zhou, Chunhui , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s11005-017-1006-3 , URL =

2018 doi
[33]

Liu, Si-Qi and Yang, Di and Zhang, Youjin and Zhou, Chunhui , TITLE =. J. Differential Geom. , FJOURNAL =. 2022 , NUMBER =. doi:10.4310/jdg/1659987894 , URL =

2022
[34]

Dubrovin, Boris and Liu, Si-Qi and Yang, Di and Zhang, Youjin , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.aim.2016.01.018 , URL =

2016 doi
[35]

, TITLE =

Buryak, A. , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2015 , NUMBER =. doi:10.4310/CNTP.2015.v9.n2.a1 , URL =

2015 doi
[36]

Surveys in differential geometry (

Witten, Edward , TITLE =. Surveys in differential geometry (. 1991 , ISBN =

1991
[37]

Kontsevich, Maxim , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1992 , NUMBER =

1992
[38]

Arithmetic and geometry,

Mumford, David , TITLE =. Arithmetic and geometry,. 1983 , ISBN =

1983
[39]

2026 , eprint=

Resonance transformations for the (2,2p+1) minimal string via x-y swap: a proof of Artemev's conjecture , author=. 2026 , eprint=

2026
[40]

Buryak, Alexandr and Gu\'. D. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/gt.2019.23.3537 , URL =

2019 doi
[41]

and Pandharipande, R

Faber, C. and Pandharipande, R. , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/s002229900028 , URL =

2000 doi
[42]

Moduli of curves and abelian varieties , SERIES =

Faber, Carel , TITLE =. Moduli of curves and abelian varieties , SERIES =. 1999 , ISBN =

1999
[43]

and Pandharipande, R

Getzler, E. and Pandharipande, R. , TITLE =. Nuclear Phys. B , FJOURNAL =. 1998 , NUMBER =. doi:10.1016/S0550-3213(98)00517-3 , URL =

1998 doi
[44]

, TITLE =

Givental, Alexander B. , TITLE =. Mosc. Math. J. , FJOURNAL =. 2001 , NUMBER =. doi:10.17323/1609-4514-2001-1-4-551-568 , URL =

2001 doi
[45]

Blot, Xavier and Shadrin, Sergey and Singh, Ishan Jaztar , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/blms.70117 , URL =

2025 doi
[46]

Alexandrov, Alexander , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2021 , NUMBER =. doi:10.4310/CNTP.2021.v15.n3.a6 , URL =

2021 doi
[47]

Integrability, quantization, and geometry

Takasaki, Kanehisa , TITLE =. Integrability, quantization, and geometry. [2021] 2021 , ISBN =. doi:10.1090/pspum/103.1/01844 , URL =

2021 doi
[48]

Liu, Chiu-Chu Melissa and Liu, Kefeng and Zhou, Jian , TITLE =. J. Differential Geom. , FJOURNAL =. 2003 , NUMBER =

2003
[49]

and Pandharipande, R

Okounkov, A. and Pandharipande, R. , TITLE =. Geom. Topol. , FJOURNAL =. 2004 , PAGES =. doi:10.2140/gt.2004.8.675 , URL =

2004 doi
[50]

Framed knots at large

Mari\. Framed knots at large. Orbifolds in mathematics and physics (. 2002 , ISBN =. doi:10.1090/conm/310/05404 , URL =

2002 doi
[51]

Kramer, Reinier , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2023 , NUMBER =. doi:10.4310/cntp.2023.v17.n2.a1 , URL =

2023 doi
[52]

Li, Jun and Liu, Chiu-Chu Melissa and Liu, Kefeng and Zhou, Jian , TITLE =. Geom. Topol. , FJOURNAL =. 2009 , NUMBER =. doi:10.2140/gt.2009.13.527 , URL =

2009 doi
[53]

Zhou, Jian , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2010 , NUMBER =. doi:10.1007/s11005-010-0397-1 , URL =

2010 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Arnold, V. I. and Gusein-Zade, S. M. and Varchenko, A. N. , TITLE =. 1985 , PAGES =. doi:10.1007/978-1-4612-5154-5 , URL =

doi:10.1007/978-1-4612-5154-5 1985

[2] [2]

2024 , eprint=

On the strong DR/DZ equivalence conjecture , author=. 2024 , eprint=

2024

[3] [3]

2026 , eprint=

Beyond descendants: integrable observables for cohomological field theories , author=. 2026 , eprint=

work page 2026

[4] [4]

2024 , eprint=

Rooted trees with level structures, -classes and double ramification cycles , author=. 2024 , eprint=

2024

[5] [5]

Stable tree expressions with omega-classes and double ramification cycles , JOURNAL =

Blot, Xavier and Lewa\'. Stable tree expressions with omega-classes and double ramification cycles , JOURNAL =. 2025 , PAGES =. doi:10.1016/j.geomphys.2024.105391 , URL =

arXiv 2025

[6] [6]

Blot, Xavier and Sauvaget, Adrien and Shadrin, Sergey , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/blms.13215 , URL =

doi:10.1112/blms.13215 2025

[7] [7]

, TITLE =

Buryak, A. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00220-014-2235-2 , URL =

doi:10.1007/s00220-014-2235-2 2015

[8] [8]

and Dubrovin, B

Buryak, A. and Dubrovin, B. and Gu. Tau-structure for the double ramification hierarchies , fjournal =. Comm. Math. Phys. , issn =. 2018 , language =. doi:10.1007/s00220-018-3235-4 , keywords =

doi:10.1007/s00220-018-3235-4 2018