Extend the meromorphic-differential/double-ramification correspondence from the trivial and Hodge CohFTs to general cohomological field theories satisfying the strata vanishing conditions

To every cohomological field theory (CohFT) one can associate two quantum Hamiltonian integrable hierarchies built from intersection theory on the moduli space of stable curves: the double ramification (DR) hierarchy, whose densities are integrals over DR cycles, and the meromorphic differential (MD) hierarchy of Blot and Rossi, whose densities are Hodge-class and CohFT insertions integrated over strata of meromorphic differentials. For the trivial CohFT, Blot and Rossi proved a coefficientwise identification of the strata integrals with the DR-cycle integrals, equivalently $H_d=(H_d^{\mathrm{DR}})^{[0]}$ together with degree- $2g$ polynomiality of the strata integrals; Blot, Rossi and Sauvaget established the same correspondence for the Hodge CohFT and stated explicitly (Remark 1.4 of arXiv:2508.09899) that they expect it for every CohFT satisfying a natural vanishing condition on holomorphic strata. This problem is precisely that open extension: for every CohFT with unit satisfying the vanishing conditions, prove the degree- $2g$ polynomiality of the relevant strata integrals in the pole orders and the coefficientwise identification with DR-cycle integrals, together with the equivalent hierarchy-level statement $H_d=(H_d^{\mathrm{DR}})^{[0]}$ and its consequences; alternatively, refute it by an explicit CohFT satisfying the vanishing conditions for which the polynomiality or the coefficientwise identification fails.

Background

This problem sits at the interface of the intersection theory of the moduli space of stable curves and the theory of Hamiltonian integrable hierarchies, i.e. systems of commuting evolutionary partial differential equations generated by pairwise-commuting Hamiltonian densities. $\overline{\mathcal{M}}_{g,n}$ denotes the moduli space of stable complex curves of genus $g$ with $n$ ordered marked points. Two families of tautological cohomology classes are used throughout: the $\psi$ -classes $\psi_i$ , the first Chern classes of the line bundles whose fiber at a marked curve is the cotangent line at the $i$ -th marked point; and the Hodge classes $\lambda_l$ , the $l$ -th Chern classes of the Hodge bundle, the vector bundle whose fiber at a curve is its space of holomorphic differentials. Given integers $m_0,\dots,m_{n-1}$ with $\sum_i m_i=2g-2$ , the stratum of differentials $\overline{\mathcal{H}}_g(m_0,\dots,m_{n-1})$ is the closure in $\overline{\mathcal{M}}_{g,n}$ of the locus of smooth pointed curves satisfying $\omega_C\cong\mathcal{O}_C(\sum_i m_i x_i)$ , i.e. carrying a meromorphic differential whose zeros and poles occur at the marked points with prescribed orders. If all $m_i\geq0$ the stratum is called holomorphic and has complex codimension $g-1$ ; if some $m_i<0$ it is called meromorphic and has complex codimension $g$ . A cohomological field theory (CohFT) with unit is a system $(V,\eta,\mathbf{1},c_{g,n})$ : a finite-dimensional complex vector space $V$ with a nondegenerate symmetric bilinear form $\eta$ , a unit vector $\mathbf{1}$ , and $S_n$ -equivariant maps $c_{g,n}:V^{\otimes n}\to H^*(\overline{\mathcal{M}}_{g,n})$ compatible with gluing of curves and with the unit, in the sense that inserting $\mathbf{1}$ at an extra marked point and forgetting that point reproduces $c_{g,n-1}$ . The basic examples are the trivial CohFT ( $c_{g,n}\equiv1$ ) and the Hodge CohFT (of rank one, with class given by the Chern polynomial $\Lambda(\mu)=1+\mu\lambda_1+\cdots+\mu^g\lambda_g$ of the Hodge bundle). For integers $a_0,\dots,a_{n-1}$ with $\sum a_i=0$ , the double ramification cycle $\mathrm{DR}_g(a_0,\dots,a_{n-1})$ is (up to Poincaré duality) the locus of curves carrying a meromorphic function with zeros and poles of the prescribed multiplicities at the marked points; it is computed by an explicit formula in the tautological ring (the Pixton class formula), and its dependence on the $a_i$ is polynomial of degree $2g$ , as proved by Pixton and, alternatively, by Spelier. Buryak associated to every CohFT a Hamiltonian integrable hierarchy, the DR hierarchy, whose Hamiltonian densities are generating series of intersection numbers of $\psi$ -classes and CohFT classes over DR cycles, and Buryak–Rossi constructed its deformation quantization, i.e. a quantization of the Hamiltonian structure in the parameter $\hbar$ deforming the commutative algebra of observables; for the trivial CohFT the classical DR hierarchy is the KdV (Korteweg–de Vries) hierarchy. Blot–Rossi then defined two parallel constructions with the DR cycle replaced by classes of meromorphic differentials: the MD hierarchy, whose quantum Hamiltonian densities are built from integrals of $\Lambda(-\epsilon^2/(i\hbar))\,\psi_0^{d+1}$ and the CohFT class over the strata $\overline{\mathcal{H}}_g(-1,m_1,\dots,m_n,2g-1-\sum m_i)$ , and the twisted DR hierarchy, built from the $k=1$ -twisted double ramification cycle $\mathrm{DR}^1_g$ (the Pixton-class twist $2^{-g}P^{g,1}_g(m_0+1,\dots,m_{n-1}+1)$ , which reproduces the weighted fundamental class of the moduli space of twisted canonical divisors). Here $\epsilon$ is a dispersion parameter and $\hbar$ the quantization parameter. For every CohFT these hierarchies are integrable and tau-symmetric — their Hamiltonian densities carry a symmetric tau-structure, so that one generating function, the tau function, encodes the whole hierarchy — the proofs resting on splitting formulas for the strata of differentials due to Sauvaget and for the twisted DR cycle due to Costantini–Sauvaget–Schmitt. However, for a generic CohFT the strata intersection numbers are only piecewise polynomial in the pole orders $m_i$ , so the densities generally have no interpretation as differential polynomials in the standard variables $u^\alpha(x)=\sum_m q_m^\alpha(ix)^m$ . For the trivial CohFT, Blot–Rossi proved the Main Theorem: writing strata integrals in the falling-factorial basis $m^{\underline{s}}=m(m-1)\cdots(m-s+1)$ , the coefficient of $m_1^{\underline{s_1}}\cdots m_n^{\underline{s_n}}$ with $\sum s_i=2g$ in a strata integral equals the coefficient of $a_1^{s_1}\cdots a_n^{s_n}$ in the corresponding DR-cycle integral. Equivalently, at the hierarchy level, $H_d=(H_d^{\mathrm{DR}})^{[0]}$ : with respect to the grading $\deg u_i=i$ , $\deg\epsilon=-1$ , $\deg x^{-1}=1$ , $\deg\hbar=-2$ , the quantum MD densities are the degree-zero part of the quantum DR densities, so the MD hierarchy is the top-degree part of the DR hierarchy; classically both are KdV, while quantum mechanically the MD hierarchy gives a quantization of KdV distinct from the DR quantization. Blot–Rossi–Sauvaget then established the same correspondence for the Hodge CohFT, using the star-graph decomposition of the twisted DR cycle (Conjecture A of Farkas–Pandharipande — the expression of $\mathrm{DR}^1_g$ as a sum over star-shaped stable graphs decorated by strata of differentials — proved by Bae–Holmes–Pandharipande–Schmitt–Schwarz), the vanishing of $\lambda_g\lambda_{g-1}$ away from the rational-tails locus (the locus of curves made of one genus- $g$ component with trees of rational components attached), and Faber-type intersection-number formulas for $\psi$ -classes on $\overline{\mathcal{M}}_{g}$ . Blot–Rossi–Sauvaget state (Remark 1.4 of arXiv:2508.09899) that they expect the theorem to hold for any CohFT satisfying a vanishing condition on holomorphic strata (Remark 1.2 there): $\int_{\overline{\mathcal{H}}_g(m_1,\dots,m_n)}\lambda_l\,c_{g,n}=0$ whenever $n\neq1$ and all $m_i\geq0$ . The condition holds for the trivial CohFT by a dimension count: a holomorphic stratum with $n\neq1$ marked points has complex dimension $2g-2+n$ , which strictly exceeds the maximal degree $g$ of the insertion $\lambda_l$ when $c_{g,n}\equiv1$ (for $n=0$ the condition is vacuous: for $g\geq2$ nonnegative $m_i$ cannot sum to $2g-2$ with no marked points, and for $g=1$ the stratum lies outside the stable range $2g-2+n>0$ ). The Hodge case of the correspondence is proved directly, and the vanishing condition is designed to abstract exactly the holomorphic-strata vanishing that makes that proof work; the proofs in the trivial and Hodge cases lean on degree counts and Hodge-specific vanishing phenomena that have no known analog for arbitrary CohFT insertions. Whether the correspondence holds for every CohFT satisfying the vanishing condition is the open problem recorded here.

Problem Statement

Let $(V,\eta,\mathbf{1},c_{g,n})$ be a cohomological field theory with unit satisfying the vanishing condition

$\int_{\overline{\mathcal{H}}_g(m_1,\dots,m_n)}\lambda_l\,c_{g,n}(v_1,\dots,v_n)=0$

for every $g\geq0$ , every $0\le l\le g$ , every $n\neq1$ with $2g-2+n>0$ , every choice of nonnegative $m_1,\dots,m_n$ with $\sum m_i=2g-2$ , and all $v_1,\dots,v_n\in V$ . (For the trivial CohFT the condition holds by a dimension count; the Hodge CohFT, for which the correspondence below is already proved, is the motivating anchor case, and the condition abstracts the holomorphic-strata vanishing used in its proof.)

Throughout, the marked points of a stratum $\overline{\mathcal{H}}_g(m_0,\dots,m_{n-1})$ and of a double ramification cycle $\mathrm{DR}_g(a_0,\dots,a_{n-1})$ are indexed from $0$ to $n-1$ , the $i$ -th marked point carrying the order (respectively multiplicity) $m_i$ (respectively $a_i$ ) and the cotangent-line class $\psi_i$ , and the $j$ -th listed tensor factor of $c_{g,n}$ is evaluated at the $(j-1)$ -th marked point (so the first tensor factor sits at marked point $0$ ); this fixes the marked-point bookkeeping in the formulas below. Fix a basis $e_1,\dots,e_N$ of $V$ , $N=\dim V$ .

Prove, or disprove by an explicit counterexample, that the Blot–Rossi–Sauvaget correspondence extends to every such CohFT, in the following jointly constitutive parts, where the coefficient of $m_1^{\underline{s_1}}\cdots m_n^{\underline{s_n}}$ means the coefficient after rewriting a polynomial in the falling-factorial basis $m^{\underline{s}}=m(m-1)\cdots(m-s+1)$ :

(1) Polynomiality. For $g,n\geq0$ with $2g-1+n>0$ , $d\geq0$ , $0\le l\le g$ and $v_1,\dots,v_n\in V$ , the function

$\int_{\overline{\mathcal{H}}_g(2g-2-\sum m_i,\,m_1,\dots,m_n)}\psi_0^{d}\,\lambda_l\,c_{g,n+1}(v_1\otimes\cdots\otimes v_n\otimes\mathbf{1})$

defined on the set of integer tuples $(m_1,\dots,m_n)$ with $2g-2-\sum m_i<0$ , is a polynomial of degree $2g$ in $m_1,\dots,m_n$ (so the quantum MD Hamiltonian densities become differential polynomials in the $u^\alpha(x)$ and their $x$ -derivatives, with poles in $x$ allowed).

(2) Coefficientwise identification. For $g,n\geq0$ with $2g+n>0$ , $d\geq-1$ , $0\le l\le g$ , and indices $1\le\alpha,\alpha_1,\dots,\alpha_n\le N$ (by multilinearity this covers arbitrary insertions): the integral

$\int_{\overline{\mathcal{H}}_g(-1,\,m_1,\dots,m_n,\,2g-2-\sum m_i+1)}\psi_0^{d+1}\,\lambda_l\,c_{g,n+2}(e_\alpha\otimes e_{\alpha_1}\otimes\cdots\otimes e_{\alpha_n}\otimes\mathbf{1})$

is a polynomial of degree $2g$ in $m_1,\dots,m_n$ , and its coefficient of $m_1^{\underline{s_1}}\cdots m_n^{\underline{s_n}}$ vanishes if $\sum s_i\neq2g$ , while when $\sum s_i=2g$ it equals the coefficient of $a_1^{s_1}\cdots a_n^{s_n}$ in

$\int_{\mathrm{DR}_g(0,\,a_1,\dots,a_n,\,-\sum a_i)}\psi_0^{d+1}\,\lambda_l\,c_{g,n+2}(e_\alpha\otimes e_{\alpha_1}\otimes\cdots\otimes e_{\alpha_n}\otimes\mathbf{1}).$

(3) Hierarchy formulation. Deduce from (1) and (2) the equivalent statement $H^{\mathrm{MD}}_{\alpha,d}=(H^{\mathrm{DR}}_{\alpha,d})^{[0]}$ for all $d\geq-1$ and all indices $\alpha$ , where $H^{\mathrm{MD}}_{\alpha,d}$ and $H^{\mathrm{DR}}_{\alpha,d}$ are the quantum Hamiltonian densities of the MD and of the (untwisted) DR hierarchy of the same CohFT, and $(\cdot)^{[0]}$ extracts the degree-zero part for the grading $\deg u_i^\alpha=i$ , $\deg\epsilon=-1$ , $\deg x^{-1}=1$ , $\deg\hbar=-2$ ; in particular the MD hierarchy is the top-degree part of the DR hierarchy for every such CohFT, and the resulting identities between strata integrals and DR-cycle integrals hold with arbitrary CohFT insertions.

A complete answer establishes (1)–(3) for the entire class of CohFTs satisfying the vanishing condition, or exhibits a specific such CohFT with unit for which either the polynomiality in (1) or the coefficientwise identification in (2) fails, for explicit $g,n,d,l$ and, in the latter case, a multi-index $(s_1,\dots,s_n)$ . (Blot–Rossi–Sauvaget note in Remark 1.2 of arXiv:2508.09899 that the polynomiality in (1) should already be apparent from their proof for any CohFT satisfying the vanishing condition, so the essential open content is the coefficientwise identification (2) and its hierarchy reformulation (3).) A natural further question, not required for resolution, is whether the twisted $\mathrm{DR}^1$ hierarchy of such a CohFT remains related to its MD hierarchy by a shift $u(x)\mapsto u(x)+\sum_{g\geq1}c_g/x^{2g}$ , as happens for the trivial and Hodge cases.

References

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