All-orders shifted symmetric structure of the quantum KdV eigenvalues (Conjecture 1.9 of van Ittersum–Ruzza)

The quantum KdV hierarchy of Buryak and Rossi — the quantum double ramification hierarchy of the trivial cohomological field theory — acts by commuting Hamiltonian operators on the bosonic Fock space of polynomials in power-sum variables, and its common eigenvectors deform the Schur basis, with eigenvalues $E_k(\lambda;\epsilon,c)$ indexed by partitions $\lambda$ and expanded in the dispersion parameter $\epsilon$. Order zero (Dubrovin) and order one (van Ittersum–Ruzza) of this expansion are shifted symmetric functions of $\lambda$. The open conjecture formulated by van Ittersum and Ruzza (suggested by Don Zagier, refined as Conjecture 1.9 of van Ittersum–Ruzza) asserts that every coefficient $E_k^{[m]}$ is a shifted symmetric function of homogeneous weight $k+2+m$, refined by an explicit expansion in the shifted power sums $Q_j$ with rational polynomials $f_{D,\nu}(k)$ of degree $2D$, whose construction the authors announced with Don Zagier but which has not appeared. The conjecture would yield closed polynomial formulas for the Hodge integrals $\int_{\overline{\mathcal M}_{g,1}}\lambda_s\lambda_g\psi_1^{2g-2-s}$ on the moduli space of curves and quasimodularity of the $q$-brackets of arbitrary products of the eigenvalues. The all-orders conjecture remains open as of September 2026.

Background

The Korteweg–de Vries (KdV) equation is the prototype of an infinite-dimensional integrable system: it carries an infinite hierarchy of commuting Hamiltonian flows with respect to two compatible Poisson structures. Buryak's double ramification hierarchy, built from the intersection theory of double ramification cycles on moduli spaces of stable curves, recovers the KdV Hamiltonians for the trivial cohomological field theory, and Buryak and Rossi quantized this construction: the quantum double ramification hierarchy is a family of commuting differential operators $G_k(\epsilon)$ on the bosonic Fock space $B=\mathbb Q[p_1,p_2,\ldots]$ of polynomials in the power-sum variables, obtained by representing the modes $\omega_a$ of the Heisenberg algebra as multiplication by $p_a$ ($a>0$), by the central constant $c$ ($a=0$), and by multiples of $\partial/\partial p_{-a}$ ($a<0$). In the classical limit these operators degenerate to the conserved quantities of the classical KdV hierarchy for its first Poisson structure, so this is a quantization of KdV; it is a cousin of the older conformal-field-theoretic "quantum KdV" of Bazhanov, Lukyanov, and Zamolodchikov, where the second Poisson structure is quantized instead. The parameter $\epsilon$ is the dispersion parameter of the hierarchy: at $\epsilon=0$ the classical limit is the dispersionless (hydrodynamic) KdV hierarchy.

The spectral problem for these operators was posed by Dubrovin and developed by Ruzza and Yang: the operators $\hat G_k(\epsilon,c)$ preserve the grading of $B$ by total weight and admit a common eigenbasis $\{r_\lambda(p;\epsilon)\}$ deforming the Schur functions $s_\lambda$, the standard polynomial basis of $B$ indexed by partitions $\lambda$. The eigenvalues expand in the dispersion parameter as $E_k(\lambda;\epsilon,c)=\sum_{m\ge0}\epsilon^m E_k^{[m]}(\lambda;c)$.

A shifted symmetric function is a function on the set of partitions that is symmetric in the shifted parts $\lambda_i-i+1/2$; they form the polynomial algebra $\Lambda^*=\mathbb Q[Q_2,Q_3,\ldots]$ generated by the shifted power sums $Q_k$, explicit polynomial functions of the parts of $\lambda$ (Eq. (1.14) of the source paper). Shifted symmetric functions are central to asymptotic representation theory (fluctuations of Young diagrams) and to the enumerative geometry of curves and of translation surfaces. Assigning weight $j$ to $Q_j$ and weight $1$ to $c$ makes $\Lambda^*[c]$ a graded algebra. Dubrovin proved that at $\epsilon=0$ the Schur functions are eigenvectors with shifted symmetric eigenvalues, $E_k^{[0]}(\lambda;c)=\sum_{j=0}^{k+2}\frac{c^{k+2-j}}{(k+2-j)!}\,Q_j(\lambda)$, and van Ittersum and Ruzza proved that the first-order coefficient is again shifted symmetric: at $c=0$, $E_k^{[1]}(\lambda;0)=\frac{1}{12}Q_2Q_{k+1}+\frac{k(k+3)}{24}Q_{k+3}$.

Two further structures make the higher-order question consequential. First, the Bloch–Okounkov theorem states that the $q$-bracket of a shifted symmetric function — the weighted average $\langle f\rangle_q=\sum_\lambda f(\lambda)q^{|\lambda|}\big/\sum_\lambda q^{|\lambda|}$ over all partitions, with $q$ the modular parameter of the upper half-plane — is a quasimodular form for $SL_2(\mathbb Z)$, i.e. a polynomial in the Eisenstein series $E_2,E_4,E_6$. van Ittersum and Ruzza had already proved that $\langle E_k(\cdot;\epsilon,c)\rangle_q$ is quasimodular of weight $k+2$ (in the algebra $\widetilde M[c,\epsilon]$ of quasimodular forms extended by $c$ of weight $1$ and $\epsilon$ of weight $-1$), and Okounkov and Pandharipande had earlier shown that the $\epsilon=0$ version of this quasimodularity governs the Gromov–Witten theory of an elliptic curve. Second, the $\epsilon$-expansion of the eigenvalues encodes intersection numbers on the moduli space $\overline{\mathcal M}_{g,1}$ of one-pointed stable curves: the source paper proves a closed evaluation of certain double Hodge integrals (its Corollary 1.7), where $\lambda_s$ is the $s$th Chern class of the Hodge bundle and $\psi_1$ the first Chern class of the cotangent line at the marked point.

Based on this evidence and on numerical verification up to $m\le5$, the paper formulates its principal open conjecture: all coefficients $E_k^{[m]}(\lambda;c)$ lie in $\Lambda^*[c]$ and are homogeneous of weight $k+2+m$, together with the refined Conjecture 1.9 below. The authors announce an explicit construction of the conjectural polynomials in a future publication in collaboration with Don Zagier.

Problem Statement

Prove or refute the all-orders shifted symmetric structure of the quantum KdV eigenvalues, as follows.

Let $\hat G_k(\epsilon,c)$, $k\ge0$, be the commuting Hamiltonian operators of the Buryak–Rossi quantum KdV hierarchy on $B=\mathbb Q[p_1,p_2,\ldots]$, let $r_\lambda(p;\epsilon)=s_\lambda+O(\epsilon)$ be their common eigenvectors deforming the Schur basis, and let

$$E_k(\lambda;\epsilon,c)=\sum_{m\ge0}\epsilon^m E_k^{[m]}(\lambda;c)$$

be the eigenvalue expansion in the dispersion parameter. Let $\Lambda^*[c]=\mathbb Q[c,Q_2,Q_3,\ldots]$ be the algebra of shifted symmetric functions (generators $Q_j$ of weight $j$) adjoined with the central charge $c$ of weight $1$, graded by total weight. The conventions of the source paper (Eq. (1.13)–(1.15)) are $Q_0(\lambda)=1$ and, for $j\ge1$, $Q_j(\lambda)=\frac{1}{(j-1)!}\sum_{i\ge1}\big[(\lambda_i-i+\tfrac12)^{j-1}-(-i+\tfrac12)^{j-1}\big]+\beta_j$, so that in particular $Q_1=\beta_1=0$, with $\beta_j$ given by $\sum_{j\ge0}\beta_j y^j=\frac{y/2}{\sinh(y/2)}$.

(i) Show that for all $k,m\ge0$ the coefficient $E_k^{[m]}(\lambda;c)$ belongs to $\Lambda^*[c]$ and is homogeneous of weight $k+2+m$.

(ii) Establish the refined form (Conjecture 1.9 of van Ittersum–Ruzza): for every integer $D\ge0$ and every partition $\nu$ there exists a polynomial $f_{D,\nu}(k)$ with rational coefficients of degree $2D$ such that for all $k\ge0$

$$E_k(\cdot;\epsilon,0)=\sum_{D\ge0}\sum_{\nu\in\mathscr P}\epsilon^{D+|\nu|}\,f_{D,\nu}(k)\,Q_{\,k+D+2-\ell(\nu)}\prod_i Q_{\nu_i+1},$$

where $\ell(\nu)$ is the number of parts of $\nu$; as in the source, the sum is over all $D\ge0$ and $\nu\in\mathscr P$, and only finitely many pairs contribute to each coefficient of $\epsilon^m$, namely those with $D+|\nu|=m$. The restriction to $c=0$ is no loss of generality, since $E_k(\lambda;\epsilon,c)=\sum_{\ell=0}^{k}\frac{c^\ell}{\ell!}\,E_{k-\ell}(\lambda;\epsilon,0)$.

(iii) In the affirmative case, derive the two consequences that the source shows follow from Conjecture 1.9: for all $k_1,\ldots,k_r\ge0$,

$$\left\langle\prod_{i=1}^r E_{k_i}(\cdot;\epsilon,c)\right\rangle_q\in\widetilde M[c,\epsilon]_{\sum_i(k_i+2)},$$

the graded algebra of quasimodular forms for $SL_2(\mathbb Z)$ extended by $c$ (weight $1$) and $\epsilon$ (weight $-1$); and there exist polynomials $p_{r,s}(g)\in\mathbb Q[g]$ of degree $2(s-r)$, $0\le r\le s$, such that

$$\int_{\overline{\mathcal M}_{g,1}}\lambda_s\lambda_g\psi_1^{2g-2-s}=(-1)^g\sum_{r=0}^{s}p_{r,s}(g)\,\beta_{2(g-r)}\quad(s<g),$$

with $\beta_k$ defined by $\sum_{k\ge0}\beta_k y^k=\frac{y/2}{\sinh(y/2)}$.

Since (iii) follows from (i)–(ii) by the Bloch–Okounkov theorem and the arguments the source sketches, the non-routine core of the problem is (i)–(ii); moreover, proving (ii) settles (i) (by the c-shift relation stated above), so (i) need not be shown separately. A refutation of (i) or (ii) settles the problem, whereas an affirmative answer must establish (i)–(ii) and also derive the two consequences (iii). An explicit construction of the polynomials $f_{D,\nu}$ is the authors' announced further programme in collaboration with Don Zagier and is not part of Conjecture 1.9 as stated in the source.

References

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