The Virasoro conjecture for Hodge integrals of arbitrary smooth projective varieties (Wang's Conjecture 1.1)

Wang (arXiv:2506.10033, 2025) proposed a Virasoro conjecture for Hodge integrals — integrals of Chern classes of the Hodge bundle over moduli spaces of stable maps — stating that the all-genus Hodge-integral total potential $\mathcal{D}^{\mathbb{E}}(t,s)$ of any smooth projective variety is annihilated by explicit differential operators $L_n^{\mathbb{E}}$, $n\geq -1$, forming half of the Virasoro algebra. The conjecture is proved in the paper only for targets with semisimple quantum cohomology (via Givental–Teleman reconstruction) and in genus zero for arbitrary targets, together with a genus-1 result for the $L_1^{\mathbb{E}}$-constraint with a single Hodge insertion and a higher-genus single-insertion vanishing for sufficiently large Hodge degree. Because the Hodge potential is obtained from the ordinary descendant potential by conjugation with Givental's Hodge operator, the conjecture is equivalent to the classical Eguchi–Hori–Xiong–Katz Virasoro conjecture for Gromov–Witten invariants, which remains open for general targets with non-semisimple quantum cohomology beyond smooth curves — already at genus one, where it reduces to the vanishing of a single descendant combination $\Psi_{1,1;\emptyset}$. The problem is to prove or refute the conjecture in its remaining generality: all genera, arbitrary smooth projective varieties, no semisimplicity assumption.

Background

Let $X$ be a smooth projective variety. Gromov–Witten invariants are intersection numbers on moduli spaces $\overline{\mathcal{M}}_{g,n}(X,A)$ of stable maps from genus-$g$ nodal curves with $n$ marked points into $X$ in homology class $A$; with descendant insertions one integrates, besides cohomology classes pulled back from $X$, powers of the first Chern class $\psi_i = c_1(\mathcal{L}_i)$ of the tautological line bundle $\mathcal{L}_i$ whose fiber at a stable map is the cotangent line at the $i$-th marked point. Organizing all descendant invariants into genus-$g$ potentials $F_g(t)$ and $\mathcal{D}(t)=\exp(\sum_{g\geq 0}\hbar^{2g-2}F_g(t))$ gives the total descendant potential, a formal power series in the big phase space coordinates $t$ indexed by all descendant levels. Hodge integrals are the refinement obtained by also integrating Chern classes $\lambda_k=c_k(\mathbb{E}_g)$ of the Hodge bundle $\mathbb{E}_g$ (the rank-$g$ vector bundle over $\overline{\mathcal{M}}_{g,n}(X,A)$ with fiber $H^0(C,\omega_C)$ for the source curve $C$); their generating function is the Hodge potential $\mathcal{D}^{\mathbb{E}}(t,s)$, where $s=(s_1,s_2,\dots)$ records Hodge degree. Faber and Pandharipande proved that Hodge integrals are determined by pure descendant Gromov–Witten invariants, and Givental reformulated this as the operator identity $\mathcal{D}^{\mathbb{E}}(t,s)=\exp(\sum_{k\geq 1}\frac{B_{2k}}{(2k)!}s_k\, z^{[2k-1]})\,\mathcal{D}(t)$, where $B_{2k}$ are Bernoulli numbers and $z^{[2k-1]}$ denotes a fixed infinitesimal symplectic transformation on the cohomology of $X$.

The Virasoro conjecture of Eguchi–Hori–Xiong (also proposed by Katz) predicts that $\mathcal{D}(t)$ is annihilated by a sequence of differential operators $L_n$, $n\geq -1$, satisfying the Virasoro bracket $[L_n,L_m]=(n-m)L_{n+m}$; the coefficients of $L_n$ involve the matrix of multiplication by $c_1(X)$ and elementary symmetric functions shifted by the grading. When $X$ is a point the conjecture reduces to Witten's conjecture on intersection numbers of $\psi$-classes on $\overline{\mathcal{M}}_{g,n}$, proved by Kontsevich. The conjecture is known in genus zero for all targets (Liu–Tian, with related proofs by Givental and Getzler); in all genera for targets whose quantum cohomology is semisimple (the quantum cup product has no nilpotent part after base change), via Givental's quantization formula and Teleman's classification of semisimple two-dimensional topological field theories; for genus one in the semisimple case (Dubrovin–Zhang); and for smooth target curves (Okounkov–Pandharipande). For general targets with non-semisimple quantum cohomology — beyond the smooth-curve case just cited — the conjecture is open in every genus $g\geq 1$: Xiaobo Liu reduced the genus-one case to the single vanishing $\Psi_{1,1;\emptyset}=0$ (the genus-1 coefficient of $\mathcal{D}^{-1}L_1\mathcal{D}$ with no Hodge insertions), and this reduction remains unproved in general. A recent reduction of Tseng (2026) shows that for a product target $X\times Y$ with $Y$ semisimple, the Virasoro conjecture for $X\times Y$ is equivalent to the conjecture for $X$; this settles product targets with one semisimple factor but covers no genuinely new non-semisimple target.

Wang (2025) constructs operators $L_n^{\mathbb{E}}$ as conjugates of the Eguchi–Hori–Xiong operators by the Hodge operator above; they satisfy the Virasoro bracket, and their coefficients combine Bernoulli numbers, difference operators $\Delta^k f(x)=f(x+k)-f(x)$, and shifted elementary symmetric functions. Conjecture 1.1 of that paper asserts $L_n^{\mathbb{E}}\mathcal{D}^{\mathbb{E}}(t,s)=0$ for all $n\geq -1$ and all smooth projective $X$. A crucial structural point, stated in the paper: because the Hodge potential is a conjugate of $\mathcal{D}(t)$, this conjecture is equivalent to the classical Virasoro conjecture $L_n\mathcal{D}(t)=0$; the $s=0$ sector of the Hodge conjecture is exactly the classical one. Expanding $\mathcal{D}^{\mathbb{E},-1}L_n^{\mathbb{E}}\mathcal{D}^{\mathbb{E}}=\sum_g\hbar^{2g-2}\Psi^{\mathbb{E}}_{g,n}(t,s)$ and then expanding in $s$ gives component identities $\Psi^{\mathbb{E}}_{g,n;k_1,\dots,k_m}=0$ (genus $g$, operator index $n$, Hodge degrees $k_1,\dots,k_m$), of which the $m=0$, $g=1$ case is the open classical descendant problem above. Wang proves Conjecture 1.1 for semisimple targets (Theorem 1.2), in genus zero for arbitrary targets (Theorem 1.3), the genus-one $L_1^{\mathbb{E}}$-constraint with one Hodge insertion for arbitrary targets ($\Psi^{\mathbb{E}}_{1,1;k_1}=0$ for $k_1\geq 1$, Theorem 1.4), and a higher-genus single-insertion vanishing $\Psi^{\mathbb{E}}_{g,1;k_1}=0$ for $g\geq 2$ and $k_1\geq\max\{g+1,(3g-1)/2\}$ (Theorem 1.5), the latter via genus-$g$ tautological relations on $\overline{\mathcal{M}}_{g,n}$. The paper is also motivated by the Hodge hierarchy of Dubrovin–Liu–Yang–Zhang — the tau-symmetric integrable deformation of the principal hierarchy of a Frobenius manifold (the flat structure underlying genus-zero Gromov–Witten theory) — for which Virasoro constraints are expected to give a new construction route beyond the semisimple case; Blot–Carlet–Makris–Shadrin (2026) recently settled the rank-one (triple-Hodge, Calabi–Yau) case of the related Dubrovin–Zhang constant-Poisson-bracket conjecture, one piece of that program, but the general construction remains open.

Problem Statement

Determine the truth of Conjecture 1.1 of Wang (2025): is it true that for every smooth projective variety $X$ — including those whose quantum cohomology is non-semisimple — and every integer $n\geq -1$, the all-genus total potential of Hodge integrals satisfies

$L_n^{\mathbb{E}}\,\mathcal{D}^{\mathbb{E}}(t,s)=0,$

where $\mathcal{D}^{\mathbb{E}}(t,s)$ is the Hodge-integral descendant potential of $X$ and $L_n^{\mathbb{E}}$ are the Virasoro operators defined in that paper (equivalently, in the coefficient language, $\Psi^{\mathbb{E}}_{g,n;k_1,\dots,k_m}=0$ for all genera $g$, all operator indices $n\geq -1$ — the index of the corresponding $L_n^{\mathbb{E}}$-constraint, not a marked-point count — and all Hodge-degree tuples $(k_1,\dots,k_m)$)? By the conjugation identity of Section 3.2 of the source, this is equivalent to the classical Eguchi–Hori–Xiong–Katz Virasoro conjecture $L_n\mathcal{D}(t)=0$ for arbitrary targets, whose $s=0$ sector it contains; in particular the conjecture includes, at genus one, the still-open descendant-only vanishing $\Psi_{1,1;\emptyset}=0$. The conjecture is currently known only for semisimple quantum cohomology and in genus zero; the question is open in all remaining cases, i.e. arbitrary targets without a semisimplicity assumption at every genus $g\geq 1$. The conjecture is equivalent to the classical Virasoro conjecture for arbitrary smooth projective varieties, and it fails exactly when there exist a smooth projective variety $X$, an index $n\geq -1$, and a genus and Hodge-degree tuple for which the corresponding coefficient $\Psi^{\mathbb{E}}_{g,n;k_1,\dots,k_m}$ is nonzero.

References

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