The $\lambda_g$ conjecture with target varieties: unconditional universal Hodge-integral constraints for arbitrary smooth projective targets

The $\lambda_g$ conjecture with target varieties (X. Wang, arXiv:2412.05287, 2024) generalizes the classical $\lambda_g$ conjecture of Getzler–Pandharipande from the moduli space of curves to the Gromov–Witten theory of an arbitrary smooth projective target $X$: it predicts two explicit infinite families of universal identities, $\Theta_{g,n,m,\beta}=0$ and $\Psi_{g,n}\equiv 0$, constraining genus-$g$ Hodge integrals with top (and next-to-top) Hodge-class insertions over moduli spaces of stable maps to $X$. By a conjugation formula of the source, the two families together are equivalent to the fiber degree-0 Virasoro constraints for $X\times\mathbb{P}^1$, i.e. Virasoro-type constraints on the total Hodge-integral potential of $X$. Both families are proved when the Virasoro conjecture holds for $X$ (in particular for targets with semisimple quantum cohomology and for smooth curves, in all genera) and in genus 0 for arbitrary $X$; the $\Theta$-family is also proved in genus 1 for arbitrary $X$ via Pixton's double ramification formula. The problem asks for an unconditional proof, or an explicit counterexample, for arbitrary smooth projective targets — the open core being the $\Theta$-family in genus $\ge 2$ and the $\Psi$-family in genus $\ge 1$ for targets with non-semisimple quantum cohomology. No unconditional proof and no counterexample is known: the problem is open.

Background

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable genus-$g$ curves with $n$ marked points. Its tautological classes include the $\psi$-classes $\psi_i=c_1(\mathcal{L}_i)$, where $\mathcal{L}_i$ is the line bundle whose fiber over a curve $(C;x_1,\dots,x_n)$ is the cotangent line $T^*_{x_i}C$, and the Hodge bundle $\mathbb{E}$, the rank-$g$ vector bundle with fiber $H^0(C,\omega_C)$; write $\lambda_k=c_k(\mathbb{E})$. The classical $\lambda_g$ conjecture, formulated by Getzler and Pandharipande as a consequence of the Virasoro conjecture for $\mathbb{P}^1$, predicts a remarkably simple closed form — a single multinomial coefficient — for $\psi$-class integrals coupled with the top Hodge class $\lambda_g$ over $\overline{\mathcal{M}}_{g,n}$; it was proved by Faber and Pandharipande through the Gromov–Witten theory of $\mathbb{P}^1$ and reproved by Goulden, Jackson and Vakil through the ELSV formula, which expresses Hurwitz numbers (counts of branched covers of $\mathbb{P}^1$) as Hodge integrals. It is a basic tool for computing Hodge integrals.

Gromov–Witten theory replaces the varying domain curve by a varying map. For a smooth projective variety $X$ of complex dimension $d$, let $\overline{\mathcal{M}}_{g,n}(X,B)$ be the moduli space of stable maps from genus-$g$ $n$-pointed curves to $X$ in homology class $B\in H_2(X;\mathbb{Z})$. Descendant Gromov–Witten invariants are integrals of evaluation-map pullbacks of cohomology classes of $X$ and powers of the $\psi$-classes against the virtual fundamental class; they are organized into generating functions $F_g$ of the big phase space coordinates $t^\alpha_n$, where $\alpha$ indexes a basis $\{\varphi_\alpha\}$ of $H^*(X;\mathbb{C})$ with $\varphi_\alpha$ of Hodge type $(p_\alpha,q_\alpha)$. Hodge integrals with target insert, in addition, the classes $\lambda_k$ pulled back from $\overline{\mathcal{M}}_{g,n}$ by the map that forgets the stable map; their correlation functions are denoted $\langle\!\langle W_1\cdots W_k;\lambda_g\rangle\!\rangle_g$ for vector fields $W_i$ on the big phase space.

The Virasoro conjecture of Eguchi, Hori and Xiong predicts that the total descendant potential of $X$ is annihilated by an explicit sequence of differential operators $L_n$ ($n\ge -1$) forming half of the Virasoro algebra. It is proved for genus $0$ for every target (Liu–Tian, Dubrovin–Zhang, Getzler), for all genera when the quantum cohomology of $X$ is semisimple — semisimplicity means that, generically, the deformed cup product on $H^*(X)$ (the quantum product) turns $H^*(X)$ into a direct sum of copies of $\mathbb{C}$ — via Givental's quantization formalism and Teleman's classification of two-dimensional semisimple field theories, for smooth target curves (Okounkov–Pandharipande), and for toric bundles (Coates–Givental–Tseng); for general targets it remains a central open problem.

Wang (arXiv:2412.05287) constructed the $\lambda_g$-analogue of the Virasoro constraints for an arbitrary target. Writing $b_\alpha=p_\alpha-\tfrac{1}{2}(d-1)$, $b^\alpha=1-b_\alpha$, $\tilde t^\alpha_r=t^\alpha_r-\delta^1_r\delta^1_\alpha$ (the dilaton shift), $e_k$ for the $k$-th elementary symmetric polynomial, $\varphi^\alpha=\sum_\gamma\eta^{\alpha\gamma}\varphi_\gamma$ for raising an index by the inverse Poincaré pairing, and $c_1(X)$ for the first Chern class, the paper defines explicit expressions $\Theta_{g,n,m,\beta}$ and $\Psi_{g,n}$ in the Hodge correlation functions of $X$ and proves a conjugation formula (Theorem 3.7): for the fiber degree-0 descendant potential $\widetilde{\mathcal{D}}$ of $X\times\mathbb{P}^1$ — which the same paper identifies with the total Hodge-integral potential of $X$ (Proposition 3.5) —

$$\widetilde{\mathcal{D}}^{-1}L_n\widetilde{\mathcal{D}}=\sum_{g\ge0}\hbar^{2g-2}(-1)^g\left((-2)\Psi_{g,n}+\sum_{m,\beta}u^\beta_m\,\Theta_{g,n,m,\beta}\right).$$

In this display $L_n$ denotes the Virasoro operators of $X\times\mathbb{P}^1$ (Proposition 3.3 of the source), $H\in H^2(\mathbb{P}^1;\mathbb{C})$ is the hyperplane class, and $\{u^\beta_m\}$ are the flat coordinates of the big phase space of $X\times\mathbb{P}^1$ corresponding to the $\varphi_\beta\otimes H$ directions of the canonical basis $\{\tau_m(\varphi_\alpha\otimes 1),\tau_m(\varphi_\alpha\otimes H)\}$; they are formal variables independent of the $t^\alpha_r$, and neither $\Theta_{g,n,m,\beta}$ nor $\Psi_{g,n}$ involves them.

Thus the pair of conjectures $\Theta_{g,n,m,\beta}=0$ and $\Psi_{g,n}\equiv 0$ is exactly the assertion that these Virasoro-type constraints hold for the Hodge-integral theory of an arbitrary smooth projective target. What is proved in the source: both conjectures hold whenever the Virasoro conjecture holds for $X$ (Theorem 1.2) — hence in all genera for semisimple targets and smooth curves; both hold in genus $0$ for arbitrary $X$ (Theorem 1.3); and the $\Theta$-family holds in genus $1$ for arbitrary $X$ (Theorem 1.6), the proof inserting Pixton's double ramification formula $\lambda_g=\tfrac{(-1)^g}{2^g}\mathsf{P}^g_g(0,\dots,0)$ in $R^g(\overline{\mathcal{M}}_{g,n})$, where $\mathsf{P}^g_g(a_1,\dots,a_n)$ is the codimension-$g$ Pixton class on $\overline{\mathcal{M}}_{g,n}$ representing cycles of curves carrying a meromorphic differential with prescribed zero and residue data. What remains open is the subject of this problem.

Problem Statement

Prove, or disprove, the following two conjectures of Wang (arXiv:2412.05287) for every smooth projective variety $X$ over $\mathbb{C}$, with no semisimplicity hypothesis on the quantum cohomology of $X$ and without assuming the Virasoro conjecture.

(1) The $\lambda_g$ conjecture with target varieties (Conjecture 1.1 of the source). For every $g\ge 0$, $m\ge 0$, $n\ge -1$ and basis index $\beta$ (with $\varphi_\beta$ ranging over the fixed homogeneous basis of $H^*(X;\mathbb{C})$), the identity

$$\Theta_{g,n,m,\beta}=0$$

holds as an identity of formal power series in the coordinates $t^\alpha_r$ and the curve-class (Novikov) variables, where

$$\Theta_{g,n,m,\beta}=\sum_{j=0}^{n+1}\sum_{r,\alpha}e_{n+1-j}\big(r+b_\alpha-\tfrac{1}{2},\dots,r+n+b_\alpha-\tfrac{1}{2}\big)\,\tilde t^\alpha_r\,\langle\!\langle\tau_{r+n-j}(c_1(X)^{j}\cup\varphi_\alpha)\,\tau_m(\varphi_\beta);\lambda_g\rangle\!\rangle_g$$ $$+\sum_{j=0}^{n+1}e_{n+1-j}\big(m+b_\beta+\tfrac{1}{2},\dots,m+n+b_\beta+\tfrac{1}{2}\big)\,\langle\!\langle\tau_{m+n-j}(c_1(X)^{j}\cup\varphi_\beta);\lambda_g\rangle\!\rangle_g$$ $$+\sum_{j=0}^{n+1}\sum_{s,\alpha}(-1)^{-s-1}\,e_{n+1-j}\big(-s+b^\alpha-\tfrac{3}{2},\dots,-s+n+b^\alpha-\tfrac{3}{2}\big)\sum_{g_1+g_2=g}\langle\!\langle\tau_m(\varphi_\beta)\,\tau_{-s-1+n-j}(c_1(X)^{j}\cup\varphi^\alpha);\lambda_{g_1}\rangle\!\rangle_{g_1}\,\langle\!\langle\tau_s(\varphi_\alpha);\lambda_{g_2}\rangle\!\rangle_{g_2}$$ $$+\delta^0_g\,\delta^0_m\sum_{\alpha}t^\alpha_0\int_X c_1(X)^{n+1}\cup\varphi_\alpha\cup\varphi_\beta.$$

Here $e_k$ is the $k$-th elementary symmetric polynomial, evaluated at the displayed list of $n+1$ arguments (with $e_0=1$); $b_\alpha=p_\alpha-\tfrac{1}{2}(d-1)$ and $b^\alpha=1-b_\alpha$ with $d=\dim_{\mathbb{C}}X$ and $\varphi_\alpha\in H^{p_\alpha,q_\alpha}(X;\mathbb{C})$; $\tilde t^\alpha_r=t^\alpha_r-\delta^1_r\delta^1_\alpha$ is the dilaton-shifted coordinate; $\varphi^\alpha=\sum_\gamma\eta^{\alpha\gamma}\varphi_\gamma$ raises an index by the inverse Poincaré pairing; $\tau_k(\varphi_\alpha)$ denotes the $k$-th descendant of $\varphi_\alpha$, and $\langle\!\langle\dots;\lambda_g\rangle\!\rangle_g$ are the genus-$g$ Hodge correlation functions of $X$ (generating functions of Hodge integrals over $\overline{\mathcal{M}}_{g,\bullet}(X,\bullet)$ with a $\lambda_g$ insertion). The sums over $r$ and $s$ run over the nonnegative integers (terms with a negative descendant level vanish by convention), and the sum over $\alpha$ runs over the basis of $H^*(X;\mathbb{C})$. The identity $\Theta_{g,n,m,\beta}=0$ means that every coefficient of this formal power series vanishes.

(2) The companion vanishing (Conjecture 3.8 of the source). For every $g\ge 0$ and $n\ge -1$,

$$\Psi_{g,n}\equiv 0,$$

where $\Psi_{g,n}$ is the companion expression defined in Theorem 3.7 of the source: a six-summand linear combination of the same Hodge correlation functions carrying one $\lambda_{g-1}$ or $\lambda_g$ insertion, with coefficients given by the same shifted elementary symmetric polynomials $e_{n+1-j}(\dots)$ and the dilaton-shifted coordinates, genus-splitting products over $g_1+g_2=g$, and explicit genus-0 and genus-1 correction terms involving integrals of powers of $c_1(X)$ and of $c_d(X)$, $c_{d-1}(X)$ over $X$:

$$\Psi_{g,n}=\sum_{j=0}^{n+1}\sum_{r,\alpha}e_{n+1-j}\big(r+b_\alpha-\tfrac{1}{2},\dots,r+n+b_\alpha-\tfrac{1}{2}\big)\,\tilde t^\alpha_r\,\langle\!\langle\tau_{r+n-j}(c_1(X)^{j}\cup\varphi_\alpha);\lambda_{g-1}\rangle\!\rangle_g$$ $$-\sum_{j=0}^{n+1}\sum_{r,\alpha}j\cdot e_{n+1-j}\big(r+b_\alpha-\tfrac{1}{2},\dots,r+n+b_\alpha-\tfrac{1}{2}\big)\,\tilde t^\alpha_r\,\langle\!\langle\tau_{r+n-j}(c_1(X)^{j-1}\cup\varphi_\alpha);\lambda_{g}\rangle\!\rangle_g$$ $$+\sum_{j=0}^{n+1}\sum_{s,\alpha}\sum_{g_1+g_2=g}(-1)^{-s-1}e_{n+1-j}\big(-s-\tfrac{3}{2}+b^\alpha,\dots,-s-\tfrac{3}{2}+n+b^\alpha\big)\,\langle\!\langle\tau_{-s-1+n-j}(c_1(X)^{j}\cup\varphi^\alpha);\lambda_{g_1-1}\rangle\!\rangle_{g_1}\,\langle\!\langle\tau_s(\varphi_\alpha);\lambda_{g_2}\rangle\!\rangle_{g_2}$$ $$+\tfrac{1}{2}\sum_{j=0}^{n+1}\sum_{s,\alpha}(-1)^{-s-1}e_{n+1-j}\big(-s-\tfrac{3}{2}+b^\alpha,\dots,-s-\tfrac{3}{2}+n+b^\alpha\big)\,\langle\!\langle\tau_{-s-1+n-j}(c_1(X)^{j}\cup\varphi^\alpha)\,\tau_s(\varphi_\alpha);\lambda_{g-1}\rangle\!\rangle_{g-1}$$ $$-\tfrac{1}{2}\sum_{j=0}^{n+1}\sum_{s,\alpha}\sum_{g_1+g_2=g}j\cdot(-1)^{-s-1}e_{n+1-j}\big(-s-\tfrac{3}{2}+b^\alpha,\dots,-s-\tfrac{3}{2}+n+b^\alpha\big)\,\langle\!\langle\tau_s(\varphi_\alpha);\lambda_{g_1}\rangle\!\rangle_{g_1}\,\langle\!\langle\tau_{-s-1+n-j}(c_1(X)^{j-1}\cup\varphi^\alpha);\lambda_{g_2}\rangle\!\rangle_{g_2}$$ $$-\delta^0_g\,\frac{n+1}{2}\sum_{\alpha,\beta}t^\alpha_0t^\beta_0\int_X c_1(X)^{n}\cup\varphi_\alpha\cup\varphi_\beta+\frac{\delta^1_g\delta^0_n}{48}\int_X\big((-d)c_d(X)-2c_1(X)\cup c_{d-1}(X)\big).$$

The two identities form one coupled system: by the conjugation formula quoted in the Background, they are together equivalent to the fiber degree-0 Virasoro constraints for $X\times\mathbb{P}^1$, and the genus-splitting sums tie each genus $g$ to all lower genera, so a complete solution settles both families.

The following cases are already settled and are not the target: genus 0 for arbitrary $X$ (both families); all genera for targets with semisimple quantum cohomology and for smooth algebraic curves (both families, via the Virasoro conjecture); genus 1 for arbitrary $X$ (the $\Theta$-family, via Pixton's double ramification formula). The unresolved content is the $\Theta$-family for $g\ge 2$ and the $\Psi$-family for $g\ge 1$ for smooth projective varieties with non-semisimple quantum cohomology. A complete answer is either (i) an unconditional proof for arbitrary smooth projective targets, or (ii) an explicit counterexample: a specific smooth projective variety $X$ and explicit indices $(g,n,m,\beta)$ together with a computed nonzero coefficient of $\Theta_{g,n,m,\beta}$ or $\Psi_{g,n}$. The adjacent extensions explicitly postponed by the source — a Givental-type reconstruction formula for the $\lambda_g$ ancestor potential beyond semisimple points of the theory, and an analogue of the conjecture for semisimple F-cohomological field theories (cohomological field theories without the metric axiom) and flat F-manifolds — are separate problems and not part of this one.

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