The λg\lambda_g conjecture with target varieties (X. Wang, arXiv:2412.05287, 2024) generalizes the classical λg\lambda_g conjecture of Getzler–Pandharipande from the moduli space of curves to the Gromov–Witten theory of an arbitrary smooth projective target XX: it predicts two explicit infinite families of universal identities, Θg,n,m,β=0\Theta_{g,n,m,\beta}=0 and Ψg,n≡0\Psi_{g,n}\equiv 0, constraining genus-gg Hodge integrals with top (and next-to-top) Hodge-class insertions over moduli spaces of stable maps to XX. By a conjugation formula of the source, the two families together are equivalent to the fiber degree-0 Virasoro constraints for X×P1X\times\mathbb{P}^1, i.e. Virasoro-type constraints on the total Hodge-integral potential of XX. Both families are proved when the Virasoro conjecture holds for XX (in particular for targets with semisimple quantum cohomology and for smooth curves, in all genera) and in genus 0 for arbitrary XX; the Θ\Theta-family is also proved in genus 1 for arbitrary XX 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 Θ\Theta-family in genus ≥2\ge 2 and the Ψ\Psi-family in genus ≥1\ge 1 for targets with non-semisimple quantum cohomology. No unconditional proof and no counterexample is known: the problem is open.

Background

Let M‾g,n\overline{\mathcal{M}}_{g,n} be the moduli space of stable genus-gg curves with nn marked points. Its tautological classes include the ψ\psi-classes ψi=c1(Li)\psi_i=c_1(\mathcal{L}_i), where Li\mathcal{L}_i is the line bundle whose fiber over a curve (C;x1,…,xn)(C;x_1,\dots,x_n) is the cotangent line Txi∗CT^*_{x_i}C, and the Hodge bundle E\mathbb{E}, the rank-gg vector bundle with fiber H0(C,ωC)H^0(C,\omega_C); write λk=ck(E)\lambda_k=c_k(\mathbb{E}). The classical λg\lambda_g conjecture, formulated by Getzler and Pandharipande as a consequence of the Virasoro conjecture for P1\mathbb{P}^1, predicts a remarkably simple closed form — a single multinomial coefficient — for ψ\psi-class integrals coupled with the top Hodge class λg\lambda_g over M‾g,n\overline{\mathcal{M}}_{g,n}; it was proved by Faber and Pandharipande through the Gromov–Witten theory of P1\mathbb{P}^1 and reproved by Goulden, Jackson and Vakil through the ELSV formula, which expresses Hurwitz numbers (counts of branched covers of P1\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 XX of complex dimension dd, let M‾g,n(X,B)\overline{\mathcal{M}}_{g,n}(X,B) be the moduli space of stable maps from genus-gg nn-pointed curves to XX in homology class B∈H2(X;Z)B\in H_2(X;\mathbb{Z}). Descendant Gromov–Witten invariants are integrals of evaluation-map pullbacks of cohomology classes of XX and powers of the ψ\psi-classes against the virtual fundamental class; they are organized into generating functions FgF_g of the big phase space coordinates tnαt^\alpha_n, where α\alpha indexes a basis {φα}\{\varphi_\alpha\} of H∗(X;C)H^*(X;\mathbb{C}) with φα\varphi_\alpha of Hodge type (pα,qα)(p_\alpha,q_\alpha). Hodge integrals with target insert, in addition, the classes λk\lambda_k pulled back from M‾g,n\overline{\mathcal{M}}_{g,n} by the map that forgets the stable map; their correlation functions are denoted ⟨ ⁣⟨W1⋯Wk;λg⟩ ⁣⟩g\langle\!\langle W_1\cdots W_k;\lambda_g\rangle\!\rangle_g for vector fields WiW_i on the big phase space.

The Virasoro conjecture of Eguchi, Hori and Xiong predicts that the total descendant potential of XX is annihilated by an explicit sequence of differential operators LnL_n (n≥−1n\ge -1) forming half of the Virasoro algebra. It is proved for genus 00 for every target (Liu–Tian, Dubrovin–Zhang, Getzler), for all genera when the quantum cohomology of XX is semisimple — semisimplicity means that, generically, the deformed cup product on H∗(X)H^*(X) (the quantum product) turns H∗(X)H^*(X) into a direct sum of copies of C\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 λg\lambda_g-analogue of the Virasoro constraints for an arbitrary target. Writing bα=pα−12(d−1)b_\alpha=p_\alpha-\tfrac{1}{2}(d-1), bα=1−bαb^\alpha=1-b_\alpha, t~rα=trα−δr1δα1\tilde t^\alpha_r=t^\alpha_r-\delta^1_r\delta^1_\alpha (the dilaton shift), eke_k for the kk-th elementary symmetric polynomial, φα=∑γηαγφγ\varphi^\alpha=\sum_\gamma\eta^{\alpha\gamma}\varphi_\gamma for raising an index by the inverse Poincaré pairing, and c1(X)c_1(X) for the first Chern class, the paper defines explicit expressions Θg,n,m,β\Theta_{g,n,m,\beta} and Ψg,n\Psi_{g,n} in the Hodge correlation functions of XX and proves a conjugation formula (Theorem 3.7): for the fiber degree-0 descendant potential D~\widetilde{\mathcal{D}} of X×P1X\times\mathbb{P}^1 — which the same paper identifies with the total Hodge-integral potential of XX (Proposition 3.5) —

D~−1LnD~=∑g≥0ℏ2g−2(−1)g((−2)Ψg,n+∑m,βumβ Θg,n,m,β).\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 LnL_n denotes the Virasoro operators of X×P1X\times\mathbb{P}^1 (Proposition 3.3 of the source), H∈H2(P1;C)H\in H^2(\mathbb{P}^1;\mathbb{C}) is the hyperplane class, and {umβ}\{u^\beta_m\} are the flat coordinates of the big phase space of X×P1X\times\mathbb{P}^1 corresponding to the φβ⊗H\varphi_\beta\otimes H directions of the canonical basis {τm(φα⊗1),τm(φα⊗H)}\{\tau_m(\varphi_\alpha\otimes 1),\tau_m(\varphi_\alpha\otimes H)\}; they are formal variables independent of the trαt^\alpha_r, and neither Θg,n,m,β\Theta_{g,n,m,\beta} nor Ψg,n\Psi_{g,n} involves them.

Thus the pair of conjectures Θg,n,m,β=0\Theta_{g,n,m,\beta}=0 and Ψg,n≡0\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 XX (Theorem 1.2) — hence in all genera for semisimple targets and smooth curves; both hold in genus 00 for arbitrary XX (Theorem 1.3); and the Θ\Theta-family holds in genus 11 for arbitrary XX (Theorem 1.6), the proof inserting Pixton's double ramification formula λg=(−1)g2gPgg(0,…,0)\lambda_g=\tfrac{(-1)^g}{2^g}\mathsf{P}^g_g(0,\dots,0) in Rg(M‾g,n)R^g(\overline{\mathcal{M}}_{g,n}), where Pgg(a1,…,an)\mathsf{P}^g_g(a_1,\dots,a_n) is the codimension-gg Pixton class on M‾g,n\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 XX over C\mathbb{C}, with no semisimplicity hypothesis on the quantum cohomology of XX and without assuming the Virasoro conjecture.

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

Θg,n,m,β=0\Theta_{g,n,m,\beta}=0

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

Θg,n,m,β=∑j=0n+1∑r,αen+1−j(r+bα−12,…,r+n+bα−12) t~rα ⟨ ⁣⟨τr+n−j(c1(X)j∪φα) τm(φβ);λg⟩ ⁣⟩g\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 +∑j=0n+1en+1−j(m+bβ+12,…,m+n+bβ+12) ⟨ ⁣⟨τm+n−j(c1(X)j∪φβ);λg⟩ ⁣⟩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 +∑j=0n+1∑s,α(−1)−s−1 en+1−j(−s+bα−32,…,−s+n+bα−32)∑g1+g2=g⟨ ⁣⟨τm(φβ) τ−s−1+n−j(c1(X)j∪φα);λg1⟩ ⁣⟩g1 ⟨ ⁣⟨τs(φα);λg2⟩ ⁣⟩g2+\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} +δg0 δm0∑αt0α∫Xc1(X)n+1∪φα∪φβ.+\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 eke_k is the kk-th elementary symmetric polynomial, evaluated at the displayed list of n+1n+1 arguments (with e0=1e_0=1); bα=pα−12(d−1)b_\alpha=p_\alpha-\tfrac{1}{2}(d-1) and bα=1−bαb^\alpha=1-b_\alpha with d=dim⁡CXd=\dim_{\mathbb{C}}X and φα∈Hpα,qα(X;C)\varphi_\alpha\in H^{p_\alpha,q_\alpha}(X;\mathbb{C}); t~rα=trα−δr1δα1\tilde t^\alpha_r=t^\alpha_r-\delta^1_r\delta^1_\alpha is the dilaton-shifted coordinate; φα=∑γηαγφγ\varphi^\alpha=\sum_\gamma\eta^{\alpha\gamma}\varphi_\gamma raises an index by the inverse Poincaré pairing; τk(φα)\tau_k(\varphi_\alpha) denotes the kk-th descendant of φα\varphi_\alpha, and ⟨ ⁣⟨… ;λg⟩ ⁣⟩g\langle\!\langle\dots;\lambda_g\rangle\!\rangle_g are the genus-gg Hodge correlation functions of XX (generating functions of Hodge integrals over M‾g,∙(X,∙)\overline{\mathcal{M}}_{g,\bullet}(X,\bullet) with a λg\lambda_g insertion). The sums over rr and ss run over the nonnegative integers (terms with a negative descendant level vanish by convention), and the sum over α\alpha runs over the basis of H∗(X;C)H^*(X;\mathbb{C}). The identity Θg,n,m,β=0\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≥0g\ge 0 and n≥−1n\ge -1,

Ψg,n≡0,\Psi_{g,n}\equiv 0,

where Ψg,n\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 λg−1\lambda_{g-1} or λg\lambda_g insertion, with coefficients given by the same shifted elementary symmetric polynomials en+1−j(… )e_{n+1-j}(\dots) and the dilaton-shifted coordinates, genus-splitting products over g1+g2=gg_1+g_2=g, and explicit genus-0 and genus-1 correction terms involving integrals of powers of c1(X)c_1(X) and of cd(X)c_d(X), cd−1(X)c_{d-1}(X) over XX:

Ψg,n=∑j=0n+1∑r,αen+1−j(r+bα−12,…,r+n+bα−12) t~rα ⟨ ⁣⟨τr+n−j(c1(X)j∪φα);λg−1⟩ ⁣⟩g\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 −∑j=0n+1∑r,αj⋅en+1−j(r+bα−12,…,r+n+bα−12) t~rα ⟨ ⁣⟨τr+n−j(c1(X)j−1∪φα);λg⟩ ⁣⟩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 +∑j=0n+1∑s,α∑g1+g2=g(−1)−s−1en+1−j(−s−32+bα,…,−s−32+n+bα) ⟨ ⁣⟨τ−s−1+n−j(c1(X)j∪φα);λg1−1⟩ ⁣⟩g1 ⟨ ⁣⟨τs(φα);λg2⟩ ⁣⟩g2+\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} +12∑j=0n+1∑s,α(−1)−s−1en+1−j(−s−32+bα,…,−s−32+n+bα) ⟨ ⁣⟨τ−s−1+n−j(c1(X)j∪φα) τs(φα);λg−1⟩ ⁣⟩g−1+\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} −12∑j=0n+1∑s,α∑g1+g2=gj⋅(−1)−s−1en+1−j(−s−32+bα,…,−s−32+n+bα) ⟨ ⁣⟨τs(φα);λg1⟩ ⁣⟩g1 ⟨ ⁣⟨τ−s−1+n−j(c1(X)j−1∪φα);λg2⟩ ⁣⟩g2-\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} −δg0 n+12∑α,βt0αt0β∫Xc1(X)n∪φα∪φβ+δg1δn048∫X((−d)cd(X)−2c1(X)∪cd−1(X)).-\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×P1X\times\mathbb{P}^1, and the genus-splitting sums tie each genus gg 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 XX (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 XX (the Θ\Theta-family, via Pixton's double ramification formula). The unresolved content is the Θ\Theta-family for g≥2g\ge 2 and the Ψ\Psi-family for g≥1g\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 XX and explicit indices (g,n,m,β)(g,n,m,\beta) together with a computed nonzero coefficient of Θg,n,m,β\Theta_{g,n,m,\beta} or Ψg,n\Psi_{g,n}. The adjacent extensions explicitly postponed by the source — a Givental-type reconstruction formula for the λg\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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