The Virasoro conjecture of Eguchi-Hori-Xiong predicts that the total descendent potential of the Gromov-Witten theory of a smooth projective variety is annihilated by a half-branch of differential operators forming the Virasoro algebra. It is known for genus zero for arbitrary targets (Liu-Tian) and for all genera when the quantum cohomology is semisimple (Dubrovin-Zhang, Givental-Teleman), but the genus-one part for a general (non-semisimple) target remains open. Lin (2019) applied Getzler's genus-one relation to quantum powers of the Euler vector field and proved that all composites of the quantum volume element with cohomology-class directions annihilate the obstruction ⟨ ⁣⟨Ek⟩ ⁣⟩1−Φk\langle\!\langle E^{k}\rangle\!\rangle_{1}-\Phi_{k}, yet the full vanishing needed for the conjecture is not established. This problem asks for a proof, or a refutation, of the genus-one Virasoro conjecture for every smooth projective variety without any semisimplicity assumption; a 2025 paper explicitly records that this vanishing is still open for general target varieties.

Background

Gromov-Witten (GW) invariants of a smooth projective variety XX are the rational numbers obtained by integrating cohomology classes over the moduli spaces M‾g,n(X,d)\overline{\mathcal{M}}_{g,n}(X,d) of stable maps from genus-gg, nn-pointed curves into XX; they are organized into generating functions FgF_{g} (the genus-gg potential) of so-called descendent invariants, in which the insertion at each marked point may be multiplied by powers of the cotangent-line classes ψi\psi_{i}. The Eguchi-Hori-Xiong Virasoro conjecture predicts that the total descendent potential Z=exp⁡(∑g≥0ℏg−1Fg)Z=\exp(\sum_{g\ge 0}\hbar^{g-1}F_{g}), a formal function on the big phase space (the informally infinite-dimensional space with coordinates labeled by a basis γ1=1,γ2,…,γN\gamma_{1}=\mathbf{1},\gamma_{2},\dots,\gamma_{N} of H∗(X;C)H^{*}(X;\mathbb{C}) and by all descendent degrees), is annihilated by a sequence of differential operators LnL_{n}, n≥−1n\ge -1, satisfying [Lm,Ln]=(m−n)Lm+n[L_{m},L_{n}]=(m-n)L_{m+n}, one half of the Virasoro algebra. When XX is a point this reduces to the Witten conjecture on intersection numbers on the moduli space of curves, proved by Kontsevich. Writing LnZ=∑g≥0ℏg−2Ωg,nL_{n}Z=\sum_{g\ge 0}\hbar^{g-2}\Omega_{g,n}, the genus-gg Virasoro conjecture is the claim Ωg,n=0\Omega_{g,n}=0 for all n≥−1n\ge -1; the cases n=−1n=-1 (the string equation) and n=0n=0 (Hori) hold for every target, so the genus-one conjecture concerns Ω1,n=0\Omega_{1,n}=0 for n≥1n\ge 1.

The genus-zero part holds for arbitrary targets (Liu-Tian), and the conjecture is completely settled for targets whose quantum cohomology is semisimple — meaning the quantum product, the deformation of the cup product defined by genus-zero three-point GW invariants through the WDVV associativity relation, admits a basis of idempotents — by Dubrovin-Zhang in low genera and by Givental's formula as established by Teleman in all genera; Okounkov and Pandharipande proved it for all nonsingular target curves. Beyond these classes, already at genus one the conjecture is open, and Xiaobo Liu reduced it to an equation on the small phase space (the finite-dimensional space with coordinates on H∗(X;C)H^{*}(X;\mathbb{C})): let EE be the Euler vector field on the small phase space, E=c1(X)+∑α(b1+1−bα)tαγαE=c_{1}(X)+\sum_{\alpha}(b_{1}+1-b_{\alpha})t^{\alpha}\gamma_{\alpha}, where tαt^{\alpha} are the coordinates dual to the basis, γα∈Hpα,qα(X;C)\gamma_{\alpha}\in H^{p_{\alpha},q_{\alpha}}(X;\mathbb{C}), bα=pα−d−12b_{\alpha}=p_{\alpha}-\frac{d-1}{2}, and d=dim⁡CXd=\dim_{\mathbb{C}}X; let quantum powers of EE be defined recursively by E0=γ1E^{0}=\gamma_{1} and Ek+1=Ek∘EE^{k+1}=E^{k}\circ E using the quantum product, let ⟨ ⁣⟨Ek⟩ ⁣⟩1\langle\!\langle E^{k}\rangle\!\rangle_{1} denote the genus-one potential with the kk-th quantum power of EE inserted, and let Φk\Phi_{k} be the explicit universal expressions in genus-zero correlators defined recursively in Liu's work; then the genus-one Virasoro conjecture holds for XX if and only if ⟨ ⁣⟨E2⟩ ⁣⟩1=Φ2\langle\!\langle E^{2}\rangle\!\rangle_{1}=\Phi_{2} (Liu 2001), and equivalently, for any single fixed k≥2k\ge 2, if and only if ⟨ ⁣⟨Ek⟩ ⁣⟩1=Φk\langle\!\langle E^{k}\rangle\!\rangle_{1}=\Phi_{k} (Liu, Seoul proceedings 2001). Since EE acts on the obstruction ⟨ ⁣⟨E2⟩ ⁣⟩1−Φ2\langle\!\langle E^{2}\rangle\!\rangle_{1}-\Phi_{2} by homogeneity, the conjecture is further reduced to the vanishing of its first derivatives υ(⟨ ⁣⟨E2⟩ ⁣⟩1−Φ2)=0\upsilon(\langle\!\langle E^{2}\rangle\!\rangle_{1}-\Phi_{2})=0 along every class υ∈H∗(X;C)\upsilon\in H^{*}(X;\mathbb{C}). Liu verified the unit direction (the cases k=0,1k=0,1), and later the quantum volume element direction, where the quantum volume element Δ=∑μγμ∘γμ\Delta=\sum_{\mu}\gamma_{\mu}\circ\gamma^{\mu} (the sum over a basis and its metric dual, with the quantum product) is a distinguished vector field on the small phase space.

Getzler's genus-one relation is the universal linear relation among genus-one descendent GW invariants induced by a tautological relation on the moduli space M‾1,4\overline{\mathcal{M}}_{1,4}; it holds for every target. Lin (2019) applied this relation systematically to quantum powers of EE, derived explicit universal equations for primary GW invariants, and proved (Theorem 4.6) that (Δ∘γα)(⟨ ⁣⟨Ek⟩ ⁣⟩1−Φk)=0(\Delta\circ\gamma_{\alpha})(\langle\!\langle E^{k}\rangle\!\rangle_{1}-\Phi_{k})=0 for all k≥0k\ge 0 and all α\alpha, generalizing Liu's Δ\Delta-direction result to all composites of Δ\Delta with cohomology directions. Lin also observed that invertibility of the symmetric matrix Aαβ=∑σ,μ⟨ ⁣⟨γαγσγμ⟩ ⁣⟩0⟨ ⁣⟨γσγμγβ⟩ ⁣⟩0A_{\alpha\beta}=\sum_{\sigma,\mu}\langle\!\langle\gamma_{\alpha}\gamma^{\sigma}\gamma^{\mu}\rangle\!\rangle_{0}\langle\!\langle\gamma_{\sigma}\gamma_{\mu}\gamma_{\beta}\rangle\!\rangle_{0} built from genus-zero three-point functions would immediately force γα(⟨ ⁣⟨Ek⟩ ⁣⟩1−Φk)=0\gamma_{\alpha}(\langle\!\langle E^{k}\rangle\!\rangle_{1}-\Phi_{k})=0 for all α\alpha, and hence the conjecture, but invertibility is known only in the semisimple situation (via idempotents), where it recovers Dubrovin-Zhang's theorem. Controlling higher directional derivatives, or equivalently closing the resulting system of Getzler-derived universal equations on the full set of genus-one primary correlators in the non-semisimple case, is left open; Wang (2025) observes that genus-one tautological relations alone appear insufficient to obtain the missing vanishing.

Problem Statement

Prove, or refute, the genus-one Virasoro conjecture for arbitrary targets: for every smooth projective variety XX (with no assumption that its quantum cohomology is semisimple), the genus-one Virasoro constraints Ω1,n=0\Omega_{1,n}=0 hold for all n≥1n\ge 1. By Xiaobo Liu's published reductions, an equivalent and fully concrete target is any one of the following, each to be established for all XX: (i) ⟨ ⁣⟨E2⟩ ⁣⟩1=Φ2\langle\!\langle E^{2}\rangle\!\rangle_{1}=\Phi_{2}, or ⟨ ⁣⟨Ek⟩ ⁣⟩1=Φk\langle\!\langle E^{k}\rangle\!\rangle_{1}=\Phi_{k} for any single fixed k≥2k\ge 2; (ii) υ(⟨ ⁣⟨E2⟩ ⁣⟩1−Φ2)=0\upsilon(\langle\!\langle E^{2}\rangle\!\rangle_{1}-\Phi_{2})=0 for every cohomology class υ∈H∗(X;C)\upsilon\in H^{*}(X;\mathbb{C}). Currently the unit direction, the quantum volume element direction Δ\Delta, and all composite directions Δ∘γα\Delta\circ\gamma_{\alpha} (Lin 2019) are known to annihilate the obstruction, and invertibility of the genus-zero matrix Aαβ=∑σ,μ⟨ ⁣⟨γαγσγμ⟩ ⁣⟩0⟨ ⁣⟨γσγμγβ⟩ ⁣⟩0A_{\alpha\beta}=\sum_{\sigma,\mu}\langle\!\langle\gamma_{\alpha}\gamma^{\sigma}\gamma^{\mu}\rangle\!\rangle_{0}\langle\!\langle\gamma_{\sigma}\gamma_{\mu}\gamma_{\beta}\rangle\!\rangle_{0} would close the problem but is established only in the semisimple case. Results restricted to semisimple targets, nonsingular curves, or products with a semisimple factor are known.

References

  1. T. Eguchi, K. Hori, C. Xiong (1997). Quantum cohomology and Virasoro algebra. Physics Letters B 402(1-2), 71-80. DOI 10.1016/S0370-2693(97)00401-2, https://doi.org/10.1016/S0370-2693(97)00401-2
  2. E. Witten (1990). Two-dimensional gravity and intersection theory on moduli space. Surveys in Differential Geometry 1, 243-310. DOI 10.4310/sdg.1990.v1.n1.a5, https://doi.org/10.4310/sdg.1990.v1.n1.a5
  3. M. Kontsevich (1992). Intersection theory on the moduli space of curves and the matrix Airy function. Communications in Mathematical Physics 147(1), 1-23. DOI 10.1007/BF02099526, https://doi.org/10.1007/BF02099526
  4. X. Liu, G. Tian (1998). Virasoro constraints for quantum cohomology. Journal of Differential Geometry 50(3), 537-590. DOI 10.4310/jdg/1214424970, https://doi.org/10.4310/jdg/1214424970
  5. E. Getzler (1997). Intersection theory on M‾1,4\overline{\mathcal{M}}_{1,4} and elliptic Gromov-Witten invariants. Journal of the American Mathematical Society 10(4), 973-998. DOI 10.1090/S0894-0347-97-00246-4, https://doi.org/10.1090/S0894-0347-97-00246-4
  6. E. Getzler (1999). The Virasoro conjecture for Gromov-Witten invariants. Contemporary Mathematics 241, 147-176. DOI 10.1090/conm/241/03634, https://doi.org/10.1090/conm/241/03634
  7. B. Dubrovin, Y. Zhang (1999). Frobenius manifolds and Virasoro constraints. Selecta Mathematica (N.S.) 5, 423-466. DOI 10.1007/s000290050053, https://doi.org/10.1007/s000290050053
  8. X. Liu (2001). Elliptic Gromov-Witten invariants and Virasoro conjecture. Communications in Mathematical Physics 216(3), 705-728. DOI 10.1007/s002200000357, https://doi.org/10.1007/s002200000357
  9. X. Liu (2001). Genus-1 Virasoro conjecture on the small phase space. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), 265-279. World Scientific, River Edge, NJ. DOI 10.1142/9789812799821_0008, https://doi.org/10.1142/9789812799821_0008
  10. C. Teleman (2012). The structure of 2D semi-simple field theories. Inventiones Mathematicae 188(3), 525-588. DOI 10.1007/s00222-011-0352-5, https://doi.org/10.1007/s00222-011-0352-5
  11. A. Okounkov, R. Pandharipande (2006). Virasoro constraints for target curves. Inventiones Mathematicae 163(1), 47-108. DOI 10.1007/s00222-005-0455-y, https://doi.org/10.1007/s00222-005-0455-y
  12. X. Liu (2013). Genus-1 Virasoro conjecture along quantum volume direction. International Mathematics Research Notices 2013(8), 1747-1760. DOI 10.1093/imrn/rns023, https://doi.org/10.1093/imrn/rns023
  13. Y. Lin (2019). Getzler relation and Virasoro conjecture for genus one. Letters in Mathematical Physics 109(9), 2083-2132. DOI 10.1007/s11005-019-01176-w, https://doi.org/10.1007/s11005-019-01176-w (preprint arXiv:1805.10952, https://arxiv.org/abs/1805.10952)
  14. S. Guo, Q. Zhang (2025). Cohomological Field Theory with vacuum and its Virasoro constraints. arXiv:2502.18895, https://arxiv.org/abs/2502.18895
  15. X. Wang (2025). Virasoro constraints for Hodge integrals. arXiv:2506.10033, https://arxiv.org/abs/2506.10033
  16. H.-H. Tseng (2026). A note on Virasoro constraints for products. arXiv:2603.22486, https://arxiv.org/abs/2603.22486