Genus-1 Virasoro conjecture for Gromov–Witten invariants without semisimplicity: vanishing of the obstruction $\Psi$ on the small phase space

The Virasoro conjecture of Eguchi–Hori–Xiong and Katz predicts that the generating functions of Gromov–Witten invariants of every smooth projective variety are annihilated by a half-branch of Virasoro operators. Its genus-1 part is equivalent (X. Liu, 2001) to a single scalar equation $\Psi=0$ on the small phase space, where $\Psi$ is an explicit combination of genus-0 and genus-1 primary correlators. The genus-1 conjecture is known for targets with semisimple quantum cohomology (Dubrovin–Zhang; Teleman), and the source paper (X. Liu, IMRN 2013) proved that the quantum volume direction annihilates $\Psi$ ($\Delta\Psi=0$) for all smooth projective varieties without any semisimplicity assumption, later extended by Lin (2019). The open problem is to prove $\Psi=0$ for every smooth projective variety without semisimplicity — equivalently, to exhibit a generating family of tangent directions $v$ with $v\Psi=0$ — or to refute the conjecture by a certified computation exhibiting $\Psi\neq0$.

Background

Gromov–Witten (GW) invariants of a smooth projective variety $M$ are rational numbers obtained as integrals over moduli spaces of stable maps from algebraic curves to $M$; the genus-$g$ primary invariants assemble into a generating function $F_g$ depending on coordinates $t_1,\dots,t_N$ on the cohomology space $H^*(M;\mathbb{C})$, called the small phase space. Throughout this line of work the standing convention is $H^{\mathrm{odd}}(M;\mathbb{C})=0$, with a fixed basis $\{\gamma_1=1,\gamma_2,\dots,\gamma_N\}$, $\gamma_\alpha\in H^{p_\alpha,q_\alpha}(M)$; genus-1 descendant invariants (i.e. invariants with cotangent-line classes on the moduli spaces of stable maps as insertions) reduce to primary ones by the genus-1 topological recursion relation, so the primary correlators $\langle\!\langle v_1\cdots v_k\rangle\!\rangle_g=\sum f_{\alpha_1}\cdots f_{\alpha_k}\,\partial^k F_g/\partial t_{\alpha_1}\cdots\partial t_{\alpha_k}$ (for vector fields $v_i=\sum_\alpha f_{\alpha i}\gamma_\alpha$) carry all genus-1 information. The quantum product $v_1\circ v_2:=\langle\!\langle v_1 v_2 \gamma^\alpha\rangle\!\rangle_0\gamma_\alpha$, with indices raised by the Poincaré intersection form, deforms the cup product and gives $H^*(M;\mathbb{C})$ the structure of a Frobenius manifold (a commutative algebra with a flat compatible metric and an Euler field, the standard axiomatization of two-dimensional topological field theory); the associativity of $\circ$ is the WDVV equation. The Euler vector field is $E=c_1(M)+\sum_\alpha (b_1+1-b_\alpha)t_\alpha\gamma_\alpha$ with $b_\alpha=p_\alpha-\frac{d-1}{2}$ and $d=\dim_{\mathbb{C}}M$. Quantum cohomology is called semisimple when the algebra $(H^*(M;\mathbb{C}),\circ)$ is semisimple at a generic point of the small phase space; in that case idempotents give canonical coordinates, the backbone of the Dubrovin–Zhang and Teleman classification machinery.

The Virasoro conjecture (Eguchi–Hori–Xiong, also S. Katz) predicts that the generating functions of descendant GW invariants of every smooth projective variety are annihilated by a sequence of differential operators $L_n$, $n\geq -1$, forming a half-branch of the Virasoro algebra; when $M$ is a point it reduces to Witten's conjecture on intersection numbers of $\overline{\mathcal{M}}_{g,n}$, proved by Kontsevich. The genus-0 part holds for all compact symplectic manifolds (Liu–Tian 1998), and for semisimple quantum cohomology the full conjecture in all genera follows from Teleman's classification of semisimple 2D field theories (the genus-1 semisimple case was proved earlier by Dubrovin–Zhang using canonical coordinates). Beyond these cases nothing general is known; in particular the physically motivated Calabi–Yau targets (smooth projective varieties with trivial canonical class) are expected to be non-semisimple.

For genus 1 the conjecture admits a drastic reduction (X. Liu, Comm. Math. Phys. 2001): it is equivalent to the single scalar equation

$$\Psi=0$$

on the small phase space, where

$$\Psi:=\langle\!\langle E\circ E\rangle\!\rangle_1+\frac{1}{24}\sum_\alpha\langle\!\langle E\, E\, \gamma_\alpha\gamma^\alpha\rangle\!\rangle_0-\frac{1}{2}\sum_\alpha\Big(b_\alpha(1-b_\alpha)-\frac{b_1+1}{6}\Big)\langle\!\langle\gamma_\alpha\gamma^\alpha\rangle\!\rangle_0 .$$

Moreover $E\Psi=\Psi$, so $\Psi=0$ follows once $v\Psi=0$ is known for vector fields $v$ generating the tangent directions of the small phase space. The string equation gives the identity direction $v=\gamma_1$; the source paper of this problem (X. Liu, IMRN 2013) proved the quantum volume direction $v=\Delta:=\gamma^\alpha\circ\gamma_\alpha$ — the cup-product volume direction deformed by the quantum product — with no semisimplicity assumption. The universal ingredient behind such identities is Getzler's elliptic relation, the tautological relation on the moduli space $\overline{\mathcal{M}}_{1,4}$ of genus-1 stable curves with 4 marked points. In the semisimple case Getzler's relation can be solved in canonical coordinates; without semisimplicity no method is known to extract from it, or from any other tautological relation, enough independent directional identities to close the conjecture.

Problem Statement

Prove, or disprove, the genus-1 Virasoro conjecture for arbitrary smooth projective varieties without any semisimplicity assumption, in the reduced form established in the source line of work. Let $M$ be a smooth projective variety with $H^{\mathrm{odd}}(M;\mathbb{C})=0$ (the standing convention of this literature; extending the statement beyond this convention is part of the general conjecture), with basis $\{\gamma_\alpha\}$, Euler vector field $E$, quantum product $\circ$, and the obstruction function

$$\Psi:=\langle\!\langle E\circ E\rangle\!\rangle_1+\frac{1}{24}\sum_\alpha\langle\!\langle E\, E\, \gamma_\alpha\gamma^\alpha\rangle\!\rangle_0-\frac{1}{2}\sum_\alpha\Big(b_\alpha(1-b_\alpha)-\frac{b_1+1}{6}\Big)\langle\!\langle\gamma_\alpha\gamma^\alpha\rangle\!\rangle_0,$$

with $b_\alpha=p_\alpha-\frac{d-1}{2}$, $d=\dim_{\mathbb{C}}M$, and repeated indices summed over the basis (X. Liu's normalization; cf. the Background for all conventions). Establish that $\Psi=0$ identically on the small phase space $H^*(M;\mathbb{C})$ for every such $M$. Equivalently: exhibit a family of vector fields $v$ whose values generate the tangent space of the small phase space at every point, and prove $v\Psi=0$ for each of them, going beyond the known annihilators — the identity direction $\gamma_1$ (string equation), the quantum volume element $\Delta=\gamma^\alpha\circ\gamma_\alpha$ (Liu 2013), and Lin's 2019 extension of these vanishings to the directions $\Delta\circ\gamma_\alpha$ acting on the higher $E^k$-obstructions. The conjecture is refuted by an explicit smooth projective variety $M$ and a point of its small phase space at which $\Psi\neq0$ is established from certified genus-0 and genus-1 primary Gromov–Witten correlators — that is, from the finitely many primary correlators entering $\Psi$ at that point, computed exactly or with rigorous error bounds certifying $\Psi\neq0$.

References

1. Xiaobo 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 (preprint: arXiv:1106.3735, https://arxiv.org/abs/1106.3735)
2. Tohru Eguchi, Kentaro Hori, Chuan-Sheng Xiong (1997). Quantum cohomology and Virasoro algebra. Physics Letters B 402, 71–80. DOI: 10.1016/S0370-2693(97)00401-2. https://doi.org/10.1016/S0370-2693(97)00401-2
3. E. Getzler (1997). Intersection theory on $\overline{\mathcal{M}}_{1,4}$ and elliptic Gromov–Witten invariants. Journal of the American Mathematical Society 10, 973–998. DOI: 10.1090/S0894-0347-97-00246-4. https://doi.org/10.1090/S0894-0347-97-00246-4
4. Boris Dubrovin, Youjin Zhang (1998). Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation. Communications in Mathematical Physics 198, 311–361. DOI: 10.1007/s002200050480. https://doi.org/10.1007/s002200050480
5. Xiaobo Liu (2001). Elliptic Gromov–Witten invariants and Virasoro conjecture. Communications in Mathematical Physics 216, 705–728. DOI: 10.1007/s002200000357. https://doi.org/10.1007/s002200000357 (preprint: arXiv:math/9907113, https://arxiv.org/abs/math/9907113)
6. Xiaobo Liu, Gang Tian (1998). Virasoro constraints for quantum cohomology. Journal of Differential Geometry 50, 537–590. DOI: 10.4310/jdg/1214424970. https://doi.org/10.4310/jdg/1214424970 (preprint: arXiv:math/9806028, https://arxiv.org/abs/math/9806028)
7. Constantin Teleman (2012). The structure of 2D semi-simple field theories. Inventiones Mathematicae 188, 525–588. DOI: 10.1007/s00222-011-0352-5. https://doi.org/10.1007/s00222-011-0352-5
8. Yijie Lin (2019). Getzler relation and Virasoro conjecture for genus one. Letters in Mathematical Physics 109, 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)
9. Tom Coates, Alexander Givental, Hsian-Hua Tseng (2024). Virasoro Constraints for Toric Bundles. Forum of Mathematics, Pi 12, e2. DOI: 10.1017/fmp.2024.2. https://doi.org/10.1017/fmp.2024.2 (preprint: arXiv:1508.06282, https://arxiv.org/abs/1508.06282)
10. Shuai Guo, Qingsheng Zhang (2025). Cohomological Field Theory with vacuum and its Virasoro constraints. arXiv:2502.18895. https://arxiv.org/abs/2502.18895
11. Xin Wang (2024). Hodge integrals and $\lambda_g$ conjecture with target varieties. arXiv:2412.05287. https://arxiv.org/abs/2412.05287
12. Xin Wang (2025). Virasoro constraints for Hodge integrals. arXiv:2506.10033. https://arxiv.org/abs/2506.10033
13. Hsian-Hua Tseng (2026). A note on Virasoro constraints for products. arXiv:2603.22486. https://arxiv.org/abs/2603.22486
14. Xiaobo Liu, Chongyu Wang (2026). Complexity of quantum cohomology. arXiv:2603.02736. https://arxiv.org/abs/2603.02736
15. Maxim Kontsevich (1992). Intersection theory on the moduli space of curves and the matrix Airy function. Communications in Mathematical Physics 147, 1–23. DOI: 10.1007/BF02099526. https://doi.org/10.1007/BF02099526
16. E. Witten (1990). Two-dimensional gravity and intersection theory on moduli space. Surveys in Differential Geometry 1(1), 243–310. DOI: 10.4310/SDG.1990.v1.n1.a5. https://doi.org/10.4310/SDG.1990.v1.n1.a5