The Virasoro conjecture predicts that the total descendent Gromov–Witten potential of a smooth projective variety is annihilated by an explicit half-chain of Virasoro-algebra differential operators, giving universal linear recurrences among its descendent invariants. It is proven in genus 0 for all targets and in all genera for targets with generically semisimple quantum cohomology, but is open in general. Oberdieck's determination of all local Enriques Gromov–Witten invariants and unconditional proof of the Klemm–Mariño formula (arXiv:2305.11115; J. Eur. Math. Soc. 2025) explicitly records the Virasoro constraints for the Enriques surface as an interesting open question: the surface's torsion canonical class, purely even 12-dimensional rational cohomology, and expected non-semisimple quantum cohomology place it outside every class of targets where the constraints are known. The problem is to prove that the descendent Gromov–Witten potential of every Enriques surface satisfies the Eguchi–Hori–Xiong–Katz Virasoro constraints in all genera; the genus-2 case alone would remove the conjectural input from Maulik–Pandharipande's conditional proof of the genus-1 Klemm–Mariño formula.

Background

An Enriques surface is a smooth complex projective surface YY whose canonical bundle ωY\omega_Y has order two (ωY⊗2≅OY\omega_Y^{\otimes 2}\cong\mathcal{O}_Y but ωY≇OY\omega_Y\not\cong\mathcal{O}_Y) with H1(Y,OY)=0H^1(Y,\mathcal{O}_Y)=0; equivalently, it is the quotient of a K3 surface by a fixed-point-free involution. Its rational cohomology is purely even and 12-dimensional, H2(Y,Q)H^2(Y,\mathbb{Q}) carries the intersection form of the lattice U⊕E8(−1)U\oplus E_8(-1) (the hyperbolic plane plus the negative-definite E8E_8 root lattice), the Euler characteristic is χ(Y)=12\chi(Y)=12, and c1(Y)c_1(Y) is a torsion class, hence zero in rational cohomology.

Gromov–Witten invariants are defined on the moduli space M‾g,n(Y,β)\overline{M}_{g,n}(Y,\beta) of genus-gg, nn-marked stable maps (maps from nodal curves with finite automorphism group) representing a curve class β∈H2(Y,Z)\beta\in H_2(Y,\mathbb{Z}); this space carries a virtual fundamental class, a homology class of the 'expected' dimension produced by the deformation–obstruction theory, and integrating cohomology classes over it yields rational numbers. The insertions τk(γ)\tau_k(\gamma) (descendents, with γ∈H∗(Y,Q)\gamma\in H^*(Y,\mathbb{Q}) and k≥0k\ge 0) combine the insertion of γ\gamma at a marked point with the kk-th Chern class of the cotangent line there; the generating function of all such invariants is the total descendent potential ZYZ_Y. The Virasoro conjecture of Eguchi–Hori–Xiong and Katz predicts that ZYZ_Y is annihilated by explicit differential operators LkL_k, k≥−1k\ge-1, in the descendent variables: L−1L_{-1} encodes the string equation, L0L_0 encodes the dilaton and dimension equations and is a first-order operator together with a constant term built from characteristic numbers, and each LkL_k with k≥1k\ge1 is an explicit second-order operator built from the intersection pairing on H∗(Y,Q)H^*(Y,\mathbb{Q}), the cohomological degrees, and cup-product with c1(Y)c_1(Y); the operators satisfy the Virasoro algebra relations, and the complete formula is written out in Getzler's survey. The coefficient of ℏg−1\hbar^{g-1} in ZY−1LkZYZ_Y^{-1}L_kZ_Y depends only on invariants of genus at most gg, so the constraints can be pursued genus by genus. They are proven in genus 0 for all smooth projective varieties, and in all genera for targets whose quantum cohomology — the deformation of the cup product determined by genus-0 counts — is generically semisimple, the case of toric varieties and flag varieties via Givental's program completed by Teleman's classification of semisimple two-dimensional field theories; for targets with non-semisimple quantum cohomology the conjecture is, in the words of a 2025 treatment, 'widely open'.

The Enriques surface is a natural test case outside the semisimple world. Klemm and Mariño conjectured from string theory an explicit formula for the invariants Ng,β=∫[M‾g(Y,β)]vir(−1)g−1λg−1N_{g,\beta}=\int_{[\overline{M}_g(Y,\beta)]^{\mathrm{vir}}}(-1)^{g-1}\lambda_{g-1}, where λi\lambda_i is the ii-th Chern class of the Hodge bundle over the moduli space of stable maps; these compute the Gromov–Witten theory of the local Calabi–Yau threefold KYK_Y (total space of the canonical bundle). Maulik and Pandharipande proved the genus-1 case of this formula by torus localization on Y×P1Y\times\mathbb{P}^1, but the key relation for the genus-1 invariants of YY evaluates one term 'by the Virasoro constraints' — specifically the genus-2 constraints for YY, which are unproven — so their proof is conditional. Oberdieck later determined all local Enriques Gromov–Witten invariants (all genera and curve classes), proved the Klemm–Mariño formula unconditionally, and determined the Vafa–Witten invariants (invariants counting stable sheaves on the surface), by showing that the relative potentials of an elliptic Enriques fibration are quasi-Jacobi forms — elliptic analogues of quasimodular forms for the E8E_8 lattice, transforming like Jacobi forms up to quasimodular corrections — satisfying a holomorphic anomaly equation (a differential equation for the dependence on the quasimodular parameter), combined with a Toda-type wall-crossing formula (an identity tracking how sheaf-counting invariants change with the choice of stability condition) relating Pandharipande–Thomas stable-sheaf counts to two-dimensional Donaldson–Thomas invariants (the parallel enumerative theory counting stable sheaves). In that paper he nevertheless records that 'the proof of the Virasoro constraints for the Enriques surface is an interesting open question', noting that Maulik–Pandharipande's genus-1 argument 'remains conditional at this point'. No proof of generic semisimplicity of the quantum cohomology of Enriques surfaces is known, so the semisimple route cannot supply the constraints.

Problem Statement

Let YY be a smooth complex projective Enriques surface, and let ZY(t,ℏ)=exp⁡(∑g≥0ℏg−1FgY(t))Z_Y(\mathbf{t},\hbar)=\exp\big(\sum_{g\ge0}\hbar^{g-1}F_g^Y(\mathbf{t})\big) be its total descendent Gromov–Witten potential in the descendent variables tkat_k^a (k≥0k\ge0, aa indexing a basis {γa}\{\gamma_a\} of H∗(Y,Q)H^*(Y,\mathbb{Q})), the genus-gg potential FgYF_g^Y summing over all curve classes β∈H2(Y,Z)\beta\in H_2(Y,\mathbb{Z}), including the degree-zero class β=0\beta=0, modulo torsion. Let LkL_k, k≥−1k\ge-1, be the Virasoro operators of Eguchi–Hori–Xiong and Katz in the normalization written out by Getzler: L−1L_{-1} is the string operator, L0L_0 the dilaton/dimension operator, for k≥1k\ge1 the operator is an explicit second-order differential operator while L0L_0 is a first-order differential operator together with the constant term 148∫Y((3−r)cr(Y)−2c1(Y)cr−1(Y))\frac{1}{48}\int_Y\big((3-r)c_r(Y)-2c_1(Y)c_{r-1}(Y)\big) with r=dim⁡C(Y)=2r=\dim_{\mathbb{C}}(Y)=2, and these operators involve the intersection pairing gab=∫Yγa∪γbg_{ab}=\int_Y\gamma_a\cup\gamma_b, the degree shifts μa=pa−dim⁡C(Y)/2\mu_a=p_a-\dim_{\mathbb{C}}(Y)/2, and the matrix RR of cup-product with c1(Y)c_1(Y); for an Enriques surface c1(Y)c_1(Y) is 2-torsion and hence vanishes in H2(Y,Q)H^2(Y,\mathbb{Q}), so R=0R=0, the constant term of L0L_0 reduces to 148∫Yc2(Y)=148χ(Y)=14\frac{1}{48}\int_Yc_2(Y)=\frac{1}{48}\chi(Y)=\frac{1}{4}, the operators satisfy the Virasoro algebra relations, and they depend only on the pairing and the cohomological degrees.

Prove that ZYZ_Y satisfies the Virasoro constraints: for every k≥−1k\ge-1 and every genus g≥0g\ge0, the coefficient of ℏg−1\hbar^{g-1} in ZY−1LkZYZ_Y^{-1}L_kZ_Y vanishes. The claim is required for every Enriques surface YY and for arbitrary descendent insertions — not only for the special Hodge integrals Ng,β=∫[M‾g(Y,β)]vir(−1)g−1λg−1N_{g,\beta}=\int_{[\overline{M}_g(Y,\beta)]^{\mathrm{vir}}}(-1)^{g-1}\lambda_{g-1} that Oberdieck determined. Since the genus-0 constraints hold for all smooth projective varieties, the essential new content is genus g≥1g\ge1; the historically critical case is genus 2, which is exactly the conjectural input assumed in Maulik–Pandharipande's conditional proof of the genus-1 Klemm–Mariño formula, so a proof of the genus-2 constraints would in particular make that argument unconditional. This is the concrete target-specific instance of the general Virasoro conjecture recorded as an interesting open question by Oberdieck in the source paper.

References

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