Stationary closed formula for the reduced descendent Gromov-Witten theory of a K3 surface and its polynomiality structure (Oberdieck's Conjectures 0.2, 0.4 and 3.10)

The reduced Gromov-Witten theory of a K3 surface, defined against the reduced virtual class forced by the holomorphic symplectic form, is effectively computable in primitive curve classes by the Maulik-Pandharipande-Thomas algorithm, and its generating series are quasi-modular forms satisfying a proven holomorphic anomaly equation; yet essentially no closed formulas are known beyond the Bryan-Leung evaluation of point-insertion invariants. Oberdieck (2023 preprint; published 2025) conjectured an explicit closed formula for the entire stationary descendent theory: the partition function should be obtained by a single coefficient extraction from $(1/\Delta)\exp(\sum_{k\ge0}(\gamma_k,\beta)A_k+\sum_{k\ge0}(\gamma_k,1)B_k+\frac12\sum_{k,\ell\ge0}(\gamma_k\cdot\gamma_\ell)C_{k\ell})$, built from three universal quasi-modular series $A_k$, $B_k$, $C_{k\ell}$ defined by residues of the Weierstrass $\wp$-function, and, as a derived prediction, a striking polynomiality structure on normalized descendent invariants. Both structures remain unproved. The problem is to prove or refute the stationary closed formula (Conjecture 0.2 of the source) and to settle the polynomiality conjectures it predicts (Conjectures 0.4 and 3.10).

Background

A K3 surface is a simply connected compact complex surface with trivial canonical bundle; it carries a nowhere-vanishing holomorphic symplectic form. Its Gromov-Witten (GW) theory counts stable maps to the surface: for an effective curve class $\beta\in H_2(S,\mathbb Z)$, the descendent invariant is

$$\langle\tau_{k_1}(\gamma_1)\cdots\tau_{k_n}(\gamma_n)\rangle^S_{g,\beta}=\int_{[\overline M_{g,n}(S,\beta)]^{\mathrm{red}}}\prod_{i=1}^n\mathrm{ev}_i^*(\gamma_i)\psi_i^{k_i},$$

where $\overline M_{g,n}(S,\beta)$ is the moduli space of $n$-marked genus-$g$ degree-$\beta$ stable maps, $\mathrm{ev}_i$ evaluates a marked point, $\psi_i$ is the cotangent-line class at the $i$-th marking, $\gamma_i\in H^*(S)$, and $\tau_{k_i}(\gamma_i)$ is called a descendent insertion. Because the holomorphic 2-form of $S$ produces a trivial summand in the obstruction theory, the ordinary virtual fundamental class vanishes, and one integrates against the reduced virtual class obtained by removing that summand. The resulting reduced invariants are deformation invariant and depend on $(S,\beta)$ only through the square $\beta^2$, the divisibility of $\beta$ (the largest positive integer dividing $\beta$), and elementary pairings of the insertions. An invariant is stationary if every insertion has positive cohomological degree (no $\tau_k(1)$ factors), and primitive if $\beta$ is not a positive multiple of another integral class.

For primitive classes the theory is effectively determined: Maulik-Pandharipande-Thomas gave an algorithm, based on degenerations of elliptically fibered K3 surfaces and tautological relations on moduli spaces of curves, which computes every primitive descendent invariant and shows that the natural generating series are quasi-modular forms (polynomials in the Eisenstein series $G_{2k}(q)$, the classical modular $q$-series $-B_{2k}/(4k)+\sum_{n\ge1}\sigma_{2k-1}(n)q^n$ built from divisor-power sums; here $\Delta(q)=q\prod_{n\ge1}(1-q^n)^{24}$ is the modular discriminant). Oberdieck-Pixton proved that these series satisfy a holomorphic anomaly equation governing their $G_2$-derivative. In practice, however, the algorithm is extremely slow (the available implementation handles genus at most 3), and the only general closed formula known remained the Bryan-Leung evaluation of the point-insertion invariants

$$\langle\tau_0(p)^n\rangle^S_\beta=\mathrm{Coeff}_{q^{\beta^2/2}}\left[\frac{1}{\Delta(q)}\big(q\frac{d}{dq}G_2(q)\big)^n\right],$$

where $p\in H^4(S)$ is the class of a point. This contrasts sharply with the GW theory of an elliptic curve, where the Bloch-Okounkov theorem (which evaluates the generating series of all descendent invariants by an explicit closed formula) and the Virasoro constraints for target curves (an explicit system of differential constraints, coming from the Virasoro algebra of vector fields on the circle, that recursively remove the descendent insertions $\tau_k(1)$ and thus determine the whole theory) yield explicit closed formulas and a clean polynomiality structure for all descendent invariants. For K3 surfaces the standard Virasoro constraints fail for the reduced theory, and no replacement principle is known, even conjecturally in general; in particular the dependence on descendents of the unit, the factors $\tau_k(1)$, is a priori out of control. What the source paper proves is a recursion (its Theorem 0.4) that removes $\tau_k(1)$ insertions, reducing a large class of non-stationary invariants to stationary ones, subject to a linear-algebraic hypothesis: the existence of sufficiently many isotropic classes (classes of self-intersection $0$) in $H^2(S)$ orthogonal to $\beta$ and to the $H^2$-insertions (satisfied, for example, whenever the number of $H^2$-insertions of degree 1 is at most 5 and all insertions lie in $\{1,\beta,p\}$).

The central open prediction of the paper concerns the stationary theory. Write the Weierstrass $\wp$-function (the basic doubly-periodic meromorphic function on an elliptic curve) as a formal Laurent series in $z$ with quasi-modular coefficients, $\wp(z)=z^{-2}+2\sum_{k\ge4}G_k(q)z^{k-2}/(k-2)!$ (with the convention $G_k=0$ for $k$ odd), and define universal series by formal residues

$$A_k(q)=\frac{(-1)^k}{(2k+1)!!}\operatorname{Res}_{z=0}\big(\wp(z)-4G_2\big)^{k+1/2},\quad B_k(q)=\frac{(-1)^k}{(2k+3)!!}\operatorname{Res}_{z=0}\big(\wp(z)-4G_2\big)^{k+3/2}\big(\wp+2G_2\big),$$ $$C_{k\ell}(q)=\frac{(-1)^{k+\ell-1}}{(2k+1)!!(2\ell+1)!!}\operatorname{Res}_{z_1=z_2=0}\big(\wp(z_1)-4G_2\big)^{k+1/2}\big(\wp(z_2)-4G_2\big)^{\ell+1/2}\big(\wp(z_1-z_2)+2G_2\big),$$

where fractional powers are computed formally from the square root of $\wp-4G_2=z^{-2}(1-4G_2z^2+\cdots)$, and $(\alpha,\gamma)=\int_S\alpha\cup\gamma$ is the intersection pairing. Conjecture 0.2 of the paper asserts that the partition function $Z_\beta(\gamma_0,\gamma_1,\ldots)=\langle\exp(\sum_{k\ge0}\tau_k(\gamma_k))\rangle^S_\beta$, which encodes all stationary invariants in class $\beta$, is obtained by a single coefficient extraction from $(1/\Delta)\exp(\sum_k(\gamma_k,\beta)A_k+\sum_k(\gamma_k,1)B_k+\frac12\sum_{k,\ell}(\gamma_k\cdot\gamma_\ell)C_{k\ell})$; the Bryan-Leung formula is the special case where only point insertions appear, since $B_0=q\,dG_2/dq$. The paper proves uniqueness characterizations of the series $A_k$, $B_k$, $C_{k\ell}$ (quasi-modularity, the differential equations $dA_k/dG_2=2A_{k-1}$ and $dB_k/dG_2=2B_{k-1}-2A_{k+1}$, initial terms, and polynomiality of normalized coefficients), which reduce the conjecture to verifying a list of properties of the GW partition function. As a derived prediction, the paper conjectures a striking polynomiality (its Conjecture 0.4): if the descendent invariant is normalized by the explicit combinatorial prefactor $\prod_i(-4)^{k_i-1}(2k_i-1)!!\prod_i(-4)^{\ell_i}(2\ell_i+1)!!\prod_i(-4)^{m_i-1}(2m_i-1)!!\prod_i(-4)^{n_i}(2n_i+1)!!$ (with $k_i,m_i\ge1$), then in the stable range of large insertion indices the normalized stationary and, via the proven recursion, general descendent invariants agree with a polynomial of explicitly predicted degree in the indices; a strengthening (Conjecture 3.10 of the published version, numbered 3.8 in the first arXiv version) asserts polynomiality already when only a subset of the indices varies. Extensive numerical evaluations in the paper confirm both structures, but every one of them is conditional on Conjecture 0.2.

Problem Statement

Let $S$ be a K3 surface and $\beta\in H_2(S,\mathbb Z)$ a primitive effective curve class. For classes $\gamma_0,\gamma_1,\gamma_2,\ldots\in H^*(S)$ of positive cohomological degree, let

$$Z_\beta(\gamma_0,\gamma_1,\ldots)=\Big\langle\exp\Big(\sum_{k\ge0}\tau_k(\gamma_k)\Big)\Big\rangle^S_\beta=\sum_{m_0,m_1,\ldots\ge0}\frac{1}{m_0!m_1!\cdots}\langle\tau_0(\gamma_0)^{m_0}\tau_1(\gamma_1)^{m_1}\cdots\rangle^S_\beta$$

be the partition function encoding all stationary descendent invariants in class $\beta$, and let $A_k(q)$, $B_k(q)$, $C_{k\ell}(q)$ be the universal quasi-modular series defined in the Background by residues of the Weierstrass $\wp$-function. The problem is to determine the truth of the following conjectures of Oberdieck (arXiv:2308.09074; Port. Math. 82 (2025), 357-386):

(i) Conjecture 0.2 (stationary closed formula):

$$Z_\beta(\gamma_0,\gamma_1,\ldots)=\operatorname{Coeff}_{q^{\beta^2/2}}\left[\frac{1}{\Delta(q)}\exp\left(\sum_{k\ge0}(\gamma_k,\beta)A_k(q)+\sum_{k\ge0}(\gamma_k,1)B_k(q)+\frac12\sum_{k,\ell\ge0}(\gamma_k\cdot\gamma_\ell)C_{k\ell}(q)\right)\right],$$

where $(\alpha,\gamma)=\int_S\alpha\cup\gamma$.

(ii) Conjecture 0.4 (polynomiality): for $\beta^2\neq0$ and classes $\delta_1,\ldots,\delta_t\in H^2(S)$ with $\delta_i\cdot\beta=0$, the descendent invariants $\langle\prod_i\tau_{k_i}(1)\prod_i\tau_{\ell_i}(\beta)\prod_i\tau_{m_i}(\delta_i)\prod_i\tau_{n_i}(p)\rangle^S_\beta$ (with $k_i,m_i\ge1$), normalized by the prefactor $\prod_i(-4)^{k_i-1}(2k_i-1)!!\prod_i(-4)^{\ell_i}(2\ell_i+1)!!\prod_i(-4)^{m_i-1}(2m_i-1)!!\prod_i(-4)^{n_i}(2n_i+1)!!$, agree with a single polynomial $p$ of degree $\beta^2+2-2u-t+r$ (where $r,s,t,u$ count the insertions of $1$, $\beta$, the $\delta_i$, and the point class respectively) for all indices in the stable range $k_i\ge\beta^2/2+3-(u+t/2)$ and $\ell_i,m_i,n_i\ge\beta^2/2+1-(u+t/2)$.

(iii) Conjecture 3.10 of the published version (upgrade of (ii)): the same polynomiality holds when an arbitrary subset of the descendent indices is allowed to vary while the remaining indices are held fixed (at values $\ge1$ for the $k_i$ and $m_i$), with the same degree bound.

Specifically: prove or refute (i); and settle (ii) and (iii) (note that (i) implies (ii) in the stationary case by the paper's Remark 3.9, and (i) together with the paper's proven recursion for removing $\tau_k(1)$ insertions is the source of the full polynomiality prediction). A complete resolution supplies the first unconditional closed-form evaluation of the reduced descendent Gromov-Witten theory of K3 surfaces in primitive classes, and makes the paper's high-genus numerical evaluations unconditional.

References

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