The Virasoro conjecture for Gromov-Witten descendent potentials of arbitrary smooth projective varieties

The Virasoro conjecture of Eguchi, Hori and Xiong (1997) predicts that the total descendent potential of every smooth projective variety X is annihilated by a half-branch of Virasoro differential operators $L_n^X$, $n \geq -1$, generalizing Witten's conjecture (Kontsevich's theorem) from a point target to arbitrary targets.

Background

Gromov-Witten theory associates to a smooth projective variety X its stable maps: morphisms to X from nodal n-pointed genus-g curves representing a fixed curve class. These form a proper moduli space carrying a virtual fundamental class (the homology class that stands in for the fundamental class when the moduli space has excess dimension), and intersection numbers against this class are the Gromov-Witten invariants, the basic enumerative invariants of curves in X. The descendent invariants additionally insert powers of the classes $\psi_i = c_1(\mathcal{L}_i)$, where $\mathcal{L}_i$ is the line bundle on the moduli space whose fiber at a stable map is the cotangent line at the $i$-th marked point; these gravitational descendent classes record the geometry of the source curve, not just the target. All descendent invariants of X are packaged into a formal generating series, the total descendent potential $D_X = \exp\left(\sum_{g \geq 0} \hbar^{2g-2} F_g\right)$, where $F_g$ is the genus-$g$ Gromov-Witten free energy and $\hbar$ a formal parameter.

For a point target the moduli space reduces to the moduli space of stable curves and the descendent invariants are the intersection numbers of psi-classes. Witten conjectured, and Kontsevich proved, that the resulting potential is governed by the Korteweg-de Vries (KdV) hierarchy of integrable differential equations, equivalently annihilated by differential operators $L_n$, $n \geq -1$, forming a half-branch of the Virasoro algebra, the infinite-dimensional Lie algebra of conformal symmetry in two dimensions. Eguchi, Hori and Xiong proposed in 1997 the generalization to arbitrary (originally Fano) targets: for each smooth projective $X$ they introduced differential operators $L_n^X$ on the phase space of descendents, built from multiplication by the first Chern class $c_1(X)$ in cohomology, the grading (dilaton) data, and Bernoulli-number corrections, again forming a Virasoro half-branch, and conjectured that $L_n^X D_X = 0$ for all $n \geq -1$. The case $n = -1$ is the classical string equation and the case $n = 0$ follows from the classical divisor and dilaton equations together with the selection rule; both are known in general; the higher constraints form an infinite tower of universal linear relations among all-genus descendent invariants.

The conjecture's known scope is sharply divided by the structure of the quantum cohomology $QH^*(X)$, the cohomology ring deformed by genus-zero three-point invariants, which is a commutative Frobenius algebra (a commutative ring with an invariant nondegenerate bilinear form). When this algebra is semisimple, i.e. decomposes into a product of fields, Givental's quantization formula, completed by Teleman's classification of semisimple cohomological field theories, yields the conjecture in all genera. Genus zero is known for all compact symplectic targets, genus one and two for semisimple targets, and all genera is also known for toric Fano manifolds and for nonsingular curves. For targets with non-semisimple quantum cohomology, however (Calabi-Yau threefolds such as the quintic hypersurface in $\mathbf{P}^4$, abelian varieties, and most Fano complete intersections), no all-genus structural theorem of any kind is known, and even the formulation is simplest under the common simplifying assumption that the odd cohomology of X vanishes, since the operator normalization requires extra care otherwise.

Okounkov and Pandharipande (2001) connected the Gromov-Witten theory of the projective line $\mathbf{P}^1$ to Hurwitz numbers (enumerations of branched covers of a curve by ramification data), infinite-wedge/Fock-space methods, matrix models (integrals over matrix spaces whose perturbative expansions generate combinatorial series), and the Toda lattice hierarchy of integrable equations. For a general target $X$, the corresponding matrix model or integrable hierarchy is not known, and the Virasoro constraints of Eguchi, Hori and Xiong are among the most fundamental open questions in Gromov-Witten theory.

Problem Statement

Prove or refute the Virasoro conjecture for arbitrary smooth projective targets, the precise general conjecture, one of the most fundamental open questions in Gromov-Witten theory: for every smooth projective variety $X$, the total descendent potential $D_X$ satisfies

$$L_n^X D_X = 0 \quad \text{for all } n \geq -1,$$

where the operators $L_n^X$, $n \geq -1$, are the Virasoro differential operators of Eguchi, Hori and Xiong in their conventional normalization; equivalently, every genus-$g$ component of $D_X$ with $g \geq 0$ satisfies the genus-$g$ $L_n$-constraint for every $n \geq -1$. The fully general conjecture places no restriction on the cohomology of $X$, but it is commonly stated under the simplifying assumption that the odd cohomology of $X$ vanishes (an assumption imposed explicitly by Eguchi, Hori and Xiong in their general-manifold operator construction, and not automatic even for Fano targets), since the operator normalization requires extra care otherwise; a proof covering only the vanishing-odd-cohomology formulation counts as a partial result rather than a complete resolution.

The conjecture is established for genus 0 for all targets, and in all genera for targets with semisimple quantum cohomology (including $\mathbf{P}^1$ and toric Fano manifolds) and for nonsingular curves; the surviving open core is the case of targets whose quantum cohomology is non-semisimple, such as Calabi-Yau threefolds, abelian varieties, and most Fano complete intersections, beyond the known genus-0 results and the recent genus-1 results on the ambient state space of complete intersections. What remains open is the constraint for all smooth projective $X$ in all genera, and a single target variety, genus, and constraint index at which a coefficient of the corresponding $L_n^X D_X$ is nonzero refutes the conjecture.

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