Forbes–Jinzenji conjecture: $QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k)))\cong QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1)))$ for all integers $k$

For each integer $k$, let $G_k=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k))$ be the $\mathbb{P}^2$-bundle over $\mathbb{P}^1$. Forbes and Jinzenji (2007) conjectured that for all $k\in\mathbb{Z}$ the small quantum cohomology rings satisfy $QH^*(G_k)\cong QH^*(G_{-1})$, the isomorphism being taken after a mirror-map change of the two quantum variables and a linear change of the natural cohomology basis. The cases $k=-1,0$ follow from Givental's mirror theorem for toric varieties, and $k=1,2$ were verified by Forbes–Jinzenji through Birkhoff factorization of the I-function; since $G_k\cong G_{-2-k}$, the open content is $k\ge 3$. The conjecture is the compact remnant of the physics observation that local mirror symmetry of the curves $\mathcal{O}(k)\oplus\mathcal{O}(-2-k)\to\mathbb{P}^1$ is independent of $k$, and it sits outside the reach of the known invariance theorems: the Lee–Lin–Qu–Wang flop-invariance series applies to K-equivalent pairs (varieties related by crepant birational maps, which this family is not), McLean's theorem requires Calabi–Yau varieties, and the Iritani–Koto splitting of the quantum cohomology D-module of a projective bundle requires the dual bundle to be globally generated, which fails exactly for $k\ge 1$.

Background

Let $X$ be a smooth complex projective variety. The small quantum cohomology ring $QH^*(X)$ deforms the classical cohomology ring: its structure constants are genus-zero three-point Gromov–Witten invariants, which are intersection numbers on moduli spaces of stable maps $f:\mathbb{P}^1\to X$ of fixed homology class $\beta$, recorded in formal Novikov variables $q^\beta$ that keep track of the curve class. For toric varieties — varieties containing a dense algebraic torus, and therefore describable by explicit combinatorial data — Givental's mirror theorem computes $QH^*(X)$ from an explicit hypergeometric I-function: after a change of variables called the mirror map, the I-function becomes the J-function, whose coefficients determine the quantum product and hence the connection form of the quantum connection, the flat connection on the trivial bundle with fiber $H^*(X)$ used to organize genus-zero Gromov–Witten theory.

For a rank-3 vector bundle $E\to\mathbb{P}^1$, the projectivization $\mathbb{P}(E)$ is a $\mathbb{P}^2$-bundle over $\mathbb{P}^1$; by the projective bundle formula its cohomology is a free module over $H^*(\mathbb{P}^1)$ on $1,\xi,\xi^2$, where $\xi$ is the relative hyperplane class, with a relation determined by the Chern classes of $E$. Write $G_k=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k))$. Two such bundles are isomorphic exactly when their splitting degrees agree up to a common twist, so $G_k\cong G_{-2-k}$ and the distinct members are $k=-1,0,1,2,\dots$; each is a smooth projective toric threefold with two Kähler parameters, with divisor classes $p_1,p_2$ dual to the base and fiber curve classes. A computation in the source paper gives $c_1(G_k)=-3k\,p_1+3\,p_2$, so $G_{-1}$ and $G_0$ satisfy the semi-positivity condition $c_1\ge 0$ (equivalently, the relevant divisor classes are nef, i.e. numerically effective — non-negative on every effective curve) under which Givental's original framework applies, while for $k\ge 1$ the class $-3k\,p_1$ is not nef, the I-function contains positive powers of the parameter $\hbar$, and the J-function can only be reached through Birkhoff factorization — a gauge normalization introduced by Coates and Givental in their quantum Riemann–Roch theorem. Removing these positive $\hbar$-powers and controlling the resulting mirror map is the technical source of difficulty.

The conjecture arose from local mirror symmetry of curves. The non-compact Calabi–Yau threefolds (threefolds with trivial canonical class) $X_k=\operatorname{tot}(\mathcal{O}(k)\oplus\mathcal{O}(-2-k)\to\mathbb{P}^1)$ are the standard local curve models, and both physics calculations (Chiang–Klemm–Yau–Zaslow) and the local Gromov–Witten theory of curves (Bryan–Pandharipande) expect their A-models to be independent of $k$; $G_k$ is a natural projective compactification of $X_k$. Givental's mirror theorem gives $I_{-1}=I_0$ up to the mirror map, hence $QH^*(G_{-1})\cong QH^*(G_0)$, and Forbes–Jinzenji conjectured the same isomorphism type for every $k$, verifying the first two non-semi-positive cases $k=1,2$ by explicit computation of J-functions and of $\hbar$-independent connection matrices for the quantum product.

The statement should not be confused with Ruan's K-equivalence conjecture — that varieties related by crepant birational maps, such as flops, have isomorphic quantum cohomology — which Lee–Lin–Wang and Lee–Lin–Qu–Wang proved for simple and ordinary flops in the big-ring, analytic-continuation form. The $G_k$ are birational to one another through elementary transformations (the standard birational modifications of a projective bundle: blow up a section along a divisor and blow down the other ruling), but they do not form a K-equivalence class: only $G_{-1}$, which is the compactified local model $\mathbb{P}(N_{Z/X}\oplus\mathcal{O})$ of an Atiyah flop (a flop contracting a $\mathbb{P}^1$ with normal bundle $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$), carries a floppable curve, while for $k\ge 1$ the canonical-trivial section has normal bundle $\mathcal{O}(k)\oplus\mathcal{O}(-2-k)$. The Forbes–Jinzenji conjecture is thus a mirror-symmetric $k$-independence phenomenon stronger than and not predicted by the flop-invariance program — and, accordingly, not resolved by it.

Problem Statement

Prove or refute Conjecture 2 of Forbes–Jinzenji (Int. J. Mod. Phys. A 22 (2007) 2327–2360, arXiv:math/0603728): for every integer $k$ there is an isomorphism of small quantum cohomology rings

$QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k)))\cong QH^*(\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1)\oplus\mathcal{O}(-1))),$

where both projective bundles are over $\mathbb{P}^1$. The isomorphism is meant in the same sense as in the settled cases: an isomorphism of $\mathbb{C}$-algebras after (i) a linear change of the natural cohomology bases generated by the two divisor classes and (ii) the change of the two quantum variables induced by the mirror map — equivalently, an identification under which the quantum multiplication matrices (the $\hbar$-independent connection matrices of the quantum connection) of the two rings coincide, as was verified explicitly for $k=1,2$. The cases $k=-1$ (trivial), $k=0$ (Givental) and $k=1,2$ (Forbes–Jinzenji) are settled, and $G_k\cong G_{-2-k}$, so the open content of the conjecture is $k\ge 3$; a complete answer establishes the isomorphism, or its impossibility, for all remaining integers $k\ge 3$, uniformly in $k$ or case by case. A refutation must exhibit a specific $k$ and prove that no choice of basis change and quantum-variable identification makes the two rings isomorphic; differing structure constants in one particular basis alone do not suffice.

References

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