For each integer kk, let Gk=P(O⊕O(k)⊕O(−2−k))G_k=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(k)\oplus\mathcal{O}(-2-k)) be the P2\mathbb{P}^2-bundle over P1\mathbb{P}^1. Forbes and Jinzenji (2007) conjectured that for all k∈Zk\in\mathbb{Z} the small quantum cohomology rings satisfy QH∗(Gk)≅QH∗(G−1)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=−1,0k=-1,0 follow from Givental's mirror theorem for toric varieties, and k=1,2k=1,2 were verified by Forbes–Jinzenji through Birkhoff factorization of the I-function; since Gk≅G−2−kG_k\cong G_{-2-k}, the open content is k≥3k\ge 3. The conjecture is the compact remnant of the physics observation that local mirror symmetry of the curves O(k)⊕O(−2−k)→P1\mathcal{O}(k)\oplus\mathcal{O}(-2-k)\to\mathbb{P}^1 is independent of kk, 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≥1k\ge 1.

Background

Let XX be a smooth complex projective variety. The small quantum cohomology ring QH∗(X)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:P1→Xf:\mathbb{P}^1\to X of fixed homology class β\beta, recorded in formal Novikov variables qβq^\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)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)H^*(X) used to organize genus-zero Gromov–Witten theory.

For a rank-3 vector bundle E→P1E\to\mathbb{P}^1, the projectivization P(E)\mathbb{P}(E) is a P2\mathbb{P}^2-bundle over P1\mathbb{P}^1; by the projective bundle formula its cohomology is a free module over H∗(P1)H^*(\mathbb{P}^1) on 1,ξ,ξ21,\xi,\xi^2, where ξ\xi is the relative hyperplane class, with a relation determined by the Chern classes of EE. Write Gk=P(O⊕O(k)⊕O(−2−k))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 Gk≅G−2−kG_k\cong G_{-2-k} and the distinct members are k=−1,0,1,2,…k=-1,0,1,2,\dots; each is a smooth projective toric threefold with two Kähler parameters, with divisor classes p1,p2p_1,p_2 dual to the base and fiber curve classes. A computation in the source paper gives c1(Gk)=−3k p1+3 p2c_1(G_k)=-3k\,p_1+3\,p_2, so G−1G_{-1} and G0G_0 satisfy the semi-positivity condition c1≥0c_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≥1k\ge 1 the class −3k p1-3k\,p_1 is not nef, the I-function contains positive powers of the parameter ℏ\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 ℏ\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) Xk=tot⁡(O(k)⊕O(−2−k)→P1)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 kk; GkG_k is a natural projective compactification of XkX_k. Givental's mirror theorem gives I−1=I0I_{-1}=I_0 up to the mirror map, hence QH∗(G−1)≅QH∗(G0)QH^*(G_{-1})\cong QH^*(G_0), and Forbes–Jinzenji conjectured the same isomorphism type for every kk, verifying the first two non-semi-positive cases k=1,2k=1,2 by explicit computation of J-functions and of ℏ\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 GkG_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−1G_{-1}, which is the compactified local model P(NZ/X⊕O)\mathbb{P}(N_{Z/X}\oplus\mathcal{O}) of an Atiyah flop (a flop contracting a P1\mathbb{P}^1 with normal bundle O(−1)⊕O(−1)\mathcal{O}(-1)\oplus\mathcal{O}(-1)), carries a floppable curve, while for k≥1k\ge 1 the canonical-trivial section has normal bundle O(k)⊕O(−2−k)\mathcal{O}(k)\oplus\mathcal{O}(-2-k). The Forbes–Jinzenji conjecture is thus a mirror-symmetric kk-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 kk there is an isomorphism of small quantum cohomology rings

QH∗(P(O⊕O(k)⊕O(−2−k)))≅QH∗(P(O⊕O(−1)⊕O(−1))),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 P1\mathbb{P}^1. The isomorphism is meant in the same sense as in the settled cases: an isomorphism of C\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 ℏ\hbar-independent connection matrices of the quantum connection) of the two rings coincide, as was verified explicitly for k=1,2k=1,2. The cases k=−1k=-1 (trivial), k=0k=0 (Givental) and k=1,2k=1,2 (Forbes–Jinzenji) are settled, and Gk≅G−2−kG_k\cong G_{-2-k}, so the open content of the conjecture is k≥3k\ge 3; a complete answer establishes the isomorphism, or its impossibility, for all remaining integers k≥3k\ge 3, uniformly in kk or case by case. A refutation must exhibit a specific kk 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

  1. Brian Forbes, Masao Jinzenji, 'J functions, non-nef toric varieties and equivariant local mirror symmetry of curves', Int. J. Mod. Phys. A 22 (2007) 2327–2360, DOI 10.1142/S0217751X0703649X, arXiv:math/0603728, https://arxiv.org/abs/math/0603728
  2. Alexander Givental, 'A mirror theorem for toric complete intersections', in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progress in Mathematics 160, Birkhäuser Boston, 1998, pp. 141–175, DOI 10.1007/978-1-4612-0705-4_5, arXiv:alg-geom/9701016, https://arxiv.org/abs/alg-geom/9701016
  3. Tom Coates, Alexander Givental, 'Quantum Riemann-Roch, Lefschetz and Serre', Ann. of Math. (2) 165 (2007) 15–53, DOI 10.4007/annals.2007.165.15, arXiv:math/0110142, https://arxiv.org/abs/math/0110142
  4. Jim Bryan, Rahul Pandharipande, 'The local Gromov-Witten theory of curves', J. Amer. Math. Soc. 21 (2008) 101–136, DOI 10.1090/S0894-0347-06-00545-5, arXiv:math/0411037, https://arxiv.org/abs/math/0411037
  5. T.-M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow, 'Local Mirror Symmetry: Calculations and Interpretations', Adv. Theor. Math. Phys. 3 (1999) 495–565, DOI 10.4310/ATMP.1999.v3.n3.a3, arXiv:hep-th/9903053, https://arxiv.org/abs/hep-th/9903053
  6. Brian Forbes, Masao Jinzenji, 'Extending the Picard-Fuchs system of local mirror symmetry', J. Math. Phys. 46 (2005) 082302, DOI 10.1063/1.1996441, arXiv:hep-th/0503098, https://arxiv.org/abs/hep-th/0503098
  7. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Flops, motives and invariance of quantum rings', Ann. of Math. (2) 172 (2010) 243–290, DOI 10.4007/annals.2010.172.243, arXiv:math/0608370, https://arxiv.org/abs/math/0608370
  8. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Invariance of Quantum Rings under Ordinary Flops I: Quantum corrections and reduction to local models', Algebr. Geom. 3 (2016) 578–614, DOI 10.14231/ag-2016-026, arXiv:1109.5540, https://arxiv.org/abs/1109.5540
  9. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Invariance of Quantum Rings under Ordinary Flops II: A quantum Leray-Hirsch theorem', Algebr. Geom. 3 (2016) 615–653, DOI 10.14231/ag-2016-027, arXiv:1311.5725, https://arxiv.org/abs/1311.5725
  10. Yuan-Pin Lee, Hui-Wen Lin, Feng Qu, Chin-Lung Wang, 'Invariance of quantum rings under ordinary flops III: A quantum splitting principle', Camb. J. Math. 4 (2016) 333–401, DOI 10.4310/CJM.2016.v4.n3.a2, arXiv:1401.7097, https://arxiv.org/abs/1401.7097
  11. Hiroshi Iritani, 'Ruan's conjecture and integral structures in quantum cohomology', Adv. Stud. Pure Math. 59 (2010) 111–166, DOI 10.2969/aspm/05910111, arXiv:0809.2749, https://arxiv.org/abs/0809.2749
  12. Mark McLean, 'Birational Calabi-Yau manifolds have the same small quantum products', Ann. of Math. (2) 191 (2020) 439–579, DOI 10.4007/annals.2020.191.2.4, arXiv:1806.01752, https://arxiv.org/abs/1806.01752
  13. Hiroshi Iritani, Yuki Koto, 'Quantum cohomology of projective bundles', arXiv:2307.03696 (preprint; latest version v4, 31 January 2026), https://arxiv.org/abs/2307.03696
  14. Artur Elezi, 'A mirror conjecture for projective bundles', Int. Math. Res. Not. 2005 (2005), no. 55, 3445–3458, DOI 10.1155/imrn.2005.3445, arXiv:math/0609509, https://arxiv.org/abs/math/0609509
  15. Artur Elezi, 'Mirror symmetry and quantum cohomology of projective bundles', arXiv:math/0610758 (preprint), https://arxiv.org/abs/math/0610758