Let h be the hyperplane class of CP^2, and let w(cal O_h^acal O_h^b|(x^0,x^1,x^2))_0 denote the generating function of multi-point virtual structure constants. Define the mirror map by t^j(x^0,x^1,x^2):=w(cal O_h^2-jcal O_1|(x^0,x^1,x^2))_0. Let x^j=x^j(t^0,t^1,t^2) be its inverse. Jinzenji–Shimizu CP^2 mirror conjecture. For the relevant indices a,b, langlecal O_h^acal O_h^b(t^0,t^1,t^2)rangle_0 =w(cal O_h^acal O_h^b|(x^0(t),x^1(t),x^2(t)))_0. Equivalently, before inversion, the Gromov–Witten generating function evaluated at the mirror map equals the virtual-structure-constant generating function. This is proposed as a mirror formula for computing rational genus-0 Gromov–Witten invariants of CP^2 from multi-point virtual structure constants.

Jinzenji–Shimizu categorical mirror formula for CP^2

Let $h$ be the hyperplane class of $CP^2$ , and let $w({\cal O}_{h^a}{\cal O}_{h^b}\mid(x^0,x^1,x^2))_0$ denote the generating function of multi-point virtual structure constants. Define the mirror map by

t^j(x^0,x^1,x^2):=w({\cal O}_{h^{2-j}}{\cal O}_{1}\mid(x^0,x^1,x^2))_0.

Let $x^j=x^j(t^0,t^1,t^2)$ be its inverse. Jinzenji–Shimizu $CP^2$ mirror conjecture. For the relevant indices $a,b$ ,

\langle{\cal O}_{h^a}{\cal O}_{h^b}(t^0,t^1,t^2)\rangle_0 =w({\cal O}_{h^a}{\cal O}_{h^b}\mid(x^0(t),x^1(t),x^2(t)))_0.

Equivalently, before inversion, the Gromov–Witten generating function evaluated at the mirror map equals the virtual-structure-constant generating function. This is proposed as a mirror formula for computing rational genus- $0$ Gromov–Witten invariants of $CP^2$ from multi-point virtual structure constants.

The Jinzenji–Shimizu conjecture identifies genus- $0$ Gromov–Witten generating functions for $CP^2$ with multi-point virtual-structure-constant generating functions after the specified mirror-map substitution. The retrieved literature gives no separate account of who first posed this exact formulation or when.

Known results

The generalized mirror transformation for projective hypersurfaces had rigorous proofs by Iritani and Guest; the 2017 paper treats a related two-point statement, not explicitly the exact multi-point $CP^2$ formula.
By April 2024, the genus- $0$ formula was still called a conjecture and had been confirmed numerically in low degrees, including the $CP^2$ case.

March 2025 claimed proof

A paper by Jinzenji states a theorem for multi-point virtual structure constants of projective hypersurfaces and says that, after mirror-map expansion, its theorem is equivalent to the conjecture, including the $CP^2$ example with $N=4$ and $k=1$ . It therefore claims a proof, but the scan found no independent verification or referee assessment.

Current status (as of August 2026): A March 2025 preprint claims to prove the $CP^2$ formula, but the claim remains unverified and a fully established resolution is not recorded.

Jinzenji–Shimizu categorical mirror formula for $CP^2$

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