Genus-zero mirror-map conjecture for complete intersections in weighted projective space

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Let P(a1,…,aN∣k1,…,km)P(a_1,\dots,a_N\mid k_1,\dots,k_m) be a nonsingular complete intersection in weighted projective space. Let tp(x∗)t^p(x^*) be the mirror maps defined from the genus-zero virtual structure constants, and let xp(t∗)x^p(t^*) denote their inverse. Write ⟨OhaOhb⟩0(t∗)\langle{\mathcal O}_{h^a}{\mathcal O}_{h^b}\rangle_0(t^*) for the genus-zero two-point Gromov--Witten generating function and w(OhaOhb)0(x∗)w({\mathcal O}_{h^a}{\mathcal O}_{h^b})_0(x^*) for the corresponding virtual structure-constant generating function. Genus-zero mirror-map conjecture. For all admissible aa and bb,

⟨OhaOhb⟩0(t0,t1,t2,…,tN−m−1)=w(OhaOhb)0(x0(t∗),x1(t∗),x2(t∗),…,xN−m−1(t∗)).\langle{\mathcal O}_{h^a} {\mathcal O}_{h^b}\rangle_{0}\bigl(t^0,t^1,t^2,\dots,t^{N-m-1}\bigr)=w({\mathcal O}_{h^a} {\mathcal O}_{h^b})_{0}\bigl(x^0(t^*),x^1(t^*),x^2(t^*),\dots,x^{N-m-1}(t^*)\bigr).

This change-of-variables assertion is used to compute genus-zero Gromov--Witten invariants; the source states that it was already proved for projective hypersurfaces, while its status for the stated weighted complete intersections is not specified.

References

Primary source

Masao Jinzenji and Ken Kuwata, “Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space”, arXiv:2504.14273 (2026).

Additional references

2 papers in this index state this conjecture (2006–2025). The statement above is taken from the most recent of them; the others are arXiv:math/0609016.

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