Real sheaves/Gromov–Witten correspondence for KP2K_{\mathbb P^2}

Let KP2K_{\mathbb P^2} carry its natural real structure, with real locus KP2RK_{\mathbb P^2}^{\mathbb R} restricting to RP2\mathbb R\mathbb P^2. For each positive integer dd, let n0,dKP2,Rn_{0,d}^{K_{\mathbb P^2},\mathbb R} be the genus-zero degree-dd real Gopakumar–Vafa invariant, and let Md,1(R)M_{d,1}(\mathbb R) be the real locus of the moduli space Md,1M_{d,1}, with Euler characteristic e(Md,1(R))e(M_{d,1}(\mathbb R)). Real sheaves/Gromov–Witten conjecture. For every positive integer dd, one has

n0,dKP2,R=(1)d12e(Md,1(R)).n_{0,d}^{K_{\mathbb P^2},\mathbb R}=(-1)^{\frac{d-1}{2}}e(M_{d,1}(\mathbb R)).

This conjecture proposes a direct correspondence between real Gromov–Witten/Gopakumar–Vafa invariants of the local projective plane and Euler characteristics of real sheaf moduli spaces. The source gives no resolution in the provided text.

Sources & referencesView supporting material

Primary source

Pierrick Bousseau, “A proof of N.Takahashi's conjecture for (P^2,E) and a refined sheaves/Gromov-Witten correspondence”, arXiv:1909.02992 (2025).

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